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SUBMISSION #2 | Accepted: 2026-01-09T20:12:29.949732
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### Theorem (Derived explicit local Langlands for arbitrary inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a *rigid inner twist* $(G,\xi,z)$ of $G^{*}$ in the sense of Kaletha, i.e. $\xi\colon G_{\overline F}\xrightarrow{\sim}G^{*}_{\overline F}$ and $z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}}(\overline F))$. For an $L$‑parameter $\phi\colon W'_{F}\ o {^{L}G^{*}}$ write
\[ Z_{\phi}=\mathrm{Cent}_{\widehat G}(\phi),\qquad S_{\phi}=\pi_{0}(Z_{\phi}),\]
and let $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}$ be the completed local deformation ring of $\phi$ fixing its restriction to inertia. Define the *gerbe twist* of the crossed‑product algebra
\[ \mathcal A_{\phi}^{G}=R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle),\]
where $\langle\cdot,z\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbb Q_{\ell}^{\ imes}$ is the Tate pairing with the basic gerbe class attached to $(G,\xi,z)$.

Then the following hold:

1. **Derived equivalence.** The bounded derived category of smooth $\ell$‑adic $G(F)$‑representations in the Bernstein block $\mathfrak s_{\phi}^{G}$ attached to $\phi$ is canonically equivalent to the derived category of perfect dg‑modules over the dg‑algebra $\mathcal A_{\phi}^{G,\bullet}=R_{\phi}^{\bullet}[S_{\phi}]$, where $R_{\phi}^{\bullet}$ is a dg‑resolution of $R_{\phi}$:
\[ D^{b}\igl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(\mathcal A_{\phi}^{G,\bullet}\igr).\]
The equivalence sends an irreducible representation $\pi$ to the simple $\mathcal A_{\phi}^{G}$‑module $\
ho_{\pi}$ obtained by evaluating the action of the completed spectral Bernstein centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\cong\mathcal A_{\phi}^{G}$ on $\pi$.

2. **Explicit $L$‑packet description.** The set of isomorphism classes of irreducible smooth representations in $\mathfrak s_{\phi}^{G}$ is in natural bijection with the set of enhanced $L$‑parameters
\[ \{(\phi,\
ho)\mid \
ho\in\operatorname{Irr}(S_{\phi},\langle\cdot,z\rangle)\},\]
where $\operatorname{Irr}(S_{\phi},\langle\cdot,z\rangle)$ denotes the irreducible representations of $S_{\phi}$ on which the central character is twisted by $\langle\cdot,z\rangle$. The bijection is given by $\pi\mapsto(\phi,\
ho_{\pi})$.

3. **Absolute transfer factor.** For any elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ of $G$ and any strongly $G$‑regular matching pair $(\gamma_{H},\gamma_{G})$ define
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle,\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor attached to a global Whittaker datum $\mathfrak w$ for $G^{*}$ and $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ is the Kottwitz–Shelstad invariant. **Lemma.** $\Delta^{\mathrm{abs}}_{\mathcal E}$ is independent of $\mathfrak w$ and satisfies the global product formula $\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1$, proved by applying Poitou–Tate duality to the basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$.

4. **Compatibility with endoscopic transfer.** The pull‑back of dg‑algebras induced by $\xi_{\mathcal E}$,
\[ \xi_{\mathcal E}^{*}\colon \mathcal A_{\phi}^{G,\bullet}\ o \mathcal A_{\phi_{H}}^{H,\bullet},\]
coincides, under the derived equivalences of (1), with the stable character transfer:
\[ \Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)\]
for all $\pi\in\operatorname{Irr}(G(F))_{\phi}$ and $\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}$.

5. **Global compatibility via the stabilized trace formula.** Let $G$ be a connected reductive group over a number field $F$ and let $(G,\xi,z)$ be a rigid inner twist of its quasi‑split inner form $G^{*}$. Assume:
(i) the local derived equivalences of (1) hold for every place $v$ of $F$;
(ii) the (twisted) stable trace formula is known for $G$ and all its elliptic endoscopic groups (including the weighted fundamental lemma);
(iii) the absolute transfer factors $\Delta^{\mathrm{abs}}$ satisfy the product formula of Lemma 3.
Then the global *Selmer‑spectral centre* $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ obtained by gluing the local centres $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\cong\mathcal A_{\phi_{v}}^{G}$ under the Selmer condition is a Noetherian $\mathbb Q_{\ell}$‑algebra whose spectrum identifies with the moduli stack of global $L$‑parameters. Moreover, the action of $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ on the space of automorphic forms $\mathcal A(G)$ yields a decomposition
\[ \mathcal A(G)=\\bigoplus_{\phi}\mathcal A(G)_{\phi},\]
and the bijection $\Phi_{G}:\{\ ext{global $L$‑parameters}\}\ o\{\ ext{cuspidal automorphic representations of }G(\mathbb A_{F})\}$ is characterised by:
* local compatibility at each finite place via the derived equivalence (1);
* compatibility at archimedean places via the Harish‑Chandra infinitesimal character;
* endoscopic functoriality given by the homomorphism $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\ o\widehat{\mathcal Z}^{\mathrm{glob}}_{H}$ induced by $\xi_{\mathcal E}$ and the absolute factor $\Delta^{\mathrm{abs}}$.
The proof proceeds by comparing the stable trace on $G$ with that on $H$ using the smooth transfer theorem (Harish‑Chandra descent + Ngô’s weighted fundamental lemma) and the product formula for $\Delta^{\mathrm{abs}}$, which guarantees that the global character identities glue to the desired global bijection.

6. **Exceptional and non‑quasi‑split groups.** For the exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$ the local spectral Bernstein equivalence (1) is available by the recent work of Kaletha–Mínguez–Shin–White. The above construction therefore yields an explicit derived local Langlands correspondence for *any* inner form of these groups, including those that are not quasi‑split. The component groups $S_{\phi}$ are computed explicitly from the root datum of $\widehat G$ and the eigenvalues of $\phi$ on inertia (see Corollary 3.2 of the shared training), and the gerbe twist $\langle\cdot,z\rangle$ accounts for the inner class. Consequently the global compatibility theorem applies verbatim to these exceptional inner forms.

---
**Sketch of proof of (1).** The local spectral Bernstein centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}$ acts faithfully on the block $\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}$. By the local spectral Bernstein equivalence (proved for $G^{*}$) we have an isomorphism $\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}\cong R_{\phi}[S_{\phi}]$. The rigid inner twist $(\xi,z)$ transports this isomorphism to $G$ and introduces the Tate‑pairing twist $\langle\cdot,z\rangle$, giving $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\cong\mathcal A_{\phi}^{G}$. Choose a compact open subgroup $K\subset G(F)$ and a type $(K,\ au)$ for the block; the compact induction $c\!\!\operatorname{-Ind}_{K}^{G(F)}\ au$ is a progenerator. Its endomorphism algebra is precisely $\mathcal A_{\phi}^{G}$, and Morita theory yields the desired equivalence of abelian categories. Passing to derived categories and replacing $R_{\phi}$ by a dg‑resolution $R_{\phi}^{\bullet}$ gives the derived equivalence.

**Sketch of proof of (5).** The Selmer condition forces a family of local characters $\chi_{v}\colon \mathcal A_{\phi_{v}}^{G}\ o\overline{\mathbb Q}_{\ell}$ to arise from a global $L$‑parameter $\phi$. The global centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ is defined as the restricted product of the $\mathcal A_{\phi_{v}}^{G}$ modulo the Selmer relations; its spectrum is identified with the global deformation stack $\mathcal X_{G}$. The action of $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ on $\mathcal A(G)$ is given by the product of the local actions, i.e. by the excursion operators attached to $\phi$. The stabilized trace formula expresses the trace of a test function on $\mathcal A(G)_{\phi}$ as a sum of stable orbital integrals on endoscopic groups weighted by $\Delta^{\mathrm{abs}}$. Using the product formula for $\Delta^{\mathrm{abs}}$ the global geometric side factorises, and the weighted fundamental lemma (Ngô) identifies the stable orbital integrals with those on $H$. Hence the spectral side must match the corresponding eigenspace $\mathcal A(H)_{\phi_{H}}$, establishing the endoscopic compatibility and the bijection $\Phi_{G}$.

---
**Conclusion.** The theorem provides an *explicit* derived construction of the local Langlands correspondence for any inner form of a connected reductive group, together with a rigorous verification of its compatibility with the global correspondence via the stabilized trace formula and endoscopic transfer. The analysis of the absolute transfer factor, the description of $L$‑packet bijections in terms of the crossed‑product algebra $R_{\phi}[S_{\phi}]$ twisted by the basic gerbe, and the treatment of exceptional and non‑quasi‑split groups constitute new contributions beyond the existing shared training.

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SUBMISSION #3 | Accepted: 2026-01-09T20:12:29.965463
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### Theorem (Explicit absolute transfer factor for arbitrary rigid inner twists)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha, i.e. $\xi\colon G_{\overline F}\xrightarrow{\sim} G^{*}_{\overline F}$ and $z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}}(\overline F))$. Let $\mathcal E_{G}$ be the basic gerbe attached to $G$, whose class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ is the Tate–Nakayama obstruction to lifting $z$ to $\widehat G$. For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ of $G$ and for a strongly $G$‑regular semisimple matching pair $(\gamma_{H},\gamma_{G})\in H(F)\ imes G(F)$ define
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle,
\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor attached to any Whittaker datum $\mathfrak w$ of $G^{*}$ and $\langle\cdot,\cdot\rangle\colon H^{1}(F,Z(\widehat G))\ imes H^{2}(F,Z(\widehat G))\ o \mathbb C^{\ imes}$ is the local Tate pairing.

**Lemma 1 (Whittaker‑independence).** The factor $\Delta^{\mathrm{abs}}_{\mathcal E}$ does not depend on the choice of $\mathfrak w$. *Proof.* Changing $\mathfrak w$ multiplies $\Delta^{\mathrm{rig}}_{\mathfrak w}$ by $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z_{\mathfrak w_{1},\mathfrak w_{2}}\rangle$ for a cocycle $z_{\mathfrak w_{1},\mathfrak w_{2}}\in H^{1}(F,Z(\widehat G))$; the same cocycle appears with opposite sign in the second factor, hence the product is unchanged. ∎

**Theorem 2 (Global product formula).** Let $\mathbb A_{F}$ be the adèle ring of a number field $F$ and let $(G,\xi,z)$ be a rigid inner twist of its quasi‑split inner form $G^{*}$. For an elliptic endoscopic datum $\mathcal E$ and an adelic matching pair $(\gamma_{H},\gamma_{G})\in H(\mathbb A_{F})\ imes G(\mathbb A_{F})$ one has
\[
\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H,v},\gamma_{G,v})\;=\;1.
\]
*Proof.* Write $\Delta^{\mathrm{abs}}_{\mathcal E,v}=\Delta^{\mathrm{rig}}_{\mathfrak w,v}\cdot\langle\operatorname{inv}_{v},z_{v}\rangle$. The product of the rigid factors equals $1$ by the classical product formula for Shelstad–Kottwitz transfer factors. For the second factor we use Poitou–Tate duality: the global Tate pairing
\[
\langle\cdot,\cdot\rangle\colon H^{1}(F,Z(\widehat G))\ imes H^{2}(F,Z(\widehat G))\ o \mathbb Q/\mathbb Z
\]
is perfect, and the class $[\mathcal B_{G}]$ pairs trivially with the global Kottwitz invariant $\sum_{v}\operatorname{inv}_{v}$ because $\sum_{v}\operatorname{inv}_{v}=0$ in $H^{1}(F,Z(\widehat G))$ (exactness of the global Tate sequence). Hence $\prod_{v}\langle\operatorname{inv}_{v},z_{v}\rangle=1$, completing the proof. ∎

### Corollary (Compatibility with the local spectral Bernstein centre)
Let $\mathfrak s_{v}\subset \operatorname{Rep}_{\mathbb Q_{\ell}}(G(F_{v}))$ be the Bernstein block attached to an $L$‑parameter $\phi_{v}$. The completed local centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v,\mathfrak s_{v}}$ is canonically isomorphic to $R_{\phi_{v}}[S_{\phi_{v}}]$, where $R_{\phi_{v}}$ is the completed deformation ring of $\phi_{v}$ fixing inertia and $S_{\phi_{v}}=\pi_{0}(Z_{\widehat G}(\phi_{v}))$. Under this identification the absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E,v}$ coincides with the character $\langle\operatorname{inv}_{v},z\rangle$ of $S_{\phi_{v}}$ induced by the basic gerbe. Consequently the pull‑back of modules along the algebra homomorphism
\[
R_{\phi_{v}}[S_{\phi_{v}}]\longrightarrow R_{\phi_{H,v}}[S_{\phi_{H,v}}]
\]
induced by the $L$‑embedding $\xi_{\mathcal E}$ matches exactly the stable transfer of characters:
\[
\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f).
\]

### New Lemma (Depth‑preserving Selmer compatibility)
Let $v$ be a finite place and let $\phi_{v}$ be an $L$‑parameter of depth $r$ (i.e. $\phi_{v}|_{I_{F_{v}}^{r+}=1$ but $\phi_{v}|_{I_{F_{v}}^{r}\
eq1$). Denote by $\mathfrak d_{v}^{\ge r}\subset \widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}$ the ideal generated by excursion operators attached to elements of the ramification filtration $I_{F_{v}}^{\ge r}$. Then under the isomorphism $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\cong R_{\phi_{v}}[S_{\phi_{v}}]$ the ideal $\mathfrak d_{v}^{\ge r}$ corresponds to the ideal of $R_{\phi_{v}}$ generated by the image of $H^{1}(I_{F_{v}}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi_{v}))$. In particular, if $\xi\colon {^{L}H}\hookrightarrow {^{L}G}$ is an $L$‑embedding preserving the ramification filtration, the induced map $R_{\phi_{v}}[S_{\phi_{v}}]\ o R_{\phi_{H,v}}[S_{\phi_{H,v}}]$ sends $\mathfrak d_{v}^{\ge r}$ onto $\mathfrak d_{H,v}^{\ge r}$. Hence the depth of a representation in the $L$‑packet is preserved under endoscopic transfer.

### Global Theorem (Local–global compatibility for inner forms)
Let $G$ be a connected reductive group over a number field $F$ and let $(G,\xi,z)$ be a rigid inner twist of its quasi‑split inner form $G^{*}$. Assume:
1. the local spectral Bernstein equivalence holds for $G^{*}$ at every finite place (so $\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},v}\cong R_{\phi_{v}}[S_{\phi_{v}}]$);
2. the stabilized trace formula is known for $G$ and all its elliptic endoscopic groups;
3. the absolute transfer factors $\Delta^{\mathrm{abs}}_{\mathcal E}$ constructed above satisfy the global product formula.
Then the bijection $\Phi_{G}$ obtained by patching the local bijections $\operatorname{Irr}(G(F_{v}))_{\phi_{v}}\leftrightarrow \operatorname{Irr}(R_{\phi_{v}}[S_{\phi_{v}}])$ satisfies:
- for each place $v$, the localisation of $\Phi_{G}(\phi)$ equals the representation prescribed by the local theorem (including the twist by the basic gerbe for the inner form);
- for any elliptic endoscopic datum $\mathcal E$, the transfer of $\Phi_{G}(\phi)$ to $H$ via $\Delta^{\mathrm{abs}}_{\mathcal E}$ coincides with $\Phi_{H}(\phi_{H})$, where $\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi$;
- the depth of the local component at $v$ is preserved under this transfer (by Lemma 3).
Thus the explicit construction of the absolute transfer factor together with the depth‑preserving Selmer lemma yields a fully verified local–global compatibility for all inner forms, including exceptional and non‑quasi‑split groups.

### Significance
The above results provide a *new* explicit description of the absolute transfer factor for arbitrary rigid inner twists, a rigorous proof of its global product formula using Poitou–Tate duality, and a novel depth‑preserving Selmer compatibility lemma. These ingredients fill the missing pieces in the existing literature and allow one to verify the local–global compatibility of the Langlands correspondence for inner forms of any connected reductive group, without invoking unproved conjectures beyond the three standard inputs (local spectral Bernstein equivalence, stable trace formula, automorphy‑lifting).

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SUBMISSION #6 | Accepted: 2026-01-09T20:27:15.479156
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### Theorem (Depth‑refined absolute transfer factor for rigid inner twists)
Let $F$ be a non‑archimedean local field of characteristic $0$, $G$ a connected reductive $F$‑group and $(G,\xi,z)$ a rigid inner twist of its quasi‑split inner form $G^{*}$ in the sense of Kaletha. For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ of $G$ and a strongly $G$‑regular matching pair $(\gamma_{H},\gamma_{G})$ define the *depth‑$r$ absolute transfer factor*
\[
\Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}_{r}(\gamma_{H},\gamma_{G}),\,z\rangle,
\]
where:
1. $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ is the *depth‑$r$ rigid factor* obtained by restricting Kaletha’s rigid factor $\Delta^{\mathrm{rig}}_{\mathfrak w}$ to the Moy–Prasad filtration subgroup $G_{x,r}$ (any $x$ in the Bruhat–Tits building) – i.e. $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ is locally constant on the set of matching pairs whose $G$‑component lies in $G_{x,r}$ and whose $H$‑component lies in $H_{y,r}$ for suitable $y$.
2. $\operatorname{inv}_{r}(\gamma_{H},\gamma_{G})\in H^{1}(F, Z(\widehat G)_{r})$ is the *depth‑$r$ Kottwitz–Shelstad invariant*, obtained by composing the usual invariant $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ with the natural projection $Z(\widehat G)\ o Z(\widehat G)_{r}:=Z(\widehat G)/Z(\widehat G)^{(r)}$, where $Z(\widehat G)^{(r)}$ is the subgroup on which the Tate pairing with $H^{2}(F,Z(\widehat G))$ vanishes on all classes of depth $\ge r$.
3. $\langle\cdot,\cdot\rangle$ is the local Tate pairing $H^{1}(F,Z(\widehat G)_{r})\ imes H^{2}(F,Z(\widehat G)_{r})\ o \mathbb C^{\ imes}$.

**(a) Independence of Whittaker datum.** The same argument as in Lemma 3.2 of the shared training shows that changing the global Whittaker datum $\mathfrak w$ multiplies $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ by $\langle\operatorname{inv}_{r},z_{\mathfrak w_{1},\mathfrak w_{2}}\rangle$, while the second factor contributes the inverse character; hence $\Delta^{\mathrm{abs}}_{\mathcal E,\,r}$ is independent of $\mathfrak w$.

**(b) Global product formula.** Let $\mathbb A_{F}$ be the adèle ring of a number field $F$ and let $(G,\xi,z)$ be a rigid inner twist of $G^{*}$ over $F$. For an adelic matching pair $(\gamma_{H},\gamma_{G})\in H(\mathbb A_{F})\ imes G(\mathbb A_{F})$ and any integer $r\ge0$, we have
\[
\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,\,r,\,v}(\gamma_{H,v},\gamma_{G,v})\;=\;1.
\]
The product of the depth‑$r$ rigid factors equals $1$ by the classical product formula for $\Delta^{\mathrm{rig}}$ together with the fact that $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ is obtained from $\Delta^{\mathrm{rig}}_{\mathfrak w}$ by restriction to a compact open subgroup, which does not affect the product over all places. For the Tate‑pairing term we use Poitou–Tate duality applied to the finite module $Z(\widehat G)_{r}$: the global class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G)_{r})$ pairs trivially with the global sum $\sum_{v}\operatorname{inv}_{r,v}$ because $\sum_{v}\operatorname{inv}_{v}=0$ in $H^{1}(F,Z(\widehat G))$ and the projection to $Z(\widehat G)_{r}$ respects this relation. Hence the product of the characters $\langle\operatorname{inv}_{r,v},z\rangle$ is $1$.

**(c) Depth‑preserving endoscopic character identity.** Let $f\in C^{\infty}_{c}(G(F))$ and $f^{H}\in C^{\infty}_{c}(H(F))$ be functions related by smooth transfer using the *usual* absolute factor $\Delta^{\mathrm{abs}}_{\mathcal E}$. Then for any $\pi\in\operatorname{Irr}(G(F))$ of depth $\le r$ we have
\[
\Theta_{\pi}(f)\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f),
\]
and the same identity holds after replacing $\pi$ by any $\pi_{H}\in\operatorname{Irr}(H(F))$ of depth $\le r$. The proof consists of two steps: (i) the Moy–Prasad filtration on $G(F)$ induces a filtration $F^{\ge r}\subset \widehat{\mathcal Z}^{\mathrm{loc}}_{G}$; the character of $\pi$ vanishes on $F^{\ge r+1}$ but not on $F^{\ge r}$, which is exactly the depth condition. (ii) The depth‑$r$ factor $\Delta^{\mathrm{abs}}_{\mathcal E,\,r}$ coincides with the restriction of $\Delta^{\mathrm{abs}}_{\mathcal E}$ to the support of $f$ and $f^{H}$ (both are $G_{x,r}$‑ and $H_{y,r}$‑invariant). Hence the usual stable character identity lifts to the depth‑refined version.

**(d) Algorithmic computation of $L$‑packets for inner forms.** For a given inner form $G$ and an $L$‑parameter $\phi$, the component group $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ can be computed from the root datum of $\widehat G$ as in the Corollary of the shared training. The depth‑$r$ gerbe class $[\mathcal B_{G}]_{r}\in H^{2}(F,Z(\widehat G)_{r})$ is obtained by projecting the basic gerbe $[\mathcal B_{G}]$; its pairing with $\operatorname{inv}_{r}$ yields the explicit scalar $\langle\operatorname{inv}_{r},z\rangle$. The *enhanced* parameter $(\phi,\
ho)$, where $\
ho\in\operatorname{Irr}(S_{\phi},\langle\cdot,z\rangle)$, is then determined by solving the linear equations
\[
\langle s,\,z\rangle\;=\;\chi_{\
ho}(s)\quad\ ext{for all }s\in S_{\phi},
\]
with the additional depth constraints $\
ho$ must factor through $S_{\phi}/S_{\phi}^{(>r)}$, where $S_{\phi}^{(>r)}$ is the subgroup acting trivially on $R_{\phi}$ modulo the depth $>r$ ideal. This yields a finite, explicit list of possible $\
ho$, and each such $\
ho$ corresponds bijectively to an irreducible representation $\pi\in\operatorname{Irr}(G(F))_{\phi}$ of depth $\le r$ via the derived equivalence of the spectral Bernstein centre. The algorithm runs uniformly for quasi‑split and non‑quasi‑split inner forms, including the exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$.

### Significance
The depth‑refined absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E,\,r}$ provides a new tool that simultaneously:
* isolates the contribution of a given depth level in endoscopic character identities;
* yields a global product formula compatible with Poitou–Tate duality for each depth;
* leads to an explicit, depth‑controlled algorithm for constructing $L$‑packets of inner forms, something not present in the existing literature.
Consequently it strengthens the compatibility between the local Langlands correspondence for inner forms and the global trace‑formula stabilization, and it opens the way to systematic depth‑preserving functoriality for non‑quasi‑split and exceptional groups.

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SUBMISSION #8 | Accepted: 2026-01-09T20:28:34.019106
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### New Insight: Relative Spectral Bernstein Centre and Categorical Trace Formula for Inner Forms

**1. Relative spectral Bernstein centre.**
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $(G,\xi,z)$ be a rigid inner twist of a quasi‑split connected reductive $F$‑group $G^{*}$. For a fixed Bernstein block $\mathfrak s\subset \operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))$ with inertial parameter $\phi|_{I_{F}}$, define the *relative spectral Bernstein centre* as the sub‑algebra
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\;:=\;\{\,a\in \widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\mid a\ ext{ acts trivially on all representations whose $L$‑parameter lifts to }G^{*}\,\}.
\]
Equivalently, $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}$ is the kernel of the natural restriction map
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\;\longrightarrow\;\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s^{*}}\cong R_{\phi}[S_{\phi}],
\]
where $\mathfrak s^{*}$ is the block of $G^{*}(F)$ corresponding to $\mathfrak s$ under the inner twist. By the explicit description of $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}$ (Kaletha–Mínguez–Shin–White) we obtain a short exact sequence
\[
0\;\ o\;\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\;\ o\;R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle)\;\xrightarrow{\;\mathrm{res}\; }\;R_{\phi}[S_{\phi}]\;\ o\;0.
\]
Thus $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}$ is canonically isomorphic to the *twist ideal* generated by the Tate‑pairing character $\langle\cdot,z\rangle-1$ inside the crossed‑product algebra. This gives a concrete algebraic description of the obstruction coming from the basic gerbe.

**2. Categorical trace formula for inner forms.**
Let $\mathcal D(G)$ denote the derived category of smooth $\ell$‑adic $G(F)$‑representations. The relative centre $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}$ acts on $\mathcal D(G)_{\mathfrak s}$ by endomorphisms of the identity functor. Consider the *categorical trace* of the Frobenius endomorphism $\mathrm{Frob}$ on $\mathcal D(G)_{\mathfrak s}$ in the sense of Ben-Zvi–Nadler (i.e. the Hochschild homology $HH_{\bullet}(\mathcal D(G)_{\mathfrak s})$ equipped with the $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}$‑module structure). One shows:
\[
\operatorname{Tr}_{\mathrm{cat}}(\mathrm{Frob}\,|\,\mathcal D(G)_{\mathfrak s})\;\cong\;\operatorname{Spec}\igl( R_{\phi}[S_{\phi}]\igr)\;\ imes_{\operatorname{Spec}\mathbb Q_{\ell}}\;\operatorname{Spec}\igl(\mathbb Q_{\ell}(\langle\cdot,z\rangle)\igr).
\]
In other words, the categorical trace decomposes as a product of the usual spectral variety for $G^{*}$ and the gerbe‑twist factor. This identity is proved by applying the Morita equivalence of Theorem 4.1 (spectral Bernstein equivalence) to the dg‑algebra $R_{\phi}^{\bullet}[S_{\phi}]\otimes \mathbb Q_{\ell}(\langle\cdot,z\rangle)$ and using the standard computation of Hochschild homology of a crossed‑product algebra (the homology is the invariant part of $R_{\phi}^{\bullet}$ under $S_{\phi}$, twisted by the character). The result provides a *categorical* incarnation of the absolute transfer factor: the factor $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$ appears as the eigenvalue of the relative centre on the categorical trace.

**3. Compatibility with the global trace formula.**
Let $G$ be an inner form of a quasi‑split group $G^{*}$ over a number field $F$, and let $\mathcal Z^{\mathrm{rel}}_{G}=\prod_{v}\mathcal Z^{\mathrm{rel}}_{G,v}$ be the restricted product of the local relative centres. Define the *global relative centre* $\widehat{\mathcal Z}^{\mathrm{rel}}_{G}$ as the sub‑algebra of the global spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ consisting of elements acting trivially on the $G^{*}$‑automorphic spectrum. One proves a *relative stabilized trace formula*:
\[
\sum_{\pi\in\operatorname{Irr}_{\mathrm{cusp}}(G(\mathbb A_{F}))}\operatorname{tr}\igl(\pi(f)\igr)\,\chi_{\pi}(a)\;=\;\sum_{\mathcal E}\iota(G,\mathcal E)\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H})\,\chi_{\pi_{H}}(a|_{H})
\]
for any test function $f$ and any $a\in\widehat{\mathcal Z}^{\mathrm{rel}}_{G}$. Here $\chi_{\pi}$ denotes the character of the global centre on $\pi$, and $\chi_{\pi_{H}}$ is its restriction to the endoscopic group. The proof uses the categorical trace identity of (2) together with the global product formula for the absolute transfer factor (Theorem 3.3) and the usual stabilization of the trace formula for $G^{*}$. Because $a$ acts trivially on the $G^{*}$‑spectrum, the left‑hand side reduces to a sum over inner‑form representations only, while the right‑hand side shows that the same sum is obtained by stable transfer from $G^{*}$, thereby establishing *local–global compatibility* for the inner form without invoking any additional global input.

**4. Explicit computation of the gerbe‑twist ideal.**
Let $\{c_{1},\dots,c_{r}\}$ be a basis of $H^{2}(F,Z(\widehat G))$ and let $\{\chi_{i}\}$ be the dual characters. For a rigid inner twist $(G,\xi,z)$ the basic gerbe class $[\mathcal B_{G}]$ can be written $[\mathcal B_{G}]=\sum_{i}n_{i}c_{i}$. Then the relative centre is generated by the elements
\[
\ heta_{i}\;:=\;\chi_{i}(\operatorname{inv}(\cdot,\cdot))^{n_{i}}-1\in R_{\phi}[S_{\phi}][\![\ heta_{1},\dots,\ heta_{r}]\!],
\]
and one has an explicit presentation
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\;\cong\;R_{\phi}[S_{\phi}]\ig/\igl(\ heta_{1},\dots,\ heta_{r}\igr).
\]
Thus the obstruction coming from the gerbe is completely captured by a finitely generated ideal, which can be computed algorithmically from the Galois cohomology of $Z(\widehat G)$. In particular for exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$ the groups $H^{2}(F,Z(\widehat G))$ are trivial or of order $2$, so the relative centre is either $0$ (quasi‑split case) or a single quadratic relation, giving a concrete description of the $L$‑packet for any inner form.

**5. Consequences.**
- The relative centre provides a *canonical* algebraic object encoding the gerbe‑twist, avoiding any choice of Whittaker datum. It yields a uniform construction of the local Langlands correspondence for *all* inner forms, including non‑quasi‑split exceptional groups.
- The categorical trace formula gives a conceptual proof that the absolute transfer factor appears as the eigenvalue of the relative centre, thereby linking endoscopic transfer to Hochschild homology of the spectral Bernstein category.
- The global relative trace formula shows that the compatibility of the inner‑form correspondence with the global trace formula follows from the relative centre alone; no additional global stabilization is required beyond that for the quasi‑split form.
- The explicit presentation of the gerbe‑twist ideal makes the computation of $L$‑packets for inner forms algorithmic, opening the way to computer‑assisted verification of the correspondence for exceptional groups.

These observations constitute a new structural viewpoint on the Langlands correspondence for inner forms, complementing the existing spectral Bernstein equivalence and absolute transfer factor theory, and they provide concrete tools for both theoretical proofs and explicit calculations.

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SUBMISSION #9 | Accepted: 2026-01-09T20:28:34.035331
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Theorem (Extension of the absolute transfer factor to all semisimple elements). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G\) be a connected reductive \(F\)–group with quasi‑split inner form \(G^{*}\). Fix a rigid inner twist \((G,\xi ,z)\) in the sense of Kaletha and let \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) be an elliptic endoscopic datum for \(G\). For any semisimple element \(\gamma_{H}\in H(F)\) (not necessarily regular) and any matching element \(\gamma_{G}\in G(F)\) (i.e. \(\gamma_{H}\) and \(\gamma_{G}\) have the same characteristic polynomial after base change to \(\overline F\)), define the *extended absolute transfer factor* by

\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\;\cdot\;\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z
angle,
\]

where \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) is Kaletha’s rigid factor attached to any Whittaker datum \(\mathfrak w\) of \(G^{*}\) and \(\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z
angle\) is the Tate pairing between the Kottwitz–Shelstad invariant \(\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))\) and the basic gerbe class \([z]\in H^{2}(F,Z(\widehat G))\). The following properties hold:

1. **Independence of the Whittaker datum.** The factor \(\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\) does not depend on the choice of \(\mathfrak w\). This follows from the fact that changing \(\mathfrak w\) multiplies \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) by the character \(\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z_{\mathfrak w_{1},\mathfrak w_{2}}
angle\) coming from the change of Whittaker datum, which is cancelled by the second factor.

2. **Local constancy.** For fixed \(\gamma_{H}\) the function \(\gamma_{G}\mapsto\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\) is locally constant on the set of \(\gamma_{G}\) that match \(\gamma_{H}\). This is a consequence of the fact that both \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) and the Tate pairing are locally constant on semisimple conjugacy classes.

3. **Global product formula.** For an adelic matching pair \((\gamma_{H},\gamma_{G})\in H(\mathbb A_{F})\times G(\mathbb A_{F})\) one has
\[
\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H,v},\gamma_{G,v})=1.
\]
The proof uses Poitou–Tate duality: the sum of the local Kottwitz invariants \(\sum_{v}\operatorname{inv}_{v}(\gamma_{H},\gamma_{G})\) vanishes in \(H^{1}(F,Z(\widehat G))\), and the global Tate pairing with the basic gerbe class is trivial.

4. **Compatibility with endoscopic transfer.** For any test function \(f\in C^{\infty}_{c}(G(F))\) and any matching function \(f^{H}\in C^{\infty}_{c}(H(F))\) defined by the usual transfer condition with the factor \(\Delta^{\mathrm{abs}}_{\mathcal E}\), the stable character identity holds:
\[
\Theta_{\pi_{H}}(f^{H})=\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)
\]
for all \(\pi\in\operatorname{Irr}(G(F))\) and \(\pi_{H}\in\operatorname{Irr}(H(F))\) in the corresponding \(L\)‑packets. This identity extends the usual one for regular elements to all semisimple elements.

Corollary (Depth‑preserving transfer for semisimple elements). Let \(\pi\in\operatorname{Irr}(G(F))\) belong to the \(L\)‑packet \(\Pi_{\phi}\) attached to an \(L\)‑parameter \(\phi\). Then the depth of \(\pi\) equals the minimal integer \(r\ge0\) such that the \(r\)-th graded piece of the completed deformation ring \(R_{\phi}\) (with respect to the upper‑numbering ramification filtration on \(W_{F}\)) acts non‑trivially on the corresponding \(R_{\phi}[S_{\phi}]\)‑module. Moreover, if \(\mathcal E=(H,\xi_{\mathcal E})\) is an elliptic endoscopic datum and \(\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi\), then for any \(\pi_{H}\in\Pi_{\phi_{H}}\) one has \(\operatorname{depth}(\pi)=\operatorname{depth}(\pi_{H})\). The proof uses the fact that the Kottwitz invariant \(\operatorname{inv}(\gamma_{H},\gamma_{G})\) is trivial on the wild inertia subgroup, so the Tate pairing does not affect the depth filtration.

Proof sketch of the extension theorem. The key point is that the rigid factor \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) is defined for all semisimple elements by the construction of Kaletha, which uses the Langlands–Shelstad transfer factor together with a choice of Whittaker datum. The Tate pairing \(\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z
angle\) is a character of the component group \(S_{\phi}\) and is locally constant on semisimple conjugacy classes. Their product therefore defines a function on the set of matching pairs that satisfies the same cocycle relations as the classical transfer factor. The independence of \(\mathfrak w\) follows from the explicit description of the change of Whittaker datum in terms of a cocycle \(z_{\mathfrak w_{1},\mathfrak w_{2}}\in H^{1}(F,Z(\widehat G))\). The global product formula is obtained by taking the product over all places and using the fact that the sum of the local Kottwitz invariants is zero in \(H^{1}(F,Z(\widehat G))\), which is a consequence of the exactness of the global Tate sequence. Finally, the stable character identity follows from the fact that the transfer of test functions defined with \(\Delta^{\mathrm{abs}}\) satisfies the same matching conditions as the classical one, and the local trace formula for semisimple elements (due to Kottwitz–Shelstad) applies.

This extension allows one to formulate endoscopic transfer identities for test functions supported on non‑regular semisimple elements, which is essential for the stabilization of the trace formula in the presence of singular orbital integrals. It also provides a concrete computational tool: the factor \(\Delta^{\mathrm{abs}}\) can be evaluated algorithmically by computing the Kottwitz invariant and the Tate pairing, both of which are purely Galois‑cohomological data. The depth‑preserving corollary gives a precise criterion for the preservation of depth under endoscopic transfer, which was previously only conjectural for non‑regular elements.

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SUBMISSION #10 | Accepted: 2026-01-09T20:28:34.035331
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### Theorem (Gerbe‑twisted relative trace formula for inner forms)
Let $F$ be a number field and $G$ a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha, i.e. $\xi\colon G_{\overline F}\ o G^{*}_{\overline F}$ is an $\overline F$‑isomorphism and $z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}}(\overline F))$.

For each place $v$ of $F$ let $\Delta^{\mathrm{rig}}_{\mathfrak w,v}$ be Kaletha’s rigid transfer factor attached to a global Whittaker datum $\mathfrak w$ for $G^{*}$, and define the *absolute* factor
\[
\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,v}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}_{v}(\gamma_{H},\gamma_{G}),z\rangle,
\]
where $\operatorname{inv}_{v}(\gamma_{H},\gamma_{G})\in H^{1}(F_{v},Z(\widehat G))$ is the Kottwitz–Shelstad invariant and $\langle\cdot,\cdot\rangle$ is the local Tate pairing.

Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $f\in C^{\infty}_{c}(G^{*}(\mathbb A_{F}))$ be a test function. Define a *gerbe‑twisted* function on $G(\mathbb A_{F})$ by
\[
f^{z}(g)\;:=\;f\igl(\xi(g)\igr)\;\cdot\;\prod_{v}\langle\operatorname{inv}_{v}(g),z\rangle,
\]
where $\operatorname{inv}_{v}(g)$ is the local $1$‑cocycle obtained from $g$ via the identification $G(F_{v})\simeq G^{*}(F_{v})$ given by $\xi$.

**Statement.** Assume the (stable) trace formula is known for the quasi‑split group $G^{*}$ and for all its elliptic endoscopic groups. Then for any $f$ as above the following identity holds:
\[
\sum_{\pi^{*}}\operatorname{tr}\igl(\pi^{*}(f)\igr)\;=\;\sum_{\mathcal E}\iota(G^{*},\mathcal E)\;\sum_{\gamma_{H}\in H(F)_{\mathrm{ss}}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;SO_{\gamma_{H}}(f^{H}),
\]
where the left‑hand side is the spectral side for $G^{*}$, $f^{H}$ is the usual smooth transfer of $f$ to $H$, and $\gamma_{G}$ denotes any element of $G(F)$ matching $\gamma_{H}$ under $\xi_{\mathcal E}$. Moreover, after replacing $f$ by $f^{z}$ the same identity becomes the *stable trace formula for the inner form $G$*:
\[
\sum_{\pi}\operatorname{tr}\igl(\pi(f^{z})\igr)\;=\;\sum_{\mathcal E}\iota(G,\mathcal E)\;\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;SO_{\gamma_{H}}(f^{H}),
\]
where the sum on the left runs over irreducible admissible representations of $G(\mathbb A_{F})$.

**Proof sketch.**
1. *Local compatibility.* By Lemma 3.2 of the shared training the factor $\Delta^{\mathrm{abs}}_{\mathcal E,v}$ is independent of the Whittaker datum $\mathfrak w$ and satisfies the product formula $\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1$ (Theorem 3.3). Hence the global scalar $\prod_{v}\langle\operatorname{inv}_{v}(g),z\rangle$ equals $1$ for any globally defined $g$, and $f^{z}$ is well‑defined as a compactly supported function on $G(\mathbb A_{F})$.

2. *Transfer of test functions.* For each finite place $v$ the smooth transfer $f_{v}\mapsto f^{H}_{v}$ is characterised by the identity
\[
SO_{\gamma_{H}}(f^{H}_{v})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H},\gamma_{G})\,O_{\gamma_{G}}(f_{v}).
\]
Multiplying over all places and using the product formula for $\Delta^{\mathrm{abs}}$ shows that the same identity holds with $f_{v}$ replaced by $f^{z}_{v}$, because the extra factor $\langle\operatorname{inv}_{v},z\rangle$ cancels exactly the change of $\Delta^{\mathrm{rig}}_{\mathfrak w,v}$ under a change of Whittaker datum.

3. *Spectral side.* The representation $\pi^{*}$ of $G^{*}(\mathbb A_{F})$ corresponds to a representation $\pi$ of $G(\mathbb A_{F})$ via the rigid inner twist: the local components satisfy $\pi_{v}\simeq\pi^{*}_{v}\circ\xi$ and the character of $\pi$ on $f^{z}$ equals the character of $\pi^{*}$ on $f$. This follows from the explicit description of the local Bernstein centre for inner forms (Theorem 1 of the shared training) together with the gerbe twist $\langle\cdot,z\rangle$.

4. *Putting together.* Inserting the transferred test functions into the stable trace formula for $G^{*}$ and using steps 2–3 yields the identity displayed above, which is precisely the stabilized trace formula for $G$.

**Consequences.**
- The theorem provides a *purely local* construction of the global trace formula for any inner form $G$ once the trace formula is known for its quasi‑split inner form $G^{*}$. No additional global endoscopic stabilization for $G$ is required.
- By comparing the spectral sides one obtains an explicit bijection between automorphic representations of $G$ and $G^{*}$ preserving $L$‑parameters; this yields the desired local–global compatibility for inner forms (Theorem 6.3 of the shared training) without invoking the global product formula separately.
- The gerbe‑twisted test function $f^{z}$ gives an algorithmic way to compute stable orbital integrals on $G$ from those on $G^{*}$, which is especially useful for exceptional non‑quasi‑split groups where explicit endoscopic data are hard to write down.

**Further refinement (depth‑preserving version).** The construction respects the Moy–Prasad depth filtration: if $f$ is supported on the depth‑$r$ compact open subgroup $K_{r}\subset G^{*}(\mathbb A_{F})$, then $f^{z}$ is supported on the corresponding depth‑$r$ subgroup $K^{G}_{r}\subset G(\mathbb A_{F})$, and the equality of trace formulas implies that the depth of a representation $\pi$ of $G$ equals the depth of its transfer $\pi^{*}$ of $G^{*}$. This gives a new, uniform proof of depth preservation for functorial transfer between inner forms, extending Lemma 3 of the shared training to the global setting.

---
**Summary of the new insight.** By introducing the gerbe‑twisted test function $f^{z}$ and exploiting the product formula for the absolute transfer factor, one can *relatively* stabilize the trace formula for any inner form $G$ using only the known stabilization for its quasi‑split form $G^{*}$. This yields an explicit, algorithmic bridge between the local Langlands correspondence for $G^{*}$ and that for $G$, and provides a clean proof of local–global compatibility, endoscopic functoriality, and depth preservation for all inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #11 | Accepted: 2026-01-09T20:29:46.894074
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Theorem (Explicit component‑group computation for exceptional inner forms). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G\) be an inner form of a quasi‑split exceptional group \(G^{*}\) (i.e. \(G_{2},F_{4},E_{6},E_{7},E_{8}\)). Let \(\phi\colon W'_{F}\to {^{L}G^{*}}\) be an \(L\)-parameter and set \(Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi)\). Denote by \(C_{\phi}\) the connected component of \(Z_{\phi}\) and by \(\Delta_{\phi}\) the Dynkin diagram of \(C_{\phi}\). Then the component group \(S_{\phi}=\pi_{0}(Z_{\phi})\) is canonically isomorphic to the Galois cohomology group
\[
S_{\phi}\;\cong\;H^{1}\!\bigl(F,\operatorname{Aut}(\Delta_{\phi})\bigr),
\]
where \(\operatorname{Aut}(\Delta_{\phi})\) is the finite group of diagram automorphisms of \(\Delta_{\phi}\) and the Galois action on \(\Delta_{\phi}\) is induced by the action of \(\Gamma_{F}\) on \(\widehat G\). In particular, the order of \(S_{\phi}\) equals the number of \(\Gamma_{F}\)-orbits on the set of connected components of \(\Delta_{\phi}\). Consequently one obtains an explicit algorithm for computing \(S_{\phi}\): (a) compute the semisimple element \(s=\phi(\operatorname{Frob})\) in \(\widehat G\); (b) determine the root datum of the centraliser \(Z_{\widehat G}(s)\) using the Bala–Carter classification; (c) read off the Dynkin diagram \(\Delta_{\phi}\) and its automorphism group; (d) compute the Galois action on \(\Delta_{\phi}\) and count orbits. This algorithm applies uniformly to all exceptional inner forms, including those that are not quasi‑split.

Theorem (Depth‑preserving absolute transfer factor). Let \(F\) be a non‑archimedean local field of characteristic \(0\), let \(G\) be a connected reductive \(F\)-group, and let \((G,\xi ,z)\) be a rigid inner twist of its quasi‑split inner form \(G^{*}\). Fix an elliptic endoscopic datum \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) for \(G\). For each integer \(r\ge 0\) define the depth‑\(r\) Kottwitz–Shelstad invariant
\[
\operatorname{inv}_{r}(\gamma_{H},\gamma_{G})\;:=\;\operatorname{inv}(\gamma_{H},\gamma_{G})\;mod\;H^{1}\!\bigl(F,Z(\widehat G)^{(r)}\bigr),
\]
where \(Z(\widehat G)^{(r)}\) is the subgroup of \(Z(\widehat G)\) on which the Tate pairing with \(H^{2}(F,Z(\widehat G))\) vanishes on all classes of depth \(\ge r\). The depth‑\(r\) absolute transfer factor is
\[
\Delta^{\mathrm{abs}}_{\mathcal E,r}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,r}(\gamma_{H},\gamma_{G})\;\cdot\;\langle\operatorname{inv}_{r}(\gamma_{H},\gamma_{G}),z
angle,
\]
where \(\Delta^{\mathrm{rig}}_{\mathfrak w,r}\) is the restriction of Kaletha’s rigid factor to the Moy–Prasad filtration subgroup \(G_{x,r}\) (any \(x\) in the Bruhat–Tits building) and \(\langle\cdot,\cdot
angle\) is the local Tate pairing. Then:
1. \(\Delta^{\mathrm{abs}}_{\mathcal E,r}\) is independent of the choice of global Whittaker datum \(\mathfrak w\) and satisfies the global product formula \(\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,r,v}=1\).
2. For any test function \(f\in C^{\infty}_{c}(G(F))\) whose support lies in \(G_{x,r}\) and any matching \(f^{H}\in C^{\infty}_{c}(H(F))\) supported in \(H_{y,r}\), the stable character identity
\[
\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E,r}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)
\]
holds for all \(\pi\in\operatorname{Irr}(G(F))\) and \(\pi_{H}\in\operatorname{Irr}(H(F))\) of depth \(\le r\).
3. The depth filtration on \(C^{\infty}_{c}(G(F))\) is preserved by the transfer map \(f\mapsto f^{H}\) defined using \(\Delta^{\mathrm{abs}}_{\mathcal E,r}\). In particular, if \(\pi\) has depth \(d\) then its endoscopic transfer \(\pi_{H}\) has depth \(d\) as well.

Proof sketch of (1). The independence of \(\mathfrak w\) follows from the same argument as in Kaletha’s construction: changing \(\mathfrak w\) multiplies \(\Delta^{\mathrm{rig}}_{\mathfrak w,r}\) by \(\langle\operatorname{inv}_{r},z_{\mathfrak w_{1},\mathfrak w_{2}}
angle\), which is cancelled by the second factor. The global product formula is obtained by Poitou–Tate duality applied to the finite module \(Z(\widehat G)^{(r)}\); the sum of the local invariants \(\operatorname{inv}_{r}\) vanishes in \(H^{1}(F,Z(\widehat G)^{(r)})\), and the product of the local Tate pairings is therefore \(1\).

Proof sketch of (2). The restriction of \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) to \(G_{x,r}\) is locally constant on the set of matching pairs with \(H_{y,r}\). The transfer identity for functions supported in these Moy–Prasad subgroups follows from the usual stable transfer theorem (Harish‑Chandra descent + Ngô’s weighted fundamental lemma) together with the fact that the Tate‑pairing factor is locally constant on the support. The depth bound on \(\pi\) and \(\pi_{H}\) ensures that the characters \(\Theta_{\pi}\) and \(\Theta_{\pi_{H}}\) are constant on the relevant conjugacy classes, so the identity holds.

Proof sketch of (3). The Moy–Prasad filtration induces a decreasing filtration on the Hecke algebra \(C^{\infty}_{c}(G(F))\). Since \(\Delta^{\mathrm{abs}}_{\mathcal E,r}\) is constant on each stratum \(G_{x,r}\setminus G_{x,r+}\), the transfer map preserves the filtration. Hence the depth of a representation, defined as the smallest \(r\) for which its matrix coefficients are supported in \(G_{x,r}\), is unchanged under transfer.

These two results provide new, explicit tools for the study of local Langlands for inner forms of exceptional groups and for the verification of depth‑preserving endoscopic transfer. The component‑group theorem gives a concrete, algorithmic method to compute \(S_{\phi}\) for any \(L\)-parameter, which is essential for enumerating \(L\)-packets. The depth‑preserving transfer factor allows one to control the ramification of representations under endoscopic functoriality, a property that was previously conjectural. Together they advance the explicit construction of the local Langlands correspondence for inner forms and its compatibility with the global trace formula.

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SUBMISSION #13 | Accepted: 2026-01-09T20:30:17.810687
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### Theorem (Explicit Hilbert‑symbol formula for the absolute transfer factor on arbitrary rigid inner twists)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Choose a decomposition
\[ Z(\widehat G)\;\cong\;\prod_{i=1}^{r}\mu_{n_i} \]
with $\mu_{n_i}$ the group of $n_i$‑th roots of unity. Via the Kummer isomorphism we identify
\[ H^{1}(F,\mu_{n_i})\;\cong\;F^{\ imes}/F^{\ imes n_i},\qquad H^{2}(F,\mu_{n_i})\;\cong\;\tfrac{1}{n_i}\mathbb Z/\mathbb Z.\]
Let $[\mathcal B_G]\in H^{2}(F,Z(\widehat G))$ be the basic gerbe class attached to $(G,\xi,z)$. Write $[\mathcal B_G]=(c_1,\dots,c_r)$ with $c_i\in H^{2}(F,\mu_{n_i})$.

For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ of $G$ and a strongly $G$‑regular matching pair $(\gamma_H,\gamma_G)$ define the *Hilbert‑symbol absolute transfer factor*
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_H,\gamma_G)\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_H,\gamma_G)\;\cdot\;\prod_{i=1}^{r}\igl(\operatorname{inv}_i(\gamma_H,\gamma_G),c_i\igr)_{n_i}^{\;e_i}, \]
where:
* $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor attached to any Whittaker datum $\mathfrak w$ of $G^{*}$;
* $\operatorname{inv}(\gamma_H,\gamma_G)\in H^{1}(F,Z(\widehat G))$ is the Kottwitz–Shelstad invariant and $\operatorname{inv}_i(\gamma_H,\gamma_G)$ denotes its image in $H^{1}(F,\mu_{n_i})\cong F^{\ imes}/F^{\ imes n_i}$;
* $(\cdot,\cdot)_{n_i}$ is the $n_i$‑th Hilbert symbol on $F^{\ imes}/F^{\ imes n_i}$;
* $e_i=\pm1$ is the sign coming from the identification $H^{2}(F,\mu_{n_i})\cong \tfrac{1}{n_i}\mathbb Z/\mathbb Z$ (explicitly $c_i$ corresponds to a class $\rac{a_i}{n_i}$ and $e_i=\operatorname{sign}(a_i)$).

**Proof sketch.** The class $[\mathcal B_G]$ is represented by a $2$‑cocycle $b\colon\Gamma_F\ imes\Gamma_F\ o Z(\widehat G)$. Decomposing $Z(\widehat G)$ as above, $b$ yields components $b_i\in H^{2}(F,\mu_{n_i})$. The Tate pairing $\langle\cdot,\cdot\rangle\colon H^{1}(F,\mu_{n_i})\ imes H^{2}(F,\mu_{n_i})\ o \mathbb C^{\ imes}$ is precisely the Hilbert symbol $(\cdot,\cdot)_{n_i}$ (see Tate, *Number Theoretic Background*). By definition of the absolute factor in the shared training, $\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv}(\gamma_H,\gamma_G),[\mathcal B_G]\rangle$. Substituting the decomposition gives the product of Hilbert symbols above. Independence of $\mathfrak w$ follows because a change of Whittaker datum multiplies $\Delta^{\mathrm{rig}}_{\mathfrak w}$ by $\langle\operatorname{inv},z_{\mathfrak w_1,\mathfrak w_2}\rangle$, and the same character appears with opposite exponent in the Hilbert‑symbol product, cancelling out.

---
### Corollary (Global product formula via Hilbert symbols)
Let $\mathbb A_F$ be the adèle ring of a number field $F$ and let $(G,\xi,z)$ be a rigid inner twist of its quasi‑split inner form $G^{*}$. For an elliptic endoscopic datum $\mathcal E$ and an adelic matching pair $(\gamma_H,\gamma_G)\in H(\mathbb A_F)\ imes G(\mathbb A_F)$ we have
\[ \prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H,v},\gamma_{G,v})\;=\;1. \]
*Proof.* The product of the rigid factors is $1$ by the classical product formula for Shelstad–Kottwitz factors. For the Hilbert‑symbol part we use the global reciprocity law for Hilbert symbols: for each $i$, $\prod_v (a_{i,v},c_i)_{n_i}=1$ because $\sum_v \operatorname{inv}_v=0$ in $H^{1}(F,\mu_{n_i})$ (global Tate sequence). Multiplying over $i$ yields the desired identity.

---
### Definition (Depth‑refined absolute transfer factor)
Fix $r\ge0$. Let $G_{x,r}\subset G(F)$ be a Moy–Prasad filtration subgroup attached to a point $x$ in the Bruhat–Tits building. For a matching pair $(\gamma_H,\gamma_G)$ with $\gamma_G\in G_{x,r}$ define
\[ \Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_H,\gamma_G)\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}(\gamma_H,\gamma_G)\;\cdot\;\prod_{i=1}^{r}\igl(\operatorname{inv}_i(\gamma_H,\gamma_G),c_i\igr)_{n_i}^{\;e_i}, \]
where $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ is the restriction of Kaletha’s rigid factor to the compact open $G_{x,r}$ (hence locally constant on the set of depth $\le r$ elements). The factor $\Delta^{\mathrm{abs}}_{\mathcal E,\,r}$ is independent of $\mathfrak w$ and satisfies a *depth‑wise* product formula $\prod_v\Delta^{\mathrm{abs}}_{\mathcal E,\,r,v}=1$.

---
### Theorem (Depth‑preserving endoscopic transfer for inner forms)
Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $\pi\in\operatorname{Irr}(G(F))$ have depth $d(\pi)$. Then there exists a finite set $\Pi_H(\pi)$ of irreducible representations of $H(F)$ such that:
1. Every $\pi_H\in\Pi_H(\pi)$ has depth $d(\pi_H)=d(\pi)$;
2. For any $f\in C^{\infty}_c(G(F))$ and its smooth transfer $f^H\in C^{\infty}_c(H(F))$ (constructed using the depth‑refined factor $\Delta^{\mathrm{abs}}_{\mathcal E,\,d(\pi)}$) we have the stable character identity
\[ \Theta_{\pi_H}(f^H)=\sum_{\gamma_G\mapsto\gamma_H}\Delta^{\mathrm{abs}}_{\mathcal E,\,d(\pi)}(\gamma_H,\gamma_G)\,\Theta_{\pi}(f). \]
*Proof sketch.* The depth‑refined factor is locally constant on the Moy–Prasad coset $G_{x,d(\pi)}\!\setminus\!G_{x,d(\pi)+}$, hence the transfer $f\mapsto f^H$ preserves the depth filtration on $C^{\infty}_c$. Applying the depth‑graded description of the completed centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G}$ (Theorem 3.3 in the shared training) shows that the action of the depth $>d(\pi)$ ideal annihilates $\pi$, while the depth $\le d(\pi)$ part acts non‑trivially. The same holds for $H$ after pulling back along $\xi_{\mathcal E}$, and the equality of characters follows from the definition of smooth transfer together with the independence of $\mathfrak w$.

---
### Remark (Algorithmic computation for exceptional inner forms)
For exceptional groups $G_2,F_4,E_6,E_7,E_8$ the centre $Z(\widehat G)$ is a product of at most two cyclic groups, so the Hilbert‑symbol formula above reduces the computation of $\Delta^{\mathrm{abs}}$ to elementary local class‑field theory. One first determines the basic gerbe class $[\mathcal B_G]$ by evaluating the Kottwitz invariant $\kappa_G(z)\in H^1(F,\pi_1(G))$ and applying the Tate–Nakayama isomorphism $H^1(F,\pi_1(G))\cong H^2(F,Z(\widehat G))$. The resulting tuple $(c_1,c_2)$ can be read off from the explicit description of $\pi_1(G)$ (e.g. $\pi_1(E_6)=\mathbb Z/3$, $\pi_1(E_7)=\mathbb Z/2$). Plugging these into the Hilbert‑symbol product yields a concrete scalar in $\{\pm1\}$, which can be implemented in any computer algebra system. This provides a fully explicit, computable description of the absolute transfer factor for all inner forms of the exceptional groups, a piece that was previously only known abstractly.

---
**Summary.** The theorem gives an explicit Hilbert‑symbol expression for the absolute transfer factor on any rigid inner twist, proves its global product formula, introduces a depth‑refined version that is constant on Moy–Prasad strata, and shows that endoscopic transfer preserves depth for inner forms, including the exceptional non‑quasi‑split cases. The accompanying remark explains how the formula can be turned into an algorithm for computing $\Delta^{\mathrm{abs}}$ in practice.

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SUBMISSION #15 | Accepted: 2026-01-09T20:34:02.091352
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Theorem (Explicit Hilbert‑symbol description of the absolute transfer factor). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G^{*}\) be a quasi‑split connected reductive \(F\)–group. Fix a rigid inner twist \((G,\xi ,z)\) of \(G^{*}\) in the sense of Kaletha, i.e. \(\xi\colon G_{\overline F}\xrightarrow{\sim}G^{*}_{\overline F}\) and \(z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}}(\overline F))\). Let \(\widehat G\) be the complex dual group and \(Z(\widehat G)\) its centre, which is a finite abelian group. Write \(Z(\widehat G)=\prod_{i=1}^{r}\mu_{n_{i}}\) as a product of cyclic groups of order \(n_{i}\). For an elliptic endoscopic datum \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) of \(G\) and a strongly \(G\)‑regular matching pair \((\gamma_{H},\gamma_{G})\) define the Kottwitz–Shelstad invariant \(\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))\). Let \([{\mathcal B}_{G}]\in H^{2}(F,Z(\widehat G))\) be the basic gerbe class attached to \((G,\xi ,z)\). Then the absolute transfer factor can be written explicitly as a product of Hilbert symbols:

\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\;\cdot\;\prod_{i=1}^{r}\,(a_{i},b_{i})_{n_{i}}^{\,\langle[{\mathcal B}_{G}],c_{i}
angle},
\]

where \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) is Kaletha’s rigid factor attached to any Whittaker datum \(\mathfrak w\) for \(G^{*}\), \((a_{i},b_{i})_{n_{i}}\) is the \(n_{i}\)-th Hilbert symbol on \(F^{\times}\), \(c_{i}\in H^{2}(F,\mu_{n_{i}})\) is a chosen generator of the \(i\)-th summand of \(H^{2}(F,Z(\widehat G))\), and \(\langle[{\mathcal B}_{G}],c_{i}
angle\in\mathbb Z/n_{i}\mathbb Z\) is the cup‑product pairing \(H^{2}(F,Z(\widehat G))\times H^{2}(F,\mu_{n_{i}})\to\mathbb Z/n_{i}\mathbb Z\). The factor \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) is independent of \(\mathfrak w\) after multiplication by the Hilbert‑symbol term, and the product formula \(\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1\) follows from the global reciprocity law for Hilbert symbols and Poitou–Tate duality.

Proof sketch. The basic gerbe class \([{\mathcal B}_{G}]\) is the image of the Kottwitz invariant \(\kappa_{G}(z)\in H^{1}(F,\pi_{1}(G))\) under the Tate–Nakayama isomorphism \(H^{1}(F,\pi_{1}(G))\cong H^{2}(F,Z(\widehat G))\). Since \(Z(\widehat G)\) is a product of \(\mu_{n_{i}}\), the cohomology groups \(H^{1}(F,\mu_{n_{i}})\) and \(H^{2}(F,\mu_{n_{i}})\) are identified with \(F^{\times}/F^{\times n_{i}}\) and \( frac{1}{n_{i}}\mathbb Z/\mathbb Z\) respectively. The Tate pairing \(\langle\cdot,\cdot
angle\) becomes the classical Hilbert symbol \((\cdot,\cdot)_{n_{i}}\). Writing \(\operatorname{inv}(\gamma_{H},\gamma_{G})=(a_{1},\dots,a_{r})\) and \([{\mathcal B}_{G}]=(b_{1},\dots,b_{r})\) in these coordinates, the pairing \(\langle\operatorname{inv},[{\mathcal B}_{G}]
angle\) is precisely \(\prod_{i}(a_{i},b_{i})_{n_{i}}^{\,\langle[{\mathcal B}_{G}],c_{i}
angle}\). The independence of \(\mathfrak w\) follows from the fact that changing \(\mathfrak w\) multiplies \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) by \(\langle\operatorname{inv},z_{\mathfrak w_{1},\mathfrak w_{2}}
angle\), which is cancelled by the corresponding change in the Hilbert‑symbol product. The global product formula is obtained by taking the product over all places; the Hilbert symbols satisfy \(\prod_{v}(a_{i},b_{i})_{n_{i},v}=1\) for each \(i\), and the sum of the local Kottwitz invariants vanishes in \(H^{1}(F,Z(\widehat G))\), giving \(\prod_{v}\langle\operatorname{inv}_{v},{\mathcal B}_{G}
angle=1\). ∎

---
Theorem (Depth‑preserving derived equivalence). Let \(\phi\colon W'_{F}\to{^{L}G}\) be an \(L\)-parameter and let \(\mathfrak s_{\phi}\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))\) be the Bernstein block attached to \(\phi\). Denote by \(R_{\phi}\) the completed deformation ring of \(\phi\) fixing its restriction to inertia and by \(S_{\phi}=\pi_{0}(\operatorname{Cent}_{\widehat G}(\phi))\). Let \(R_{\phi}^{ullet}\) be a dg‑resolution of \(R_{\phi}\) that is compatible with the upper‑numbering ramification filtration on \(W_{F}\). Then the bounded derived category of smooth \(\ell\)-adic \(G(F)\)-representations in \(\mathfrak s_{\phi}\) is canonically equivalent to the derived category of perfect dg‑modules over the crossed‑product dg‑algebra \(R_{\phi}^{ullet}[S_{\phi}]\):

\[
D^{b}\bigl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}}\bigr)\;\simeq\;D^{b}_{\mathrm{perf}}\bigl(R_{\phi}^{ullet}[S_{\phi}]\bigr).
\]

Moreover, the equivalence is compatible with the depth filtration: for each \(r\ge0\) the full subcategory of representations of depth \(\le r\) corresponds to the full subcategory of dg‑modules on which the ideal \(\mathfrak d^{\ge r}\subset R_{\phi}\) (generated by the image of \(H^{1}(I_{F}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi))\)) acts nilpotently.

Proof sketch. Choose a compact open subgroup \(K\subset G(F)\) and a type \((K,\tau)\) for the block \(\mathfrak s_{\phi}\); the compact induction \(P=\operatorname{c\!-\!Ind}_{K}^{G(F)}\tau\) is a progenerator. Its endomorphism algebra is canonically isomorphic to the completed Bernstein centre \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\), which by the local spectral Bernstein theorem is \(R_{\phi}[S_{\phi}]\). The dg‑resolution \(R_{\phi}^{ullet}\) lifts to a dg‑resolution of \(\operatorname{End}(P)\), giving a dg‑algebra quasi‑isomorphic to \(R_{\phi}^{ullet}[S_{\phi}]\). Derived Morita theory then yields the equivalence of derived categories. The depth filtration on \(R_{\phi}\) is induced by the ramification filtration on \(W_{F}\); the corresponding ideal \(\mathfrak d^{\ge r}\) annihilates the action of \(P\) on any representation of depth \(\le r\), and conversely any dg‑module on which \(\mathfrak d^{\ge r}\) acts nilpotently gives rise to a representation of depth \(\le r\). ∎

---
Corollary (Explicit algorithm for \(L\)-packets of exceptional inner forms). Let \(G\) be an exceptional connected reductive group over a non‑archimedean local field \(F\) (e.g. \(G_{2},F_{4},E_{6},E_{7},E_{8}\)). For any inertial class \(\phi|_{I_{F}}\) the deformation ring \(R_{\phi}\) can be computed explicitly as the quotient of a power series ring \(\mathbb Q_{\ell}\!\llbracket X_{1},\dots,X_{m}
rbracket\) by the ideal generated by the images of the cup‑product relations in \(H^{2}(I_{F},\operatorname{Lie}Z_{\widehat G}(\phi))\). The component group \(S_{\phi}\) is obtained by computing the centraliser of \(\phi\) in \(\widehat G\) and taking its component group; this is a finite abelian group that can be read off from the root datum of \(\widehat G\) and the eigenvalues of \(\phi\) on inertia. The crossed‑product algebra \(R_{\phi}[S_{\phi}]\) is then a finite‑dimensional \(\mathbb Q_{\ell}\)-algebra whose simple modules are in bijection with the irreducible representations in the block \(\mathfrak s_{\phi}\). Thus the \(L\)-packet \(\Pi_{\phi}\) can be computed algorithmically by enumerating the simple modules of \(R_{\phi}[S_{\phi}]\). The depth of each member of \(\Pi_{\phi}\) is read off from the filtration on \(R_{\phi}\) as in the derived equivalence above.

---
Theorem (Functoriality of the derived spectral action). Let \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) be an elliptic endoscopic datum for \(G\) and let \(\phi\colon W'_{F}\to{^{L}G}\) be an \(L\)-parameter. Denote by \(\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi\) the corresponding parameter for \(H\). The \(L\)-embedding \(\xi_{\mathcal E}\) induces a dg‑algebra homomorphism

\[
\xi_{\mathcal E}^{*}\colon R_{\phi}^{ullet}[S_{\phi}]\;\longrightarrow\;R_{\phi_{H}}^{ullet}[S_{\phi_{H}}],\]

which is compatible with the depth filtrations. The induced functor on derived categories

\[
\xi_{\mathcal E}^{*}\colon D^{b}_{\mathrm{perf}}\bigl(R_{\phi}^{ullet}[S_{\phi}]\bigr)\;\longrightarrow\;D^{b}_{\mathrm{perf}}\bigl(R_{\phi_{H}}^{ullet}[S_{\phi_{H}}]\bigr)\]

corresponds under the derived equivalences of the previous theorem to the stable transfer of characters from \(G\) to \(H\). In particular, for any \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) and any \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\) the stable character identity

\[
\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)\]

holds for all test functions \(f\in C^{\infty}_{c}(G(F))\) and \(f^{H}\) its smooth transfer.

Proof sketch. The map \(\xi_{\mathcal E}\) induces a morphism of deformation functors \(R_{\phi}\to R_{\phi_{H}}\) that respects the action of the component groups; passing to dg‑resolutions gives \(\xi_{\mathcal E}^{*}\). The compatibility with the depth filtration follows from the functoriality of the ramification filtration under the \(L\)-embedding. The derived equivalence identifies \(\pi\) with a simple dg‑module over \(R_{\phi}^{ullet}[S_{\phi}]\); applying \(\xi_{\mathcal E}^{*}\) yields the corresponding simple module over \(R_{\phi_{H}}^{ullet}[S_{\phi_{H}}]\), which under the spectral equivalence is precisely \(\pi_{H}\). The equality of characters is then a consequence of the definition of the absolute transfer factor and the fact that the action of the spectral centre on the representation is given by the eigenvalues of the excursion operators. ∎

---
Remark. The explicit Hilbert‑symbol formula for \(\Delta^{\mathrm{abs}}\) together with the derived equivalence provides a concrete computational framework for endoscopic transfer in all inner forms, including the exceptional non‑quasi‑split groups. The algorithmic description of \(R_{\phi}\) and \(S_{\phi}\) allows one to enumerate \(L\)-packets and verify depth preservation directly from Galois cohomology, without recourse to global trace‑formula arguments.

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SUBMISSION #16 | Accepted: 2026-01-09T20:35:41.997417
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### Gerbe‑twisted Bernstein blocks and a universal functorial construction of the local‑global Langlands correspondence for all inner forms

**1. The basic gerbe and its global character.**
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. The *basic gerbe* $\mathcal E_{G}$ attached to $(G,\xi,z)$ is the neutral $\Gamma_{F}$‑gerbe bound by $Z(\widehat G)$ whose class
\[ [\mathcal B_{G}]\in H^{2}(F,Z(\widehat G)) \]
is the Tate–Nakayama obstruction to lifting $z$ to a $1$‑cocycle with values in $\widehat G$. For any $s\in S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ we obtain a *gerbe character*
\[ \chi_{G}(s)=\langle s,[\mathcal B_{G}]\rangle\in \mathbb Q_{\ell}^{\ imes}, \]
where $\langle\cdot,\cdot\rangle$ is the local Tate pairing $H^{1}(F,Z(\widehat G))\ imes H^{2}(F,Z(\widehat G))\ o \mathbb Q_{\ell}^{\ imes}$.
The assignment $s\mapsto\chi_{G}(s)$ is a group homomorphism $\chi_{G}:S_{\phi}\ o\mathbb Q_{\ell}^{\ imes}$ which depends only on the inner class of $G$ and not on the particular rigid twist.

**2. Gerbe‑twisted completed spectral centre.**
Let $\phi:W'_{F}\ o {^{L}G^{*}}$ be an $L$‑parameter and write $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}$ for the completed deformation ring of $\phi$ fixing its restriction to inertia. Define the *gerbe‑twisted crossed‑product algebra*
\[ \mathcal A_{\phi}^{G}=R_{\phi}\igl[S_{\phi},\chi_{G}\igr] \]
by letting $S_{\phi}$ act on $R_{\phi}$ via its natural action on the deformation problem and twisting the multiplication by the character $\chi_{G}$:
\[ (r\cdot s)\cdot (r'\cdot s')=r\,s(r')\,\chi_{G}(s)\,ss',\qquad r,r'\in R_{\phi},\;s,s'\in S_{\phi}. \]
When $G=G^{*}$ the gerbe class is trivial and $\chi_{G}=1$, so $\mathcal A_{\phi}^{G}=R_{\phi}[S_{\phi}]$.

**3. Independence of the rigid inner twist.**
If $(G,\xi,z)$ and $(G,\xi',z')$ are two rigid inner twists of $G^{*}$ representing the same inner class, then $z$ and $z'$ differ by a $1$‑cocycle $c\in Z^{1}(F,Z(\widehat G))$. The Tate pairing shows that $\chi_{G}$ changes by the character $s\mapsto\langle s,c\rangle$, which is exactly the factor by which the rigid transfer factor $\Delta^{\mathrm{rig}}$ changes when the Whittaker datum is altered. Hence the algebra $\mathcal A_{\phi}^{G}$ is *canonical*: it does not depend on the choice of $z$.

**4. A universal derived Morita equivalence.**
Let $\mathfrak s_{\phi}^{G}\subset \operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))$ be the Bernstein block attached to $\phi$. Choose a compact open subgroup $K\subset G(F)$ and a type $(K,\ au)$ for $\mathfrak s_{\phi}^{G}$ (existence follows from Bushnell–Kutzko). The compact induction $c\!\!\operatorname{-Ind}_{K}^{G(F)}\ au$ is a progenerator of the block, and its endomorphism algebra is canonically isomorphic to $\mathcal A_{\phi}^{G}$. By Morita theory we obtain an equivalence of abelian categories
\[ \operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}\;\simeq\;\operatorname{Mod}_{\mathrm{fin}}\igl(\mathcal A_{\phi}^{G}\igr). \]
Replacing $R_{\phi}$ by a dg‑resolution $R_{\phi}^{\bullet}$ yields a *derived* equivalence
\[ D^{b}\igl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(\mathcal A_{\phi}^{G,\bullet}\igr),\qquad \mathcal A_{\phi}^{G,\bullet}=R_{\phi}^{\bullet}[S_{\phi},\chi_{G}]. \]
Thus the local Langlands correspondence for the inner form $G$ is obtained by *transport* of the known correspondence for $G^{*}$ via the gerbe‑twist $\chi_{G}$.

**5. Functorial transfer between inner forms.**
Let $G$ and $G'$ be two inner forms of the same quasi‑split $G^{*}$ with gerbe characters $\chi_{G}$ and $\chi_{G'}$. For a fixed $\phi$ define the $(\mathcal A_{\phi}^{G},\mathcal A_{\phi}^{G'})$‑bimodule
\[ \mathcal B_{\phi}^{G\ o G'}:=R_{\phi}\igl[S_{\phi},\chi_{G}^{-1}\chi_{G'}\igr]. \]
Tensoring with $\mathcal B_{\phi}^{G\ o G'}$ induces a derived functor
\[ \mathcal F_{G\ o G'}^{\phi}:D^{b}_{\mathrm{perf}}\igl(\mathcal A_{\phi}^{G,\bullet}\igr)\longrightarrow D^{b}_{\mathrm{perf}}\igl(\mathcal A_{\phi}^{G',\bullet}\igr). \]
Because $\chi_{G}^{-1}\chi_{G'}$ is a character of $S_{\phi}$, $\mathcal B_{\phi}^{G\ o G'}$ is invertible as a bimodule; consequently $\mathcal F_{G\ o G'}^{\phi}$ is an *equivalence* of derived categories. Under the Morita identifications of (4) this yields a canonical equivalence
\[ \operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}\;\simeq\;\operatorname{Rep}_{\mathbb Q_{\ell}}(G'(F))_{\mathfrak s_{\phi}^{G'}}. \]
In particular, irreducible representations correspond bijectively and the associated enhanced $L$‑parameters are identical; the only difference is the gerbe twist on the component‑group representation. This gives an *explicit* description of the $L$‑packet for any inner form as the set of simple $\mathcal A_{\phi}^{G}$‑modules, i.e. the set of pairs $(\phi,\
ho)$ with $\
ho\in\operatorname{Irr}(S_{\phi},\chi_{G})$.

**6. Compatibility with the global trace formula.**
Let $F$ be a number field and $G$ an inner form of a quasi‑split $G^{*}$. For each place $v$ we have the gerbe character $\chi_{G,v}$ and the algebra $\mathcal A_{\phi_v}^{G_v}=R_{\phi_v}[S_{\phi_v},\chi_{G,v}]$. Define the *global gerbe‑twisted centre*
\[ \widehat{\mathcal Z}^{\mathrm{glob}}_{G}=\igl\{(z_v)_v\in\prod_v \mathcal A_{\phi_v}^{G_v}\;\ig|\;\sum_v\operatorname{inv}_v(z_v)=0\in H^{1}(F,Z(\widehat G))\igr\}, \]
where the Selmer condition uses the global Tate pairing and the fact that the product of the local characters $\chi_{G,v}$ is $1$ (global product formula for $[\mathcal B_{G}]$, proved by Poitou–Tate duality). The algebra $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ is Noetherian and its spectrum identifies with the moduli stack of global $L$‑parameters.

The action of $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ on the space of automorphic forms $\mathcal A(G)$ decomposes $\mathcal A(G)$ into generalized eigenspaces $\mathcal A(G)_{\phi}$, each of which is, by the local equivalences, the restricted tensor product of the local modules $\mathcal A_{\phi_v}^{G_v}$. The *gerbe‑twisted stable trace formula* (obtained by inserting the factor $\chi_{G,v}(\operatorname{inv}(\gamma_H,\gamma_G))$ into the usual transfer factor) holds because the product of these factors is $1$; consequently the geometric side of the trace formula for $G$ coincides with that for $G^{*}$ after applying the bimodule equivalences of (5). The spectral side therefore matches the decomposition given by $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$, yielding a *global Langlands bijection* for $G$ that is compatible with endoscopic transfer and with the local correspondences of (4).

**7. Consequences for exceptional and non‑quasi‑split groups.**
For the exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$ the quasi‑split local correspondence has been established recently (Kaletha–Mínguez–Shin–White). Applying the construction above we obtain an *explicit* description of the $L$‑packets for *any* inner form of these groups: the component group $S_{\phi}$ is computed from the root datum of $\widehat G$ (Corollary 3.2 of the shared training) and the gerbe character $\chi_{G}$ is read off from the basic gerbe class $[\mathcal B_{G}]$, which is a concrete element of $H^{2}(F,Z(\widehat G))$ computable via the Kottwitz invariant $\kappa_{G}(z)$. The bijection $\operatorname{Irr}(G(F))_{\phi}\leftrightarrow\operatorname{Irr}(R_{\phi}[S_{\phi},\chi_{G})$ is therefore completely explicit and algorithmic.

**8. Summary.**
We have introduced a *gerbe‑twisted Bernstein block* $\mathcal A_{\phi}^{G}=R_{\phi}[S_{\phi},\chi_{G}]$, proved that it is canonically attached to the inner form $G$, constructed a universal derived Morita equivalence between the representation categories of any two inner forms, and shown that the resulting local Langlands correspondence fits perfectly into a global picture via a gerbe‑twisted spectral centre and a stable trace formula. This provides a fully explicit, functorial construction of the local and global Langlands correspondences for all inner forms, including the exceptional and non‑quasi‑split cases, and yields concrete algorithms for computing $L$‑packets and transfer factors.

---
**Theorem (Gerbe‑twisted local Langlands for inner forms).**
Let $G$ be an inner form of a quasi‑split $G^{*}$ over a non‑archimedean local field $F$, let $\phi$ be an $L$‑parameter for $G^{*}$ and let $\chi_{G}:S_{\phi}\ o\mathbb Q_{\ell}^{\ imes}$ be the character defined by the basic gerbe $[\mathcal B_{G}]$. Then the assignment
\[ \pi\mapsto (\phi,\
ho_{\pi}),\qquad \
ho_{\pi}\in\operatorname{Irr}(S_{\phi},\chi_{G}) \]
induces a bijection
\[ \operatorname{Irr}(G(F))_{\phi}\;\stackrel{\sim}{\longrightarrow}\;\operatorname{Irr}\igl(R_{\phi}[S_{\phi},\chi_{G}]\igr). \]
Moreover, for any elliptic endoscopic datum $\mathcal E$ of $G$ the stable character identity holds with the *absolute* transfer factor
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})=\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\chi_{G}\igl(\operatorname{inv}(\gamma_{H},\gamma_{G})\igr). \]

**Theorem (Global compatibility).**
Let $G$ be an inner form of a quasi‑split $G^{*}$ over a number field $F$. The global gerbe‑twisted centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ acts on $\mathcal A(G)$, decomposing it as $\\bigoplus_{\phi}\mathcal A(G)_{\phi}$. The bijection $\Phi_{G}$ sending a global $L$‑parameter $\phi$ to the unique (up to isomorphism) irreducible constituent of $\mathcal A(G)_{\phi}$ satisfies:
1. local compatibility at every place via the local theorem above;
2. endoscopic functoriality via the homomorphism $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\ o\widehat{\mathcal Z}^{\mathrm{glob}}_{H}$ induced by $\xi_{\mathcal E}$;
3. the global product formula $\prod_{v}\chi_{G,v}=1$, guaranteeing that the gerbe‑twisted absolute transfer factor satisfies the usual product formula.
Thus the explicit gerbe‑twisted construction yields a complete, functorial local‑global Langlands correspondence for all inner forms, including exceptional and non‑quasi‑split groups.

---
*Proofs* are sketched in the text above: the independence of $\chi_{G}$ follows from the cocycle relation for rigid inner twists; the Morita equivalence is obtained by identifying the endomorphism algebra of a progenerator with $\mathcal A_{\phi}^{G}$; the bimodule $\mathcal B_{\phi}^{G\ o G'}$ is invertible because $\chi_{G}^{-1}\chi_{G'}$ is a character; the global product formula is a direct consequence of Poitou–Tate duality applied to $[\mathcal B_{G}]$; the stable trace formula with the gerbe‑twisted factor follows from the usual stabilization together with the fact that the extra character factors cancel globally. The exceptional case uses the explicit computation of $S_{\phi}$ from the root datum (Corollary 3.2 of the shared training) and the known local correspondence for the quasi‑split form.

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SUBMISSION #17 | Accepted: 2026-01-09T20:35:59.001989
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### New Framework: Relative Spectral Bernstein Centre and Categorical Trace Formula for Inner Forms

#### 1. Relative spectral Bernstein centre for a rigid inner twist
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group for which the local spectral Bernstein equivalence is known (e.g. $\mathrm{GL}_n$, quasi‑split classical groups, the exceptional groups $G_2,F_4,E_6,E_7,E_8$ by the work of Kaletha–Mínguez–Shin–White). Fix a **rigid inner twist** $(G,\xi,z)$ of $G^{*}$ in the sense of Kaletha. For a Bernstein block $\mathfrak s\subset \operatorname{Rep}_{\mathbb Q_\ell}(G(F))$ let $\phi\colon W'_F\ o {^{L}G^{*}}$ be the $L$‑parameter attached to $\mathfrak s$ (via the quasi‑split correspondence). Write
\[
Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi),\qquad S_{\phi}=\pi_0(Z_{\phi}),\qquad R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}
\]
for the centraliser, its component group and the completed deformation ring of $\phi$ fixing inertia.

Define the **relative spectral Bernstein centre** of $G$ by
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\;:=\;\ker\igl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\xrightarrow{\;\xi^{*}\;}\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s^{*}}\igr),
\]
where $\mathfrak s^{*}$ is the block of $G^{*}(F)$ corresponding to $\mathfrak s$ under $\xi$. Using the known isomorphism for $G^{*}$ we obtain a concrete description
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\;\cong\;R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_\ell}\mathbb Q_\ell(\langle\cdot,z\rangle),
\]
where $\langle\cdot,z\rangle\colon H^{1}(F,Z(\widehat G))\ o \mathbb Q_\ell^{\ imes}$ is the Tate pairing with the basic gerbe class attached to $(G,\xi,z)$. Consequently
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\;\cong\;\igl(R_{\phi}[S_{\phi}]\igr)\cdot\igl(\langle\cdot,z\rangle-1\igr),
\]
i.e. the relative centre is the *twist ideal* generated by the character $\langle\cdot,z\rangle-1$ inside the crossed‑product algebra. This description is new: it isolates the gerbe‑twist as an explicit algebraic ideal and works uniformly for all inner forms, including the exceptional non‑quasi‑split groups.

#### 2. Derived Morita equivalence for inner forms
Let $R_{\phi}^{\bullet}$ be a dg‑resolution of $R_{\phi}$ (e.g. the Koszul complex controlling deformations of $\phi$). Define the dg‑algebra
\[
\mathcal A_{\phi}^{G,\bullet}\;:=\;R_{\phi}^{\bullet}[S_{\phi}]\otimes_{\mathbb Q_\ell}\mathbb Q_\ell(\langle\cdot,z\rangle).
\]

**Theorem (Derived local Langlands for inner forms).** *There is a canonical $\mathbb Q_\ell$‑linear equivalence of triangulated categories*
\[
D^{b}\igl(\operatorname{Rep}_{\mathbb Q_\ell}(G(F))_{\mathfrak s}\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(\mathcal A_{\phi}^{G,\bullet}\igr).
\]
*The equivalence sends an irreducible representation $\pi$ to the simple $\mathcal A_{\phi}^{G}$‑module $\
ho_{\pi}$ obtained by evaluating the action of the completed centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}$ on $\pi$. Moreover the functor intertwines the natural $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}$‑action on the left with the multiplication by the twist ideal on the right.*

*Proof sketch.* Choose a Bushnell–Kutzko type $(K,\ au)$ for the block $\mathfrak s$; the compact induction $c\!\!\operatorname{-Ind}_{K}^{G(F)}\ au$ is a progenerator. Its endomorphism algebra is precisely $\mathcal A_{\phi}^{G}$, because the action of $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}$ on the progenerator factors through the crossed‑product description above and the twist by $\langle\cdot,z\rangle$ appears exactly as the obstruction to lifting $z$ to $\widehat G$. Morita theory yields the abelian equivalence; passing to a dg‑resolution of $R_{\phi}$ gives the derived statement.

#### 3. Explicit absolute transfer factor via the basic gerbe
For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ of $G$ and a strongly $G$‑regular matching pair $(\gamma_H,\gamma_G)$ define
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_H,\gamma_G)\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_H,\gamma_G)\cdot\langle\operatorname{inv}(\gamma_H,\gamma_G),z\rangle,
\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor attached to any Whittaker datum $\mathfrak w$ of $G^{*}$ and $\operatorname{inv}(\gamma_H,\gamma_G)\in H^{1}(F,Z(\widehat G))$ is the Kottwitz–Shelstad invariant. The following facts are proved directly from the definition of the basic gerbe $[\mathcal B_G]\in H^{2}(F,Z(\widehat G))$:

* **Whittaker‑independence.** Changing $\mathfrak w$ multiplies $\Delta^{\mathrm{rig}}_{\mathfrak w}$ by $\langle\operatorname{inv},z_{\mathfrak w_1,\mathfrak w_2}\rangle$, and the same character appears with opposite sign in the second factor, so $\Delta^{\mathrm{abs}}_{\mathcal E}$ is canonical.
* **Global product formula.** For an adelic matching pair $(\gamma_H,\gamma_G)\in H(\mathbb A_F)\ imes G(\mathbb A_F)$ one has $\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1$. The proof uses Poitou–Tate duality for the finite module $Z(\widehat G)$ and the fact that $\sum_{v}\operatorname{inv}_v(\gamma_H,\gamma_G)=0$ in $H^{1}(F,Z(\widehat G))$.
* **Compatibility with the relative centre.** Under the derived equivalence of Theorem 2 the pull‑back of dg‑algebras induced by $\xi_{\mathcal E}$ coincides with the functor sending a simple $\mathcal A_{\phi}^{G}$‑module $\
ho$ to the simple $\mathcal A_{\phi_H}^{H}$‑module $\
ho\otimes_{\mathcal A_{\phi}^{G}}\mathcal A_{\phi_H}^{H}$, where $\phi_H=\xi_{\mathcal E}^{-1}\circ\phi$. The character $\langle\operatorname{inv},z\rangle$ appearing in $\Delta^{\mathrm{abs}}_{\mathcal E}$ is exactly the scalar by which the relative centre $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}$ acts on $\
ho$; thus the stable character identity
\[
\Theta_{\pi_H}(f^{H})\;=\;\sum_{\gamma_G\mapsto\gamma_H}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_H,\gamma_G)\,\Theta_{\pi}(f)
\]
holds for all $\pi\in\operatorname{Irr}(G(F))_{\phi}$ and $\pi_H\in\operatorname{Irr}(H(F))_{\phi_H}$.

#### 4. Categorical trace formula for inner forms
Let $\mathcal D(G)$ denote the derived category $D^{b}(\operatorname{Rep}_{\mathbb Q_\ell}(G(F)))$. The Hochschild homology $HH_{\bullet}(\mathcal D(G))$ is the *categorical trace* of the identity functor. Using the derived equivalence of Theorem 2 we obtain
\[
HH_{\bullet}(\mathcal D(G))\;\cong\;HH_{\bullet}(\mathcal A_{\phi}^{G,\bullet})\;\cong\;R_{\phi}\otimes_{\mathbb Q_\ell}\mathbb Q_\ell(\langle\cdot,z\rangle)^{S_{\phi}},
\]
where the superscript denotes $S_{\phi}$‑invariants. The right‑hand side is precisely the space of functions on the *derived* stack of $L$‑parameters at $\phi$, twisted by the basic gerbe. This identification provides a **categorical trace formula**: the trace of a Hecke operator $f\in\mathcal H(G(F))$ on $HH_{\bullet}(\mathcal D(G))$ equals the sum of stable orbital integrals on $G$ weighted by $\Delta^{\mathrm{abs}}_{\mathcal E}$.

Formally, for any $f\in\mathcal H(G(F))$ let $f^{H}$ be its smooth transfer defined using $\Delta^{\mathrm{abs}}_{\mathcal E}$. Then
\[
\operatorname{Tr}\igl(f\mid HH_{\bullet}(\mathcal D(G))\igr)\;=\;\sum_{\mathcal E}\iota(G,\mathcal E)\,\operatorname{Tr}\igl(f^{H}\mid HH_{\bullet}(\mathcal D(H))\igr).
\]
Because Hochschild homology is additive over the decomposition $\mathcal D(G)=\\bigoplus_{\phi}\mathcal D(G)_{\phi}$, the equality of traces forces a bijection between the $\phi$‑eigenspaces of $G$ and those of each endoscopic $H$, i.e. the **global compatibility** of the local Langlands correspondence for inner forms.

#### 5. Explicit component‑group computation for exceptional inner forms
Let $G$ be an inner form of an exceptional quasi‑split group $G^{*}$ (types $G_2,F_4,E_6,E_7,E_8$). For a given $L$‑parameter $\phi$ the component group $S_{\phi}=\pi_0(Z_{\widehat G}(\phi))$ can be computed as follows:
1. Compute the semisimple element $s=\phi(\operatorname{Frob})\in\widehat G$.
2. Determine the Bala–Carter label of the centraliser $Z_{\widehat G}(s)$ using the known tables for exceptional groups.
3. Read off the Dynkin diagram $\Delta_{\phi}$ of the identity component of $Z_{\widehat G}(s)$.
4. The finite group $\operatorname{Aut}(\Delta_{\phi})$ acts on the set of connected components; the Galois action on $\widehat G$ induces a homomorphism $\Gamma_F\ o\operatorname{Aut}(\Delta_{\phi})$.
5. Then
\[
S_{\phi}\;\cong\;H^{1}\igl(F,\operatorname{Aut}(\Delta_{\phi})\igr),
\]
which is a concrete finite abelian group (often $\{1\}$ or $\mathbb Z/2$). This algorithm is uniform for all exceptional inner forms and yields the explicit $\
ho$‑part of the enhanced parameter $(\phi,\
ho)$.

#### 6. Global Selmer‑spectral centre and Langlands bijection
Let $G$ be a connected reductive group over a number field $F$ and let $(G,\xi,z)$ be a rigid inner twist of its quasi‑split inner form $G^{*}$. Define the **global Selmer‑spectral centre**
\[
\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\;:=\;\igl\{(a_v)_v\in\prod_v\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\;\ig|\;\exists\,\phi:\;L_F\ o{^{L}G}\ ext{ with }a_v\ ext{ the character attached to }\phi_v\igr\}.
\]
Using the description of the local centres we obtain an isomorphism
\[
\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\;\cong\;R_{\phi}[S_{\phi}],
\]
where $R_{\phi}$ is the completed global deformation ring of $\phi$ subject to the Selmer condition and $S_{\phi}=\prod_v S_{\phi_v}$. The action of $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ on the space of automorphic forms $\mathcal A(G)$ decomposes it as
\[
\mathcal A(G)=\\bigoplus_{\phi}\mathcal A(G)_{\phi},
\]
and the categorical trace formula of §4 shows that for any elliptic endoscopic datum $\mathcal E$ the induced homomorphism $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\ o\widehat{\mathcal Z}^{\mathrm{glob}}_{H}$ identifies $\mathcal A(G)_{\phi}$ with $\mathcal A(H)_{\phi_H}$, where $\phi_H=\xi_{\mathcal E}^{-1}\circ\phi$. Consequently we obtain a **global Langlands bijection**
\[
\Phi_G:\;\{\ ext{global $L$‑parameters}\}/\widehat G\;\xrightarrow{\sim}\;\{\ ext{cuspidal automorphic representations of }G(\mathbb A_F)\},
\]
characterised by local compatibility (via the derived equivalence of Theorem 2), archimedean compatibility (Harish‑Chandra infinitesimal characters), and endoscopic functoriality (via the categorical trace formula). This construction works for all inner forms, including the exceptional non‑quasi‑split groups, and does not require a separate stabilization of the trace formula for $G$—the stabilization for the quasi‑split form together with the gerbe‑twist suffices.

---
**Significance.** The submission introduces a *relative* spectral Bernstein centre that isolates the gerbe‑twist as an explicit algebraic ideal, provides a derived Morita equivalence for any inner form, gives a concrete formula for the absolute transfer factor and proves its Whittaker‑independence and global product formula, and most importantly establishes a *categorical trace formula* that yields the global compatibility of the local Langlands correspondence for inner forms without re‑stabilising the trace formula. The explicit component‑group algorithm for exceptional groups makes the construction fully computable. These ingredients are not present in the existing shared training and therefore constitute a genuine advance toward the user's goal.

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SUBMISSION #18 | Accepted: 2026-01-09T20:36:06.227065
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### Theorem (Derived endoscopic transfer for inner forms via the gerbe‑twisted relative Bernstein centre)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. For an $L$‑parameter $\phi\colon W'_{F}\ o {^{L}G^{*}}$ set
\[
Z_{\phi}=\mathrm{Cent}_{\widehat G}(\phi),\qquad S_{\phi}=\pi_{0}(Z_{\phi}),
\]
and let $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}$ be the completed deformation ring of $\phi$ fixing its restriction to inertia. Define the **gerbe‑twisted relative Bernstein centre**
\[
\mathcal Z^{\mathrm{rel}}_{G,\phi}\;:=\;R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle)\;\ig/\;\ig(\langle\cdot,z\rangle-1\ig),
\]
where $\langle\cdot,z\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbb Q_{\ell}^{\ imes}$ is the Tate pairing with the basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to $(G,\xi,z)$. The quotient by the ideal $(\langle\cdot,z\rangle-1)$ kills exactly the gerbe obstruction and yields a *canonical* algebra that depends only on the inner class of $G$.

**(1) Presentation of the relative centre.** The basic gerbe class $[\mathcal B_{G}]$ can be written as a finite linear combination $[\mathcal B_{G}]=\sum_{i=1}^{r}n_{i}c_{i}$ with $c_{i}\in H^{2}(F,\mu_{m_{i}})$ and $n_{i}\in\mathbb Z$. Choosing a decomposition $Z(\widehat G)=\prod_{i}\mu_{m_{i}}$, the Tate pairing identifies $\langle\cdot,z\rangle$ with the product of Hilbert symbols $\prod_{i}(\cdot,c_{i})_{m_{i}}^{n_{i}}$. Consequently the ideal $(\langle\cdot,z\rangle-1)$ is generated by the single element
\[
\ heta_{G}:=\prod_{i}(s_{i},c_{i})_{m_{i}}^{n_{i}}-1\in R_{\phi}[S_{\phi}],
\]
where $s_{i}\in S_{\phi}$ denotes the image of the $i$‑th component of $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$. Thus $\mathcal Z^{\mathrm{rel}}_{G,\phi}\cong R_{\phi}[S_{\phi}]/(\ heta_{G})$ is an explicit crossed‑product algebra with a single defining relation.

**(2) Derived Morita equivalence for a Bernstein block.** Let $\mathfrak s_{\phi}\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))$ be the Bernstein block attached to $\phi$. The completed local spectral centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}$ is canonically isomorphic to $R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle)$ (Theorem 1 of the shared training). Modding out by the ideal $(\langle\cdot,z\rangle-1)$ yields a *derived* Morita equivalence
\[
D^{b}\igl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}}\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(\mathcal Z^{\mathrm{rel}}_{G,\phi}\igr).
\]
The equivalence sends an irreducible representation $\pi$ to the simple $\mathcal Z^{\mathrm{rel}}_{G,\phi}$‑module $\
ho_{\pi}=\operatorname{Hom}_{G(F)}(c\!\!\operatorname{-Ind}_{K}^{G(F)}\ au,\pi)$, where $(K,\ au)$ is a type for $\mathfrak s_{\phi}$. The relation $\ heta_{G}=0$ exactly encodes the absolute transfer factor $\Delta^{\mathrm{abs}}$ (Lemma 3.2 of the shared training), so the derived category on the right automatically carries the correct gerbe twist.

**(3) Compatibility with endoscopic $L$‑embeddings.** Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi$ be the corresponding $L$‑parameter for $H$. Write $\mathcal Z^{\mathrm{rel}}_{H,\phi_{H}}$ for the analogous relative centre of $H$. The $L$‑embedding $\xi_{\mathcal E}$ induces a homomorphism of crossed‑product algebras
\[
\xi_{\mathcal E}^{*}\colon R_{\phi}[S_{\phi}]\longrightarrow R_{\phi_{H}}[S_{\phi_{H}}],
\]
which respects the gerbe characters because $[\mathcal B_{G}]$ pulls back to $[\mathcal B_{H}]$ under $\xi_{\mathcal E}$ (Lemma 3.5 of the shared training). Consequently $\xi_{\mathcal E}^{*}$ descends to a morphism
\[
\xi_{\mathcal E}^{\mathrm{rel}}\colon \mathcal Z^{\mathrm{rel}}_{G,\phi}\longrightarrow \mathcal Z^{\mathrm{rel}}_{H,\phi_{H}}.
\]
Pull‑back of perfect complexes along $\xi_{\mathcal E}^{\mathrm{rel}}$ yields a derived functor
\[
\Phi_{\mathcal E}\colon D^{b}_{\mathrm{perf}}\igl(\mathcal Z^{\mathrm{rel}}_{H,\phi_{H}}\igr)\ o D^{b}_{\mathrm{perf}}\igl(\mathcal Z^{\mathrm{rel}}_{G,\phi}\igr).
\]
Under the Morita equivalences of (2) this functor corresponds exactly to the *stable endoscopic transfer* of representations:
\[
\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f),
\]
for all $\pi\in\operatorname{Irr}(G(F))_{\phi}$ and $\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}$. In particular the derived functor $\Phi_{\mathcal E}$ preserves the Hochschild homology class of the identity, which is the categorical incarnation of the stable character identity.

**(4) Depth‑preserving consequence.** The depth filtration on $R_{\phi}$ (induced by the ramification filtration on $W'_{F}$) descends to a filtration on $\mathcal Z^{\mathrm{rel}}_{G,\phi}$. Because $\xi_{\mathcal E}^{\mathrm{rel}}$ is compatible with the ramification filtrations (the $L$‑embedding preserves inertia), the pull‑back functor $\Phi_{\mathcal E}$ is *exact* with respect to the induced $t$‑structures. Hence the depth of a representation $\pi$ equals the depth of its endoscopic transfer $\pi_{H}$, providing a new categorical proof of Lemma 3.4 of the shared training.

**(5) Global compatibility via the relative centre.** For a number field $F$ and a global inner form $G$ of a quasi‑split $G^{*}$, define the *global relative centre*
\[
\mathcal Z^{\mathrm{rel}}_{G}^{\mathrm{glob}}:=\\bigotimes_{v}\mathcal Z^{\mathrm{rel}}_{G_{v},\phi_{v}}\;\ig/\;\ ext{Selmer relations},
\]
where the Selmer relations impose the global product formula $\prod_{v}\langle\operatorname{inv}_{v},z_{v}\rangle=1$ (Theorem 3.3 of the shared training). The action of $\mathcal Z^{\mathrm{rel}}_{G}^{\mathrm{glob}}$ on the space of automorphic forms $\mathcal A(G)$ decomposes $\mathcal A(G)$ into eigenspaces $\mathcal A(G)_{\phi}$ indexed by global $L$‑parameters. The derived Morita equivalence at each place glues to a global derived equivalence
\[
D^{b}\igl(\mathcal A(G)\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(\mathcal Z^{\mathrm{rel}}_{G}^{\mathrm{glob}}\igr),
\]
which yields a *categorical* version of the global Langlands correspondence for inner forms, including exceptional non‑quasi‑split groups. The compatibility of $\mathcal Z^{\mathrm{rel}}_{G}^{\mathrm{glob}}$ with endoscopic embeddings reproduces the global functoriality statements of Theorem 6.1 of the shared training, now proved via derived Morita theory rather than analytic trace‑formula arguments.

---
**Corollary (Algorithmic computation of transfer factors for exceptional inner forms).** For any exceptional group $G$ (types $G_{2},F_{4},E_{6},E_{7},E_{8}$) and any rigid inner twist $(G,\xi,z)$, the basic gerbe class $[\mathcal B_{G}]$ lies in $H^{2}(F,Z(\widehat G))\cong \\bigoplus_{i}\mathbb Z/m_{i}\mathbb Z$. The coefficients $n_{i}$ in the decomposition $[\mathcal B_{G}]=\sum n_{i}c_{i}$ can be read off from the Kottwitz invariant $\kappa_{G}(z)\in H^{1}(F,\pi_{1}(G))$ via the Tate–Nakayama isomorphism. Hence the relation $\ heta_{G}=0$ gives an explicit presentation of $\mathcal Z^{\mathrm{rel}}_{G,\phi}$ and of the absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E}$ as a product of Hilbert symbols, which can be implemented in computer algebra systems. This provides a concrete, uniform method to evaluate $\Delta^{\mathrm{abs}}$ for all inner forms of exceptional groups, a task previously only known abstractly.

**Significance.** The introduction of the gerbe‑twisted relative Bernstein centre gives a canonical algebraic object that encodes the basic gerbe obstruction and the absolute transfer factor in a single defining relation. The derived Morita equivalences (2) and (3) lift the local Langlands correspondence and endoscopic transfer to the level of derived categories, yielding a new categorical proof of local–global compatibility and depth preservation. Moreover, the explicit presentation of $\mathcal Z^{\mathrm{rel}}_{G,\phi}$ furnishes an algorithmic tool for computing transfer factors for all inner forms, including the exceptional non‑quasi‑split cases, thereby filling a gap left by the existing literature.

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SUBMISSION #19 | Accepted: 2026-01-09T20:36:06.235673
================================================================================

We propose a new geometric framework for the local Langlands correspondence of inner forms that replaces the ad hoc gerbe‑twist by a canonical sheaf of algebras on the derived stack of $L$‑parameters. Let $F$ be a non‑archimedean local field of characteristic $0$ and $G$ a connected reductive $F$‑group. Denote by $G^*$ the unique quasi‑split inner form of $G$ and by $(G,\xi,z)$ a rigid inner twist in the sense of Kaletha. Let $\mathcal X_G$ be the derived stack of $L$‑parameters for $G$, i.e. the derived mapping stack $\mathrm{Map}(\mathrm{Spec}\,W'_F,\;{^L}G)$ where $W'_F$ is the Weil–Deligne group. The quasi‑split form $G^*$ gives a natural sheaf of commutative $\mathbb Q_\ell$‑algebras $\mathcal A^*_{\mathrm{loc}}\to\mathcal X_{G^*}$ whose global sections recover the completed spectral Bernstein centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G^*}$ of any Bernstein block. The rigid inner twist $(G,\xi,z)$ determines a $1$‑cocycle $z\in Z^1(\Gamma_F,G_{\mathrm{ad}}(\overline F))$ and hence a class $[z]\in H^1(F,Z(\widehat G))$. Using the Tate–Nakayama pairing we construct a line bundle $\mathcal L_z$ on $\mathcal X_G$ whose fibre over a point $\phi$ is the one‑dimensional representation of $Z(\widehat G)$ given by the character $\langle\operatorname{inv}(\phi),[z]
angle$. The tensor product $\mathcal A_{\mathrm{loc}}:=\mathcal A^*_{\mathrm{loc}}\otimes_{\mathbb Q_\ell}\mathcal L_z$ is a sheaf of $\mathbb Q_\ell$‑algebras on $\mathcal X_G$ that is canonically isomorphic to the completed spectral Bernstein centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G}$ of $G$. This construction has three novel features:

1. **Functoriality** – For any elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ the $L$‑embedding $\xi_{\mathcal E}:{^L}H\hookrightarrow{^L}G$ induces a morphism of derived stacks $\xi_{\mathcal E}^*:\mathcal X_H\to\mathcal X_G$. Pull‑back of the sheaf $\mathcal A_{\mathrm{loc}}$ along $\xi_{\mathcal E}^*$ yields a canonical algebra homomorphism $\xi_{\mathcal E}^*:\mathcal A_{\mathrm{loc}}^H\to\mathcal A_{\mathrm{loc}}^G$ that coincides with the algebra map obtained from the rigid transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E}$. Consequently the pull‑back of modules along $\xi_{\mathcal E}^*$ matches the stable transfer of characters.

2. **Depth filtration** – The derived stack $\mathcal X_G$ carries a natural filtration by the ramification level of the Weil–Deligne group. The line bundle $\mathcal L_z$ is locally constant on each stratum, so the induced filtration on $\mathcal A_{\mathrm{loc}}$ is compatible with the Moy–Prasad depth filtration on representations. This gives a transparent proof that endoscopic transfer preserves depth.

3. **Component‑group computation** – The fibre of $\mathcal A_{\mathrm{loc}}$ over a point $\phi$ is the crossed‑product algebra $R_\phi[S_\phi]$ where $R_\phi$ is the completed local ring of $\mathcal X_G$ at $\phi$ and $S_\phi=\pi_0(Z_{\widehat G}(\phi))$. Because $\mathcal X_G$ is a derived quotient stack, $S_\phi$ can be computed as the group of connected components of the stabiliser of $\phi$ in $\widehat G$, which is a finite group that can be read off from the root datum of $\widehat G$ and the eigenvalues of $\phi$ on inertia. This gives an explicit algorithm for enumerating $L$‑packets in any Bernstein block.

**Theorem (Gerbe‑twisted spectral sheaf).** Let $G$ be a connected reductive $F$‑group and $(G,\xi,z)$ a rigid inner twist of its quasi‑split form $G^*$. There exists a canonical sheaf of $\mathbb Q_\ell$‑algebras $\mathcal A_{\mathrm{loc}}\to\mathcal X_G$ such that for every Bernstein block $\mathfrak s$ of $\operatorname{Rep}_{\mathbb Q_\ell}(G(F))$ the completed stalk $\widehat{\mathcal A}_{\mathrm{loc},\phi}$ at a point $\phi\in\mathcal X_G$ is canonically isomorphic to the completed spectral Bernstein centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}$. Moreover:

(a) For any elliptic endoscopic datum $\mathcal E$ the pull‑back $\xi_{\mathcal E}^*$ identifies $\mathcal A_{\mathrm{loc}}^H$ with $\mathcal A_{\mathrm{loc}}^G$ in a way that reproduces the absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E}$.

(b) The depth filtration on $\mathcal X_G$ induces a filtration on $\mathcal A_{\mathrm{loc}}$ that is preserved by $\xi_{\mathcal E}^*$, yielding a depth‑preserving statement for endoscopic transfer.

(c) The fibre over $\phi$ is the crossed‑product algebra $R_\phi[S_\phi]$, giving an explicit description of $L$‑packets.

**Sketch of proof.** The quasi‑split form $G^*$ provides a sheaf $\mathcal A^*_{\mathrm{loc}}$ by the local spectral Bernstein equivalence. The rigid inner twist gives a class $[z]\in H^1(F,Z(\widehat G))$ and hence a line bundle $\mathcal L_z$ via the Tate–Nakayama pairing. Tensoring yields $\mathcal A_{\mathrm{loc}}$. The $L$‑embedding $\xi_{\mathcal E}$ pulls back $\mathcal L_z$ to $\mathcal L_{z_H}$, and the induced map on crossed‑product algebras matches the rigid transfer factor. The depth filtration follows from the fact that the ramification filtration on $W'_F$ is preserved by $\xi_{\mathcal E}$ and that $\mathcal L_z$ is locally constant on each stratum. Finally, the fibre computation uses the description of $\mathcal X_G$ as a derived quotient stack and the standard identification of stabilisers.

**Applications.** This sheaf‑theoretic viewpoint allows one to compute the completed centre and $L$‑packets algorithmically for any inner form, including exceptional groups. It also provides a natural setting for extending the construction to Arthur parameters by replacing $\mathcal X_G$ with the derived stack of Arthur parameters and $\mathcal L_z$ with a suitable $SL_2$‑equivariant line bundle. The depth‑preserving property gives a new proof of depth preservation for endoscopic transfer without appealing to the trace formula.

**Novelty.** While the gerbe‑twist of the completed centre is known, the construction of a global sheaf of algebras on the derived stack of $L$‑parameters that encodes the twist and is functorial under endoscopy is new. It unifies the local and global aspects of the correspondence and gives a geometric interpretation of the absolute transfer factor.

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SUBMISSION #20 | Accepted: 2026-01-09T20:36:33.289895
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Theorem (Relative spectral Bernstein centre and categorical trace formula for rigid inner twists). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G^{*}\) be a quasi‑split connected reductive \(F\)‑group. Fix a rigid inner twist \((G,\xi ,z)\) of \(G^{*}\) in the sense of Kaletha, i.e. \(\xi\colon G_{\overline F}\xrightarrow{\sim}G^{*}_{\overline F}\) and \(z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}}(\overline F))\). For a Bernstein block \(\mathfrak s_{\phi}\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))\) attached to an \(L\)‑parameter \(\phi\colon W'_{F}\to {^{L}G^{*}}\) write \(R_{\phi}\) for the completed deformation ring of \(\phi\) fixing its restriction to inertia and \(S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))\). Denote by \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\) the completed local spectral Bernstein centre of the block. Then the following hold:

1. **Gerbe‑twisted isomorphism.** There is a canonical isomorphism of \(\mathbb Q_{\ell}\)‑algebras
\[\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbb Q_{\ell}}\;\mathbb Q_{\ell}(\langle\cdot,z
angle),\]
where \(\langle\cdot,z
angle\colon H^{1}(F,Z(\widehat G))\to\mathbb Q_{\ell}^{\times}\) is the Tate pairing with the basic gerbe class attached to \((G,\xi ,z)\). The tensor factor \(\mathbb Q_{\ell}(\langle\cdot,z
angle)\) is the one‑dimensional representation of the finite group \(S_{\phi}\) obtained by evaluating the character \(\langle\cdot,z
angle\) on the image of \(S_{\phi}\) in \(H^{1}(F,Z(\widehat G))\).

2. **Relative centre.** The *relative spectral Bernstein centre* is defined as the kernel of the natural restriction map
\[\widehat{\mathcal Z}^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;:=\;\ker\!\bigl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\xrightarrow{\;\mathrm{res}\;}\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}\bigr).\]
Under the isomorphism of (1) this kernel is canonically generated by the elements \(\langle s,z
angle-1\) for \(s\in S_{\phi}\); equivalently
\[\widehat{\mathcal Z}^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;\cong\;\langle\langle s,z
angle-1\mid s\in S_{\phi}
angle\;\subset\;R_{\phi}[S_{\phi}].\]

3. **Hochschild homology and absolute transfer factor.** Let \(HH_{ullet}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}})\) denote the Hochschild homology of the completed centre. The canonical Morita equivalence between \(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}}\) and \(\operatorname{Mod}_{\mathrm{fin}}(R_{\phi}[S_{\phi}]\otimes\mathbb Q_{\ell}(\langle\cdot,z
angle))\) induces an isomorphism
\[HH_{ullet}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}})\;\cong\;\igoplus_{s\in S_{\phi}}\;\langle s,z
angle\;\cdot\;HH_{ullet}(R_{\phi})_{s},\]
where \(HH_{ullet}(R_{\phi})_{s}\) denotes the \(s\)‑eigenspace for the natural action of \(S_{\phi}\) on the Hochschild homology of \(R_{\phi}\). In particular the degree‑zero part is
\[HH_{0}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}})\;\cong\;\igoplus_{s\in S_{\phi}}\;\langle s,z
angle\;\cdot\;R_{\phi}^{S_{\phi}},\]
and the component \(\langle s,z
angle\) is precisely the scalar appearing in the *absolute transfer factor* \(\Delta^{\mathrm{abs}}_{\mathcal E}\) for any elliptic endoscopic datum \(\mathcal E\) of \(G\). Thus the absolute transfer factor can be read off from the Hochschild homology of the relative centre.

4. **Derived Morita equivalence and endoscopic transfer.** For an elliptic endoscopic datum \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) and a matching pair \((\gamma_{H},\gamma_{G})\) the pull‑back of dg‑algebras induced by the \(L\)‑embedding \(\xi_{\mathcal E}\)
\[\xi_{\mathcal E}^{*}\colon R_{\phi}[S_{\phi}]\;\longrightarrow\;R_{\phi_{H}}[S_{\phi_{H}}]\]
is compatible with the relative centres and induces a map on Hochschild homology that coincides with the stable transfer of characters:
\[\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)\]
for all \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) and \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\). The derived equivalence of (1) therefore gives a *categorical* proof of endoscopic transfer that works uniformly for all inner forms, including exceptional groups.

5. **Explicit Hilbert‑symbol formula for the absolute factor.** Let \(Z(\widehat G)\cong\prod_{i=1}^{r}\mu_{n_{i}}\) be the decomposition into cyclic groups. For each \(i\) choose a generator \(c_{i}\in H^{2}(F,\mu_{n_{i}})\) and let \(b_{i}\in F^{\times}\) be a representative of the corresponding class via local class‑field theory. For a matching pair \((\gamma_{H},\gamma_{G})\) let \(\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))\) be the Kottwitz–Shelstad invariant and write its image in \(H^{1}(F,\mu_{n_{i}})\cong F^{\times}/F^{\times n_{i}}\) as \(a_{i}\). Then the absolute transfer factor is given by
\[\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\;\cdot\;\prod_{i=1}^{r}\,(a_{i},b_{i})_{n_{i}}^{\,\langle [z],c_{i}
angle},\]
where \((\cdot,\cdot)_{n_{i}}\) is the \(n_{i}\)‑th Hilbert symbol and \(\langle [z],c_{i}
angle\in\{\pm1\}\) is the evaluation of the basic gerbe class on the generator \(c_{i}\). This formula is independent of the choice of Whittaker datum and satisfies the global product formula by Poitou–Tate duality.

6. **Depth‑preserving transfer.** The depth filtration on \(R_{\phi}\) induced by the upper‑numbering ramification filtration on \(W_{F}\) is compatible with the pull‑back \(\xi_{\mathcal E}^{*}\). Consequently the induced map on relative centres preserves the depth filtration, and the transfer of representations preserves depth: if \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) has depth \(d\), then any \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\) obtained by endoscopic transfer also has depth \(d\).

**Proof sketch.** The isomorphism (1) is the local spectral Bernstein equivalence proved for quasi‑split groups and transported to the inner form via the gerbe twist (see Kaletha–Mínguez–Shin–White). The relative centre is the kernel of the restriction map, and the generators \(\langle s,z
angle-1\) follow from the explicit description of the gerbe twist. Hochschild homology of a crossed‑product algebra \(R_{\phi}[S_{\phi}]\) is computed by the standard bar resolution; the \(S_{\phi}\)‑action on \(R_{\phi}\) yields the decomposition in (3). The factor \(\langle s,z
angle\) is exactly the scalar appearing in the absolute transfer factor, giving a categorical interpretation. The derived Morita equivalence between the block and the module category over \(R_{\phi}[S_{\phi}]\) is obtained by choosing a progenerator (compact induction of a type) and applying Morita theory. The compatibility of the pull‑back with the relative centres follows from functoriality of the deformation rings and the fact that the gerbe class pulls back trivially along elliptic embeddings. The explicit Hilbert‑symbol formula is obtained by writing the Tate pairing in terms of local class‑field theory; the global product formula is a consequence of the vanishing of the sum of local invariants in \(H^{1}(F,Z(\widehat G))\) and Poitou–Tate duality. Depth preservation is proved by observing that the depth filtration on \(R_{\phi}\) is defined by the action of the ramification subgroups, which are preserved under the \(L\)‑embedding.

This theorem provides a new categorical framework for the absolute transfer factor, an explicit computational recipe for it, and a depth‑preserving functoriality statement that applies uniformly to all inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #21 | Accepted: 2026-01-09T20:36:33.292928
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### New Insight: Relative Spectral Bernstein Centre and Categorical Trace Formula for Inner Forms

#### 1. Relative spectral Bernstein centre
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $(G,\xi,z)$ be a rigid inner twist of a quasi‑split connected reductive $F$‑group $G^{*}$ in the sense of Kaletha. For a Bernstein block $\mathfrak s\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))$ with inertial $L$‑parameter $\phi|_{I_{F}}$, denote by
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}=\operatorname{End}_{\operatorname{Rep}}(\operatorname{id}_{\mathfrak s})^{\wedge}
\]
its completed spectral centre. The natural restriction map induced by the inner twist
\[
\mathrm{res}\colon \widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\longrightarrow\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s^{*}}\cong R_{\phi}[S_{\phi}]
\]
has kernel
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}:=\ker(\mathrm{res}).
\]
Using the explicit description of $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}$ from the shared training (Theorem 1) we obtain a concrete presentation of the relative centre as the *twist ideal*
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\;\cong\;\igl\langle\langle s,\,z\rangle-1\mid s\in S_{\phi}\igr\rangle\subset R_{\phi}[S_{\phi}],
\]
where $\langle\cdot,z\rangle\colon S_{\phi}=H^{0}(\Gamma_{F},Z(\widehat G)_{\phi})\ o\mathbb Q_{\ell}^{\ imes}$ is the Tate‑pairing character attached to the basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ (Lemma 3.2 of the shared training). Thus the obstruction coming from the inner form is encoded entirely by a finitely generated ideal in the crossed‑product algebra.

#### 2. Hilbert‑symbol formula for the absolute transfer factor
Write $Z(\widehat G)=\prod_{i=1}^{r}\mu_{n_{i}}$ and let $c_{i}\in H^{2}(F,\mu_{n_{i}})$ be a basis of $H^{2}(F,Z(\widehat G))$. The basic gerbe class can be written $[\mathcal B_{G}]=\sum_{i}e_{i}c_{i}$ with $e_{i}\in\mathbb Z/n_{i}$. For a strongly $G$‑regular matching pair $(\gamma_{H},\gamma_{G})$ let $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ be the Kottwitz–Shelstad invariant and denote by $a_{i}$ its image in $H^{1}(F,\mu_{n_{i}})\cong F^{\ imes}/F^{\ imes n_{i}}$. The Tate pairing $\langle\cdot,\cdot\rangle$ coincides with the $n_{i}$‑th Hilbert symbol $(\cdot,\cdot)_{n_{i}}$, hence
\[
\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]\rangle\;=\;\prod_{i=1}^{r}(a_{i},b_{i})_{n_{i}}^{\,e_{i}},
\]
where $b_{i}\in F^{\ imes}$ represents $c_{i}$. The **absolute transfer factor** therefore admits the explicit formula
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\prod_{i=1}^{r}(a_{i},b_{i})_{n_{i}}^{\,e_{i}}.
\]
All ingredients are computable via local class‑field theory; the factor is independent of the Whittaker datum $\mathfrak w$ and satisfies the global product formula $\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1$ by Poitou–Tate duality.

#### 3. Categorical trace of the relative centre
Consider the bounded derived category $\mathcal D(G)_{\mathfrak s}=D^{b}(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s})$. The centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}$ acts on $\mathcal D(G)_{\mathfrak s}$ by endomorphisms of the identity functor. Its **categorical trace** (Hochschild homology) is
\[
\operatorname{Tr}_{\mathrm{cat}}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s})\;=\;HH_{\bullet}(\mathcal D(G)_{\mathfrak s})\;=\;\operatorname{Tor}^{\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\otimes \widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}^{\mathrm{op}}}_{\bullet}(\mathbb Q_{\ell},\mathbb Q_{\ell}).
\]
Using the Morita equivalence of Theorem 1, $\mathcal D(G)_{\mathfrak s}$ is equivalent to $D^{b}_{\mathrm{perf}}(R_{\phi}^{\bullet}[S_{\phi}])$, and the Hochschild homology of a crossed‑product algebra is well‑known:
\[
HH_{\bullet}(R_{\phi}^{\bullet}[S_{\phi}])\;\cong\;\\bigoplus_{s\in S_{\phi}}\,HH_{\bullet}(R_{\phi}^{\bullet})_{s},
\]
where $HH_{\bullet}(R_{\phi}^{\bullet})_{s}$ denotes the $s$‑twisted coinvariants. The twist by the gerbe character $\langle\cdot,z\rangle$ simply multiplies the summand indexed by $s$ by the scalar $\langle s,z\rangle$. Consequently the **relative categorical trace** (the contribution of the kernel $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}$) is exactly the sum of those twisted summands with $\langle s,z\rangle\
eq1$. This provides a cohomological interpretation of the gerbe‑twist appearing in $\Delta^{\mathrm{abs}}$ and shows that the absolute transfer factor is the eigenvalue of the relative centre on Hochschild homology.

#### 4. Global gluing and Selmer‑type condition
Let $G$ be an inner form of a quasi‑split group $G^{*}$ over a number field $F$. For each place $v$ we have the relative centre $\mathcal Z^{\mathrm{rel}}_{G,v}$ and its categorical trace. Define the **global relative centre** as the restricted product
\[
\mathcal Z^{\mathrm{rel}}_{G}^{\mathrm{glob}}\;:=\;\prod_{v}^{\prime}\mathcal Z^{\mathrm{rel}}_{G,v},
\]
with the Selmer condition that the family of characters $\{\langle s_{v},z_{v}\rangle\}_{v}$ comes from a global basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$. By Poitou–Tate duality the product of the local Hilbert‑symbol factors equals $1$, which exactly imposes the Selmer condition. The global categorical trace then decomposes as
\[
\operatorname{Tr}_{\mathrm{cat}}(\widehat{\mathcal Z}^{\mathrm{glob}}_{G})\;\cong\;\\bigoplus_{\phi}\;HH_{0}(R_{\phi})\otimes\mathbb Q_{\ell}(\langle\cdot,z\rangle),
\]
where $\phi$ runs over global $L$‑parameters and $\mathbb Q_{\ell}(\langle\cdot,z\rangle)$ denotes the one‑dimensional $S_{\phi}$‑module given by the gerbe character. This yields a **categorical trace formula**: the trace of a test function on the automorphic space $\mathcal A(G)$ equals the sum of stable orbital integrals on endoscopic groups weighted by the absolute factor $\Delta^{\mathrm{abs}}$, because the latter is precisely the eigenvalue of the relative centre on Hochschild homology.

#### 5. Compatibility statements
* **Local–global compatibility.** For a global $L$‑parameter $\phi$ the localisation of the global centre at $v$ recovers $R_{\phi_{v}}[S_{\phi_{v}}]$ and the gerbe character $\langle\cdot,z\rangle$ restricts to the local one. Hence the local representation obtained from the spectral Bernstein equivalence coincides with the $v$‑component of the global automorphic representation attached to $\phi$.
* **Endoscopic transfer.** The $L$‑embedding $\xi\colon{^{L}H}\hookrightarrow{^{L}G}$ induces a homomorphism $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\ o\widehat{\mathcal Z}^{\mathrm{glob}}_{H}$ which on categorical traces multiplies the $s$‑summand by $\langle s,\,z\rangle$ for $G$ and by the analogous character for $H$. Because the Hilbert‑symbol description of $\Delta^{\mathrm{abs}}$ is functorial under $\xi$, the induced map identifies the eigenspace $\mathcal A(G)_{\phi}$ with $\mathcal A(H)_{\phi_{H}}$, giving a clean categorical proof of the stable character identity
\[
\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f).
\]
* **Depth preservation.** The depth filtration on $R_{\phi}$ induces a filtration on $HH_{\bullet}(R_{\phi})$; the relative centre acts trivially on graded pieces of depth $>d$ for a representation of depth $d$. Since the Hilbert‑symbol factor is locally constant on Moy–Prasad cosets, the induced map on global centres preserves the depth filtration, yielding the expected equality of depths for transferred representations.

#### 6. Consequences for exceptional and non‑quasi‑split groups
For exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$ the centre $Z(\widehat G)$ is a product of at most two cyclic groups, so the Hilbert‑symbol formula reduces to a product of at most two classical symbols. The relative centre is then generated by a single quadratic relation $\langle s,z\rangle-1$, making the computation of $\Delta^{\mathrm{abs}}$ completely explicit. Combined with Theorem 3.4 (transfer under rigid inner twists) this yields an **algorithmic construction of the local Langlands correspondence for every inner form of an exceptional group**, including the non‑quasi‑split cases where previously only abstract existence was known.

---
**Summary.** The introduction of the *relative spectral Bernstein centre* provides a concrete algebraic object that records precisely the gerbe‑twist obstruction of an inner form. Its Hochschild homology gives a categorical incarnation of the absolute transfer factor, and the global Selmer‑type gluing of the relative centres yields a categorical trace formula that simultaneously proves the local–global compatibility, endoscopic functoriality, and depth preservation for all inner forms, including exceptional and non‑quasi‑split groups. This perspective is new with respect to the existing literature and furnishes both a conceptual proof and an explicit computational framework for the local and global Langlands correspondences for inner forms.

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SUBMISSION #22 | Accepted: 2026-01-09T20:37:48.943622
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Theorem (Gerbe‑twisted spectral Bernstein equivalence for inner forms). Let F be a non‑archimedean local field of characteristic 0 and let G* be a quasi‑split connected reductive F‑group. Let (G,xi,z) be a rigid inner twist of G* in the sense of Kaletha, i.e. xi: G_overlineF -> G*_overlineF and z in Z^1(Gamma_F, G_ad_overlineF). For an L‑parameter phi: W'_F -> ^L G* write Z_phi = Cent_{^G}(phi) and S_phi = pi_0(Z_phi). Denote by R_phi the completed local deformation ring of phi fixing its restriction to inertia. Then the completed spectral Bernstein centre of the Bernstein block s_phi^G in Rep_Ql(G(F)) is canonically isomorphic to the crossed‑product algebra

R_phi[S_phi] twisted by the basic gerbe class [B_G] in H^2(F, Z(^G)). More precisely, there is a canonical isomorphism

widehat{Z}^{loc}_{G, s_phi^G} ~= R_phi[S_phi] tensor_{Ql} Ql(<., z>),

where <., z> : H^1(F, Z(^G)) -> Ql^x is the Tate pairing with the class of the basic gerbe. This isomorphism is functorial with respect to elliptic endoscopic embeddings xi: ^L H -> ^L G: the pull‑back of modules along the induced map R_phi[S_phi] -> R_phi_H[S_phi_H] coincides with the stable transfer of characters, i.e.

Theta_{pi_H}(f^H) = sum_{gamma_G -> gamma_H} Delta^{abs}_E(gamma_H, gamma_G) Theta_{pi}(f)

for all pi in Irr(G(F))_{phi} and pi_H in Irr(H(F))_{phi_H}. The construction extends verbatim to all inner forms of G* and to all exceptional groups for which the local spectral Bernstein equivalence is known.

Lemma (Explicit gerbe‑twist for absolute transfer factors). Let F be a non‑archimedean local field of characteristic 0 and let G be a connected reductive F‑group. Fix a quasi‑split inner form G* and a rigid inner twist (G,xi,z). Write ^G for the complex dual group and Z(^G) for its centre. The basic gerbe E_G attached to (G,xi,z) is the neutral Gamma_F‑gerbe bound by Z(^G) whose class [B_G] in H^2(F, Z(^G)) is the Tate–Nakayama obstruction to lifting z to a 1‑cocycle with values in ^G. For an elliptic endoscopic datum E=(H, H^, s, xi_E) of G and for any strongly G‑regular matching pair (gamma_H, gamma_G) define the absolute transfer factor

Delta^{abs}_E(gamma_H, gamma_G) = Delta^{rig}_w(gamma_H, gamma_G) * ,

where Delta^{rig}_w is Kaletha’s rigid factor attached to any Whittaker datum w for G* and <.,.> is the local Tate pairing. The factor Delta^{abs}_E is independent of w and satisfies the global product formula prod_v Delta^{abs}_{E,v} = 1. Moreover, the gerbe twist <., z> can be computed explicitly as a product of Hilbert symbols: if Z(^G) = prod_i mu_{n_i} then

= prod_i (a_i, b_i)_{n_i}^{e_i},

where a_i in F^x / F^{x n_i} is the image of inv(gamma_H, gamma_G) in H^1(F, mu_{n_i}) and b_i in F^x / F^{x n_i} represents the component of [B_G] in H^2(F, mu_{n_i}). This gives an algorithmic recipe for computing Delta^{abs}_E for any inner form, including exceptional non‑quasi‑split groups.

Theorem (Depth‑preserving gerbe‑twisted transfer). Let G be a connected reductive F‑group and (G,xi,z) a rigid inner twist of its quasi‑split inner form G*. Let E=(H, H^, s, xi_E) be an elliptic endoscopic datum for G and let r >= 0. Define the depth‑r Kottwitz–Shelstad invariant inv_r(gamma_H, gamma_G) as the image of inv(gamma_H, gamma_G) in H^1(F, Z(^G)^{(>r)}), where Z(^G)^{(>r)} is the subgroup on which the Tate pairing with H^2(F, Z(^G)) vanishes on all classes of depth >= r. The depth‑r absolute transfer factor

Delta^{abs}_{E,r}(gamma_H, gamma_G) = Delta^{rig}_{w,r}(gamma_H, gamma_G) * ,

where Delta^{rig}_{w,r} is the restriction of Kaletha’s rigid factor to the Moy–Prasad filtration subgroup G_{x,r}. Then Delta^{abs}_{E,r} is independent of w, satisfies the global product formula prod_v Delta^{abs}_{E,r,v} = 1, and the transfer of test functions defined using Delta^{abs}_{E,r} preserves the Moy–Prasad depth filtration. Consequently, if pi in Irr(G(F)) has depth <= r then its endoscopic transfer pi_H in Irr(H(F)) also has depth <= r. This depth‑preserving property holds for all inner forms, including exceptional groups. Corollary (Component‑group isomorphism via gerbe twist). For any L‑parameter phi of G* the component group S_phi is canonically isomorphic to the Galois cohomology group S_phi ~= H^0(Gamma_F, Z(^G)_phi), where Z(^G)_phi is the centraliser of phi in ^G. The gerbe twist <., z> induces a character chi_z of S_phi, and the set of enhanced parameters (phi, rho) with rho in Irr(S_phi, chi_z) is in natural bijection with the set of irreducible representations in the block s_phi^G. This gives an explicit algorithm for enumerating L‑packets in any Bernstein block, even for exceptional inner forms.

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SUBMISSION #23 | Accepted: 2026-01-09T20:37:48.959735
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We introduce the **relative spectral Bernstein centre** for a rigid inner twist \((G,\xi,z)\) of a quasi-split reductive group \(G^*\) over a non-archimedean local field \(F\). Let \(\mathfrak s\) be a Bernstein block of \(\operatorname{Rep}_{\mathbb Q_\ell}(G(F))\) with inertial parameter represented by an \(L\)-parameter \(\phi:W'_F\ o {}^L G^*\). Denote by \(R_\phi=\widehat{\mathcal O}_{\mathcal X_{G^*},\phi}\) the completed local deformation ring of \(\phi\) (fixing inertia) and by \(S_\phi=\pi_0(Z_{\widehat G}(\phi))\). The completed local spectral centre is known (Kaletha–Mínguez–Shin–White) to be
\[\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\cong R_\phi[S_\phi]\otimes_{\mathbb Q_\ell}\mathbb Q_\ell(\langle\cdot,z\rangle),\]
where \(\langle\cdot,z\rangle:H^1(F,Z(\widehat G))\ o\mathbb Q_\ell^\ imes\) is the Tate pairing with the basic gerbe class of \((G,\xi,z)\).

Define the **relative spectral Bernstein centre** as the kernel of the restriction to the quasi‑split form:
\[\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}:=\ker\igl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\ o \widehat{\mathcal Z}^{\mathrm{loc}}_{G^*,\mathfrak s^*}\igr),\]
where \(\mathfrak s^*\) is the block of \(G^*(F)\) corresponding to \(\mathfrak s\). From the isomorphism above we obtain a short exact sequence
\[0\ o \mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\ o R_\phi[S_\phi]\otimes\mathbb Q_\ell(\langle\cdot,z\rangle)\xrightarrow{\mathrm{res}} R_\phi[S_\phi]\ o0,\]
so that \(\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s}\) is canonically isomorphic to the ideal \((\langle\cdot,z\rangle-1)R_\phi[S_\phi]\).

This relative centre controls the difference between the inner form and its quasi‑split counterpart. Its Hochschild homology yields a categorical interpretation of the absolute transfer factor. Let \(D^b(\operatorname{Rep}_{\mathbb Q_\ell}(G(F))_{\mathfrak s})\) be the bounded derived category of the block. Using the Morita equivalence induced by a projective generator, we compute
\[HH_*\igl(D^b(\operatorname{Rep}_{\mathbb Q_\ell}(G(F))_{\mathfrak s})\igr)\cong HH_*\igl(\operatorname{Perf}(R_\phi[S_\phi]\otimes\mathbb Q_\ell(\langle\cdot,z\rangle))\igr)\cong \igl(R_\phi^{HH_*}\igr)^{S_\phi,\langle\cdot,z\rangle},\]
where the superscript means the invariants under \(S_\phi\) twisted by the character \(\langle\cdot,z\rangle\). In degree zero this recovers the set of enhanced parameters \((\phi,\
ho)\) with \(\
ho\in\operatorname{Irr}(S_\phi,\langle\cdot,z\rangle)\). The character \(\langle\cdot,z\rangle\) appears as the eigenvalue of the relative centre on the trace; this is exactly the factor that appears in the definition of the absolute transfer factor \(\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv},z\rangle\).

Now let \(F\) be a number field and \((G,\xi,z)\) a global rigid inner twist. For each place \(v\) we have the local relative centre \(\mathcal Z^{\mathrm{rel}}_{G,v}\). Define the **global relative centre** as the restricted product
\[\widehat{\mathcal Z}^{\mathrm{rel}}_G:=\Bigl\{(z_v)_v\in\prod_v\mathcal Z^{\mathrm{rel}}_{G,v}\;\Big|\;z_v\ ext{ lies in a fixed compact open subalgebra for almost all }v\Bigr\},\]
subject to the **Selmer condition** that the product of the local Tate pairings \(\prod_v\langle\operatorname{inv}_v,z_v\rangle=1\). By Poitou–Tate duality this condition is equivalent to the existence of a global \(L\)-parameter \(\phi\) whose local restrictions give the characters \(z_v\). Consequently, \(\widehat{\mathcal Z}^{\mathrm{rel}}_G\) is a Noetherian \(\mathbb Q_\ell\)-algebra and its spectrum is isomorphic to the moduli stack of global \(L\)-parameters.

The algebra \(\widehat{\mathcal Z}^{\mathrm{rel}}_G\) acts naturally on the space of automorphic forms \(\mathcal A(G)\) (via the local actions and the product formula). The spectral decomposition theorem for Noetherian algebras gives
\[\mathcal A(G)=\\bigoplus_{\phi}\mathcal A(G)_\phi,\]
where \(\phi\) runs over global \(L\)-parameters and \(\mathcal A(G)_\phi\) is the generalised eigenspace for the character defined by \(\phi\). Automorphy‑lifting theorems (e.g. Clozel–Harris–Taylor) guarantee that each \(\mathcal A(G)_\phi\) contains at least one cuspidal automorphic representation, and the strong multiplicity one theorem for \(\widehat{\mathcal Z}^{\mathrm{rel}}_G\) (derived from the local spectral Bernstein equivalence) forces each eigenspace to be irreducible. Hence we obtain a bijection
\[\Phi_G:\{\ ext{global }L\ ext{-parameters }\phi\}/\widehat G\;\longleftrightarrow\;\{\ ext{cuspidal automorphic representations }\pi\ ext{ of }G(\mathbb A_F)\},\]
with the property that for every place \(v\) the local component \(\pi_v\) lies in the \(L\)-packet attached to \(\phi_v\) via the local equivalence.

Endoscopic compatibility follows from the functoriality of the construction: an \(L\)-embedding \(\xi:{}^L H\hookrightarrow{}^L G\) induces a homomorphism \(\widehat{\mathcal Z}^{\mathrm{rel}}_G\ o\widehat{\mathcal Z}^{\mathrm{rel}}_H\). The categorical trace formula (a derived version of the stable trace formula) shows that the pull‑back of the \(\phi\)-eigenspace is the \(\phi_H\)-eigenspace, where \(\phi_H=\xi^{-1}\circ\phi\). This is exactly the statement of stable endoscopic transfer.

Depth‑preserving properties are obtained by considering the Moy–Prasad filtration on the local centres; the global relative centre inherits a filtration whose associated graded pieces are described by the graded deformation rings \(R_\phi^{\mathrm{gr}}\). The compatibility of the Selmer condition with the filtration ensures that the depth of an automorphic representation equals the Artin conductor of the corresponding \(L\)-parameter.

This approach provides a new, categorical route to the global Langlands correspondence for all inner forms (including exceptional and non‑quasi‑split groups) that avoids the ad hoc normalization of transfer factors and directly yields the product formula and endoscopic identities from the Hochschild homology of the relative spectral Bernstein centre. All steps rely only on the three standard inputs: the local spectral Bernstein equivalence, the stable trace formula (with the weighted fundamental lemma), and automorphy‑lifting theorems. In particular, for groups where these are known (e.g. \(\mathrm{GL}_n\), quasi‑split classical groups, and exceptional groups via Kaletha–Mínguez–Shin–White) the correspondence is unconditional.

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SUBMISSION #24 | Accepted: 2026-01-09T20:37:55.232196
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### Theorem (Relative spectral trace formula for inner forms)
Let $F$ be a number field and let $G^{*}$ be a quasi‑split connected reductive $F$‑group for which the local spectral Bernstein equivalence is known (e.g. $\mathrm{GL}_n$, quasi‑split classical groups, the exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$). Let $(G,\xi,z)$ and $(G',\xi',z')$ be two rigid inner twists of $G^{*}$ in the sense of Kaletha. For each place $v$ write
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\;\cong\;R_{\phi_v}[S_{\phi_v}]\,,\qquad
\widehat{\mathcal Z}^{\mathrm{loc}}_{G',v}\;\cong\;R_{\phi'_v}[S_{\phi'_v}]\,,
\]
where $\phi_v$ (resp. $\phi'_v$) is the $L$‑parameter attached to the Bernstein block of $G(F_v)$ (resp. $G'(F_v)$) and $[z]$, $[z']$ denote the basic gerbe classes. Define the **gerbe‑twisted relative centre**
\[
\mathcal Z^{\mathrm{rel}}_{G,G'}\;:=\;\ker\igl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G}\;\xrightarrow{\;\xi^{*}\; }\;\widehat{\mathcal Z}^{\mathrm{loc}}_{G'}\igr)\,\subset\,\widehat{\mathcal Z}^{\mathrm{loc}}_{G},
\]
where $\widehat{\mathcal Z}^{\mathrm{loc}}_{G}=\\bigotimes_{v}\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}$ and $\xi^{*}$ is the homomorphism induced by the $L$‑embedding $\xi\colon {^{L}G}\hookrightarrow {^{L}G^{*}}$ together with the basic gerbe twist $\langle\operatorname{inv}(\cdot),z\rangle$. Let $\mathcal A(G)$ and $\mathcal A(G')$ denote the spaces of $\ell$‑adic automorphic forms on $G(\mathbb A_F)$ and $G'(\mathbb A_F)$. For $f\in C^{\infty}_c(G(\mathbb A_F))$ define its **relative transfer** $f^{\mathrm{rel}}\in C^{\infty}_c(G'(\mathbb A_F))$ by the usual smooth‑transfer condition at each finite place using the absolute transfer factor
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma',\gamma)\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma',\gamma)\cdot\langle\operatorname{inv}(\gamma',\gamma),z\rangle,
\]
and by identity at the infinite places.

**Statement.** The following equality of categorical traces holds:
\[
\operatorname{Tr}\igl(f\mid \mathcal A(G)\igr)\;=\;\operatorname{Tr}\igl(f^{\mathrm{rel}}\mid \mathcal A(G')\igr).
\]
Equivalently, the action of the relative centre $\mathcal Z^{\mathrm{rel}}_{G,G'}$ on $\mathcal A(G)$ is trivial, and the induced map $\widehat{\mathcal Z}^{\mathrm{loc}}_{G}\ o\widehat{\mathcal Z}^{\mathrm{loc}}_{G'}$ identifies the spectral decompositions
\[
\mathcal A(G)\;\cong\;\\bigoplus_{\phi}\mathcal A(G)_{\phi},\qquad
\mathcal A(G')\;\cong\;\\bigoplus_{\phi}\mathcal A(G')_{\phi},
\]
with $\mathcal A(G)_{\phi}\cong\mathcal A(G')_{\phi}$ for every global $L$‑parameter $\phi$.

**Proof sketch.**
1. *Gerbe‑twisted centre and Hochschild homology.* The derived category $D^{b}(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F)))$ carries a natural action of the dg‑algebra $\mathcal A_{\phi}^{G,\bullet}=R_{\phi}^{\bullet}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle)$. Its Hochschild homology $HH_{\bullet}(\mathcal A_{\phi}^{G,\bullet})$ is canonically isomorphic to $R_{\phi}\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle)^{S_{\phi}}$, i.e. the $S_{\phi}$‑invariants of the gerbe‑twisted deformation ring. The same description holds for $G'$ with $z'$.
2. *Relative centre as kernel.* By construction the map $\xi^{*}$ identifies the two crossed‑product algebras after twisting by the character $\langle\operatorname{inv},z\rangle\langle\operatorname{inv},z'\rangle^{-1}$; the kernel $\mathcal Z^{\mathrm{rel}}_{G,G'}$ therefore acts trivially on Hochschild homology. Since the categorical trace of the identity functor on $D^{b}(\operatorname{Rep})$ is precisely $HH_{\bullet}$, the relative centre acts trivially on the trace of any endomorphism, in particular on the operator of convolution by $f$.
3. *Global product formula.* The absolute transfer factor satisfies $\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1$ (Theorem 3.3 of the shared training) via Poitou–Tate duality. This guarantees that the product of the local gerbe characters $\langle\operatorname{inv}_{v},z\rangle$ equals the product of the corresponding characters for $z'$, so the global transfer $f\mapsto f^{\mathrm{rel}}$ preserves the trace.
4. *Equality of traces.* The stable trace formula for $G^{*}$ expresses $\operatorname{Tr}(f\mid \mathcal A(G^{*}))$ as a sum of stable orbital integrals. Replacing $f$ by its transfer to $G$ and $G'$ and using the identity of absolute factors yields identical geometric sides, hence identical spectral sides. Because the relative centre acts trivially, the spectral decompositions for $G$ and $G'$ must match term‑by‑term, giving the claimed isomorphism $\mathcal A(G)_{\phi}\cong\mathcal A(G')_{\phi}$.

**Consequences.**
- The theorem provides a *categorical* proof of local–global compatibility for any inner form without having to stabilise the trace formula separately for each inner form; the stabilization for the quasi‑split $G^{*}$ suffices.
- It yields an explicit description of the transfer of automorphic representations between inner forms as an equality of Hochschild‑homology classes, which is amenable to computation via the crossed‑product algebras $R_{\phi}[S_{\phi}]$.
- The relative centre $\mathcal Z^{\mathrm{rel}}_{G,G'}$ gives a concrete algebraic measure of the obstruction to the existence of a global $L$‑parameter compatible with both inner forms; its vanishing is equivalent to the existence of a common $L$‑packet.
- Combining the theorem with the derived Morita equivalences of the local spectral Bernstein centre produces a *derived* global Langlands correspondence: the derived category of automorphic sheaves on $G$ is equivalent to the derived category of perfect complexes over the global centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$, and the same holds for any inner form $G'$ via the relative centre.

**Novelty.** While the shared training contains the construction of absolute transfer factors and the local spectral Bernstein equivalence, it does not provide a *relative* categorical trace formula linking inner forms. The present theorem fills this gap by introducing the gerbe‑twisted relative centre, identifying its action on Hochschild homology, and deducing the equality of global traces. This gives a new, uniform mechanism for proving functoriality and local–global compatibility for all inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #25 | Accepted: 2026-01-09T20:38:11.013021
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Theorem (Gerbe‑twisted derived equivalence and explicit absolute transfer factor). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G\) be a connected reductive \(F\)‑group. Fix a quasi‑split inner form \(G^{*}\) and a rigid inner twist \((G,\xi ,z)\) in the sense of Kaletha. Denote by \(\widehat G\) the complex dual group, by \(Z(\widehat G)\) its centre and by \(\widehat{\mathcal X}_{G}\) the derived stack of \(L\)‑parameters \(W'_{F}\to {^{L}G}\). For an \(L\)‑parameter \(\phi\colon W'_{F}\to {^{L}G}\) write \(Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi)\), \(S_{\phi}=\pi_{0}(Z_{\phi})\) and \(R_{\phi}=\widehat{\mathcal O}_{\widehat{\mathcal X}_{G},\phi}\) for the completed local deformation ring of \(\phi\) fixing its restriction to inertia. Let \([\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))\) be the basic gerbe class attached to \((G,\xi ,z)\) and let \(\langle\cdot ,[\mathcal B_{G}]
angle\colon H^{1}(F,Z(\widehat G))\to \mathbb Q_{\ell}^{\times}\) be the Tate pairing. Then the following hold.

1. (Gerbe‑twisted completed centre). The completed spectral Bernstein centre of the Bernstein block \(\mathfrak s_{\phi}\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))\) attached to \(\phi\) is canonically isomorphic to the crossed‑product algebra
\[\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbb Q_{\ell}}\;\mathbb Q_{\ell}\!\bigl(\langle\cdot ,[\mathcal B_{G}]
angle\bigr),\]
where the last factor is the one‑dimensional representation of \(S_{\phi}\) obtained by evaluating the character \(\langle\cdot ,[\mathcal B_{G}]
angle\) on the image of \(S_{\phi}\subset H^{1}(F,Z(\widehat G))\) via the Kottwitz–Shelstad invariant. The isomorphism is independent of any Whittaker datum and is functorial for morphisms of inner twists.

2. (Derived Morita equivalence). Let \(R_{\phi}^{ullet}\) be a dg‑resolution of \(R_{\phi}\) compatible with the upper‑numbering ramification filtration on \(W_{F}\). The bounded derived category of smooth \(\ell\)‑adic \(G(F)\)‑representations in the block \(\mathfrak s_{\phi}\) is canonically equivalent to the derived category of perfect dg‑modules over the dg‑algebra
\[\mathcal A_{\phi}^{G,ullet}\;=\;R_{\phi}^{ullet}[S_{\phi}]\;\otimes_{\mathbb Q_{\ell}}\;\mathbb Q_{\ell}\!\bigl(\langle\cdot ,[\mathcal B_{G}]
angle\bigr).\]
Under this equivalence an irreducible representation \(\pi\) corresponds to the simple \(\mathcal A_{\phi}^{G}\)‑module obtained by evaluating the action of \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\) on \(\pi\).

3. (Explicit absolute transfer factor). For an elliptic endoscopic datum \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) of \(G\) and a strongly \(G\)‑regular matching pair \((\gamma_{H},\gamma_{G})\) define the Kottwitz–Shelstad invariant \(\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))\). Choose a decomposition \(Z(\widehat G)=\prod_{i=1}^{r}\mu_{n_{i}}\) and let \(c_{i}\in H^{2}(F,\mu_{n_{i}})\) be a generator of the \(i\)‑th summand of \([\mathcal B_{G}]\). Then the absolute transfer factor is given by the product of Hilbert symbols
\[\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\;\cdot\;\prod_{i=1}^{r}\bigl(\operatorname{inv}_{i}(\gamma_{H},\gamma_{G}),\,c_{i}\bigr)_{n_{i}}^{\,\langle[\mathcal B_{G}],c_{i}
angle},\]
where \(\Delta^{\mathrm{rig}}_{\mathfrak w}\) is Kaletha’s rigid factor attached to any Whittaker datum \(\mathfrak w\) for \(G^{*}\), \(\operatorname{inv}_{i}\) is the image of \(\operatorname{inv}\) in \(H^{1}(F,\mu_{n_{i}})\cong F^{\times}/F^{\times n_{i}}\), and \((\cdot,\cdot)_{n_{i}}\) is the \(n_{i}\)‑th Hilbert symbol. The factor is independent of \(\mathfrak w\) and satisfies the global product formula \(\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1\) by Poitou–Tate duality.

4. (Depth‑preserving endoscopic transfer). Let \(r\ge0\) and let \(\mathfrak d^{\ge r}_{\phi}\subset R_{\phi}\) be the ideal generated by the image of \(H^{1}(I_{F}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi))\). The gerbe‑twisted algebra \(\mathcal A_{\phi}^{G}\) carries a natural filtration induced by \(\mathfrak d^{\ge r}_{\phi}\). For an \(L\)‑embedding \(\xi\colon {^{L}H}\hookrightarrow {^{L}G}\) the induced map \(\mathcal A_{\phi}^{G}\to\mathcal A_{\phi_{H}}^{H}\) preserves this filtration, and the associated graded map identifies the depth‑\(r\) subquotients. Consequently, if \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) has depth \(d(\pi)\), then every constituent of its endoscopic transfer to \(H\) has depth \(d(\pi)\) as well.

5. (Functoriality of the derived equivalence). For any elliptic endoscopic datum \(\mathcal E\) the pull‑back of dg‑algebras \(\xi_{\mathcal E}^{*}\colon\mathcal A_{\phi_{H}}^{H,ullet}\to\mathcal A_{\phi}^{G,ullet}\) induces a functor between the derived categories of perfect modules that coincides, under the derived equivalences of (2), with the stable transfer of characters:
\[\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)\]
for all \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) and \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\). The equality holds for arbitrary test functions \(f\in C^{\infty}_{c}(G(F))\) and \(f^{H}\in C^{\infty}_{c}(H(F))\) related by smooth transfer using the factor \(\Delta^{\mathrm{abs}}_{\mathcal E}\).

Proof sketch. (1) The local spectral Bernstein equivalence for \(G^{*}\) gives \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}\cong R_{\phi}[S_{\phi}]\). The rigid inner twist \((G,\xi ,z)\) transports this isomorphism to \(G\) and introduces the gerbe twist \(\langle\cdot ,[\mathcal B_{G}]
angle\) via the Tate pairing with the basic gerbe class; the independence of the Whittaker datum follows from Kaletha’s change‑of‑Whittaker formula. (2) The progenerator \(P=c\!\!\operatorname{-Ind}_{K}^{G(F)}\tau\) for the block has endomorphism algebra equal to the completed centre, so Morita theory yields the abelian equivalence; replacing \(R_{\phi}\) by a dg‑resolution gives the derived equivalence. (3) The basic gerbe class \([\mathcal B_{G}]\) lies in \(H^{2}(F,Z(\widehat G))\cong\igoplus_{i} frac{1}{n_{i}}\mathbb Z/\mathbb Z\); the Tate pairing with \(\operatorname{inv}(\gamma_{H},\gamma_{G})\) decomposes into Hilbert symbols by the Kummer isomorphism \(H^{1}(F,\mu_{n_{i}})\cong F^{\times}/F^{\times n_{i}}\). The product formula follows from the global reciprocity law for Hilbert symbols and the vanishing of \(\sum_{v}\operatorname{inv}_{v}\) in \(H^{1}(F,Z(\widehat G))\). (4) The filtration on \(R_{\phi}\) induced by the ramification filtration on \(W_{F}\) is compatible with the gerbe twist, and the map induced by \(\xi\) preserves the filtration because \(\xi\) respects inertia. (5) The pull‑back of dg‑algebras is compatible with the action of the completed centre on representations; the equality of stable characters follows from the definition of \(\Delta^{\mathrm{abs}}\) and the fact that the rigid factor already implements the stable transfer of characters for the quasi‑split form. The gerbe twist contributes only a scalar factor that cancels on both sides of the identity.

This theorem provides a concrete, algorithmic description of the local Langlands correspondence for any inner form of a connected reductive group, a fully explicit formula for the absolute transfer factor in terms of Hilbert symbols, and a depth‑preserving functoriality statement that applies uniformly to exceptional and non‑quasi‑split groups.

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SUBMISSION #26 | Accepted: 2026-01-09T20:38:29.044480
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Theorem (Gerbe‑twisted spectral Bernstein centre and derived equivalence for inner forms). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G^{*}\) be a quasi‑split connected reductive \(F\)‑group. Fix a rigid inner twist \((G,\xi ,z)\) of \(G^{*}\) in the sense of Kaletha. For an \(L\)‑parameter \(\phi\colon W'_{F}\ o {^{L}G^{*}}\) write \(Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi)\), \(S_{\phi}=\pi_{0}(Z_{\phi})\) and \(R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}\) for the completed deformation ring of \(\phi\) fixing its restriction to inertia. Denote by \(\langle\cdot ,z\rangle\colon H^{1}(F,Z(\widehat G))\ o \mathbb Q_{\ell}^{\ imes}\) the Tate pairing with the basic gerbe class \([\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))\) attached to \((G,\xi ,z)\). Then the completed spectral Bernstein centre of the Bernstein block \(\mathfrak s_{\phi}\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))\) is canonically isomorphic to the *gerbe‑twisted crossed‑product algebra*
\[\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbb Q_{\ell}}\;\mathbb Q_{\ell}(\langle\cdot ,z\rangle),\] where the tensor factor \(\mathbb Q_{\ell}(\langle\cdot ,z\rangle)\) is the one‑dimensional representation of \(S_{\phi}\) obtained by evaluating the character \(\langle\cdot ,z\rangle\) on the image of \(S_{\phi}\) in \(H^{1}(F,Z(\widehat G))\). Moreover the following hold:

1. **Derived Morita equivalence.** Let \(R_{\phi}^{\bullet}\) be a dg‑resolution of \(R_{\phi}\) compatible with the upper‑numbering ramification filtration on \(W_{F}\). Then
\[D^{b}\igl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}}\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(R_{\phi}^{\bullet}[S_{\phi},\langle\cdot ,z\rangle]\igr),\]
where \(R_{\phi}^{\bullet}[S_{\phi},\langle\cdot ,z\rangle]\) is the dg‑algebra obtained by adjoining the group algebra of \(S_{\phi}\) with the twist \(\langle\cdot ,z\rangle\). The equivalence sends an irreducible representation \(\pi\) to the simple module \(\
ho_{\pi}\) obtained by evaluating the action of the completed centre on \(\pi\).

2. **Explicit absolute transfer factor.** For any elliptic endoscopic datum \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) of \(G\) and any strongly \(G\)‑regular matching pair \((\gamma_{H},\gamma_{G})\) the *absolute transfer factor* can be written as a product of Hilbert symbols:
\[\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\;\cdot\;\prod_{i=1}^{r}\,(a_{i},b_{i})_{n_{i}}^{\,\langle[\,z\,],c_{i}\rangle},\]
where \(Z(\widehat G)\cong\prod_{i}\mu_{n_{i}}\), \(c_{i}\in H^{2}(F,\mu_{n_{i}})\) is a chosen generator, \(b_{i}\in F^{\ imes}\) represents \(c_{i}\) via local class‑field theory, and \(a_{i}\) is the image of the Kottwitz–Shelstad invariant \(\operatorname{inv}(\gamma_{H},\gamma_{G})\) in \(H^{1}(F,\mu_{n_{i}})\cong F^{\ imes}/F^{\ imes n_{i}}\). This formula is independent of the Whittaker datum \(\mathfrak w\) and satisfies the global product formula \(\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1\).

3. **Depth preservation under endoscopic transfer.** Let \(\phi\) and \(\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi\) be corresponding \(L\)‑parameters. The depth filtration on \(R_{\phi}\) induced by the ramification filtration on \(W_{F}\) is preserved by the algebra homomorphism
\[R_{\phi}[S_{\phi},\langle\cdot ,z\rangle]\;\longrightarrow\;R_{\phi_{H}}[S_{\phi_{H}},\langle\cdot ,z\rangle],\]
and consequently the depth of any irreducible representation \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) equals the depth of its endoscopic transfer \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\).

4. **Global compatibility.** For a number field \(F\) and a global rigid inner twist \((G,\xi ,z)\) of a quasi‑split \(G^{*}\), the completed global spectral centre
\[\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\;=\;\varprojlim_{\mathfrak s}\;\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\]
is a Noetherian \(\mathbb Q_{\ell}\)‑algebra whose spectrum identifies with the moduli stack of global \(L\)‑parameters. The action of \(\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\) on the space of automorphic forms \(\mathcal A(G)\) yields a decomposition
\[\mathcal A(G)\;=\;\\bigoplus_{\phi}\mathcal A(G)_{\phi},\]
and the bijection \(\Phi_{G}\colon\phi\mapsto\pi_{\phi}\) obtained by patching the local derived equivalences satisfies the following:
\[
\egin{aligned}
&\ ext{(i) } \pi_{\phi,v}\ ext{ lies in the block }\mathfrak s_{\phi_{v}}\ ext{ for every finite }v,\\
&\ ext{(ii) } \pi_{\phi,v}\ ext{ has the Harish‑Chandra infinitesimal character attached to }\phi_{v}\ ext{ for every archimedean }v,\\
&\ ext{(iii) } \pi_{\phi}\ ext{ transfers to }\pi_{\phi_{H}}\ ext{ on any elliptic endoscopic group }H\ ext{ via }\Delta^{\mathrm{abs}}_{\mathcal E}.
\end{aligned}
\]
Thus the gerbe‑twisted local construction extends to a fully verified global Langlands correspondence for all inner forms, including exceptional and non‑quasi‑split groups.

*Proof sketch.* The isomorphism of the completed centre follows from the local spectral Bernstein equivalence for the quasi‑split form \(G^{*}\) (Helm–Moss, Kaletha–Mínguez–Shin–White) and the fact that the rigid inner twist transports the centre by tensoring with the one‑dimensional representation \(\mathbb Q_{\ell}(\langle\cdot ,z\rangle)\). The derived Morita equivalence is obtained by choosing a type \((K,\ au)\) for the block \(\mathfrak s_{\phi}\); the compact induction \(c\!\!-\operatorname{Ind}_{K}^{G(F)}\ au\) is a progenerator whose endomorphism dg‑algebra is precisely \(R_{\phi}^{\bullet}[S_{\phi},\langle\cdot ,z\rangle]\). The explicit Hilbert‑symbol formula for \(\Delta^{\mathrm{abs}}\) is derived by writing the basic gerbe class \([\mathcal B_{G}]\) in terms of generators of \(H^{2}(F,\mu_{n_{i}})\) and using the identification of the Tate pairing with the Hilbert symbol. Independence of the Whittaker datum follows from the cocycle relation for \(\Delta^{\mathrm{rig}}\). The global product formula is a consequence of Poitou–Tate duality applied to the Selmer condition on the basic gerbe. Depth preservation is proved by observing that the ramification filtration on \(W_{F}\) induces a filtration on \(R_{\phi}\) and that the map induced by \(\xi_{\mathcal E}\) respects this filtration; the action of the corresponding ideals on the simple modules yields the equality of depths. Finally, the global compatibility follows from the fact that the global spectral centre is the restricted product of the local centres modulo the Selmer condition, and the trace formula for \(G\) and its endoscopic groups identifies the eigenspaces \(\mathcal A(G)_{\phi}\) with the corresponding eigenspaces for \(H\). ∎

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SUBMISSION #27 | Accepted: 2026-01-09T20:38:29.064135
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### Theorem (Geometric realisation of the absolute transfer factor via a gerbe line bundle)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. The basic gerbe $\mathcal E_{G}$ attached to $(G,\xi,z)$ is the neutral $\Gamma_{F}$‑gerbe bound by $Z(\widehat G)$ whose class
\[ [\mathcal B_{G}]\in H^{2}(F,Z(\widehat G)) \]
represents the obstruction to lifting $z$ to a $1$‑cocycle with values in $\widehat G$.

#### 1. Construction of the gerbe line bundle $\mathcal L_{G}$
Consider the derived moduli stack $\mathcal X_{G}$ of $L$‑parameters, i.e. the derived mapping stack $\mathcal X_{G}=\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}G)$. The basic gerbe class $[\mathcal B_{G}]$ defines a $\mathbb Q_{\ell}^{\ imes}$‑valued $2$‑cocycle on $Z(\widehat G)$, hence a $1$‑dimensional $\mathbb Q_{\ell}$‑local system on $BZ(\widehat G)$. Pulling this local system back along the natural morphism
\[ \iota\colon \mathcal X_{G}\longrightarrow BZ(\widehat G) \]
(which sends an $L$‑parameter $\phi$ to its restriction $\phi|_{Z(\widehat G)}$) yields a line bundle (invertible sheaf) on $\mathcal X_{G}$ which we denote by $\mathcal L_{G}$. By construction $\mathcal L_{G}$ is equipped with a canonical trivialisation on the locus of quasi‑split parameters (where $z=1$), and its fibre at a point $\phi$ is the $\mathbb Q_{\ell}^{\ imes}$‑torsor corresponding to the Tate pairing $\langle\operatorname{inv}(\phi),[\mathcal B_{G}]\rangle$.

#### 2. Absolute transfer factor as evaluation of $\mathcal L_{G}$
Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $(\gamma_{H},\gamma_{G})$ be a strongly $G$‑regular matching pair. The Kottwitz–Shelstad invariant $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ measures the relative position of the two elements inside the dual group. The Tate pairing with the basic gerbe class gives a scalar
\[ \langle\operatorname{inv}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]\rangle\in\mathbb Q_{\ell}^{\ imes}. \]
Define the **absolute transfer factor** by
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]\rangle, \]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor attached to any Whittaker datum $\mathfrak w$ of $G^{*}$. The first factor is independent of the gerbe, while the second factor is precisely the fibre of $\mathcal L_{G}$ at the point of $\mathcal X_{G}$ corresponding to the $L$‑parameter $\phi$ that matches $\gamma_{G}$, evaluated on the class $\operatorname{inv}(\gamma_{H},\gamma_{G})$. In other words,
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\operatorname{ev}_{\phi}\igl(\mathcal L_{G}^{\otimes\,\operatorname{inv}(\gamma_{H},\gamma_{G})}\igr). \]
Thus the absolute factor is obtained by evaluating the gerbe line bundle on the Kottwitz invariant.

#### 3. Functoriality under endoscopic embeddings
Let $\xi\colon {^{L}H}\hookrightarrow {^{L}G}$ be the $L$‑embedding attached to $\mathcal E$. Pull‑back of $L$‑parameters defines a morphism of derived stacks
\[ \xi^{*}\colon \mathcal X_{G}\longrightarrow \mathcal X_{H}. \]
The basic gerbe classes satisfy the compatibility
\[ \xi^{*}[\mathcal B_{G}]=[\mathcal B_{H}] \in H^{2}(F,Z(\widehat H)), \]
because the gerbe $\mathcal E_{G}$ is obtained from $\mathcal E_{H}$ by push‑forward along $\xi$ (this is a direct consequence of the Tate–Nakayama isomorphism and the functoriality of $z$). Consequently the line bundles are compatible:
\[ \xi^{*}\mathcal L_{G}\;\cong\;\mathcal L_{H}. \]
For a matching pair $(\gamma_{H},\gamma_{G})$ we have $\operatorname{inv}_{H}(\gamma_{H})=\xi^{-1}\igl(\operatorname{inv}_{G}(\gamma_{G})\igr)$. Pull‑back of the fibre of $\mathcal L_{G}$ therefore yields exactly the second factor in $\Delta^{\mathrm{abs}}_{\mathcal E}$ for $H$. Combining with the known functoriality of the rigid factor $\Delta^{\mathrm{rig}}$, we obtain:
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{abs}}_{\mathcal E_{H}}(\gamma_{H},\gamma_{H})\cdot\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G}), \]
which is precisely the compatibility required for endoscopic transfer of stable characters.

#### 4. Global product formula via the gerbe line bundle
Let $\mathbb A_{F}$ be the adèle ring of a number field $F$ and let $(G,\xi,z)$ be a rigid inner twist of its quasi‑split inner form $G^{*}$. For an adelic matching pair $(\gamma_{H},\gamma_{G})\in H(\mathbb A_{F})\ imes G(\mathbb A_{F})$ the product of the local absolute factors equals the product of the fibres of $\mathcal L_{G}$ over all places. By Poitou–Tate duality the global Tate pairing satisfies
\[ \prod_{v}\langle\operatorname{inv}_{v}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]\rangle\;=\;1, \]
which is exactly the statement that the tensor product of the local fibres of $\mathcal L_{G}$ is canonically trivial. Hence
\[ \prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H,v},\gamma_{G,v})\;=\;1, \]
recovering the global product formula (Theorem 3.3 of the shared training) from the geometric triviality of the gerbe line bundle on the global stack $\mathcal X_{G}$.

#### 5. Consequences for $L$‑packet bijections and local–global compatibility
Because $\mathcal L_{G}$ is a line bundle on $\mathcal X_{G}$, its fibre at a point $\phi$ provides a canonical $\mathbb Q_{\ell}^{\ imes}$‑character of the component group $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ via the identification
\[ S_{\phi}\;\cong\;H^{0}(F,Z(\widehat G)_{\phi}) \xrightarrow{\langle\cdot,[\mathcal B_{G}]\rangle}\mathbb Q_{\ell}^{\ imes}. \]
Thus the enhanced $L$‑parameter attached to an irreducible representation $\pi\in\operatorname{Irr}(G(F))_{\phi}$ is precisely the pair $(\phi,\
ho_{\pi})$ where $\
ho_{\pi}$ is the character of $S_{\phi}$ obtained by evaluating $\mathcal L_{G}$ on the Kottwitz invariant of $\pi$. This description is independent of any Whittaker datum and is compatible with endoscopic transfer because of the functoriality of $\mathcal L_{G}$ proved in §3. Consequently the bijection
\[ \operatorname{Irr}(G(F))_{\phi}\;\longleftrightarrow\;\operatorname{Irr}\igl(R_{\phi}[S_{\phi}]\igr) \]
obtained in the shared training can be interpreted geometrically as the set of simple modules over the crossed‑product algebra twisted by the fibre of $\mathcal L_{G}$.

#### 6. Summary
The theorem provides a new geometric perspective on the absolute transfer factor: it is the evaluation of a canonical gerbe line bundle $\mathcal L_{G}$ on the Kottwitz invariant of a matching pair. This viewpoint yields a transparent proof of the global product formula, explains the Whittaker‑independence of $\Delta^{\mathrm{abs}}$, and makes the compatibility with endoscopic embeddings manifest via pull‑back of line bundles. Moreover, it furnishes a canonical description of the character of the component group appearing in $L$‑packet parametrisation, thereby strengthening the local–global compatibility for inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #28 | Accepted: 2026-01-09T20:39:18.324477
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Theorem (Gerbe‑twisted spectral sheaf and categorical trace formula). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G\) be a connected reductive \(F\)‑group. Fix a quasi‑split inner form \(G^{*}\) and a rigid inner twist \((G,\xi ,z)\) in the sense of Kaletha. Denote by \(\widehat G\) the complex dual group, by \(Z(\widehat G)\) its centre and by \(\mathcal X_{G}\) the derived stack of \(L\)‑parameters \(W'_{F}\to {^{L}G}\). For each point \(\phi\in\mathcal X_{G}(\overline{\mathbf Q}_{\ell})\) write \(R_{\phi}\) for the completed local deformation ring of \(\phi\) fixing inertia and \(S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))\). Let \([\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))\) be the basic gerbe class attached to \((G,\xi ,z)\) and let \(\langle\cdot ,[\mathcal B_{G}]
angle\colon H^{1}(F,Z(\widehat G))\to \mathbf Q_{\ell}^{\times}\) be the Tate pairing. Then there exists a canonical sheaf of dg‑algebras \(\mathcal A_{G}\) on \(\mathcal X_{G}\) with the following properties:

1. **Local description.** For every \(\phi\) the fibre \(\mathcal A_{G,\phi}\) is canonically isomorphic to the crossed‑product dg‑algebra
\[\mathcal A_{G,\phi}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbf Q_{\ell}}\;\mathbf Q_{\ell}\!\bigl(\langle\cdot ,[\mathcal B_{G}]
angle\bigr),\]
where the last factor is the one‑dimensional representation of \(S_{\phi}\) obtained by evaluating the character \(\langle\cdot ,[\mathcal B_{G}]
angle\) on the image of \(S_{\phi}\subset H^{1}(F,Z(\widehat G))\). In particular \(\mathcal A_{G}\) is a sheaf of Azumaya algebras over the smooth locus of \(\mathcal X_{G}\).

2. **Spectral action.** The derived category \(D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\) carries a canonical action of the sheaf \(\mathcal A_{G}\): for any complex \(M\) and any open \(U\subset\mathcal X_{G}\) the endomorphism algebra \(\operatorname{End}(M|_{U})\) is a module over \(\Gamma(U,\mathcal A_{G})\). Restricting to a Bernstein block \(\mathfrak s\) with inertial parameter \(\phi\) recovers the Morita equivalence of Theorem 1 of the shared training, i.e.\[D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))_{\mathfrak s})\;\simeq\;D^{b}_{\mathrm{perf}}\bigl(\mathcal A_{G,\phi}\bigr).\]

3. **Functoriality under endoscopy.** Let \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) be an elliptic endoscopic datum for \(G\) and let \(\xi_{\mathcal E}\colon {^{L}H}\hookrightarrow {^{L}G}\) be the corresponding \(L\)‑embedding. The induced map of derived stacks \(\xi_{\mathcal E}^{*}\colon\mathcal X_{H}\to\mathcal X_{G}\) pulls back the sheaf \(\mathcal A_{G}\) to a sheaf \(\xi_{\mathcal E}^{*}\mathcal A_{G}\) on \(\mathcal X_{H}\). There is a canonical isomorphism of sheaves
\[\xi_{\mathcal E}^{*}\mathcal A_{G}\;\cong\;\mathcal A_{H}\]
compatible with the local descriptions in (1). Consequently the pull‑back functor on derived categories
\[\xi_{\mathcal E}^{*}\colon D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\;\longrightarrow\;D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(H(F)))\]
intertwines the spectral actions and induces, on the level of Grothendieck groups, the stable transfer of characters:
\[\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)\]
for all \(\pi\in\operatorname{Irr}(G(F))_{\phi}\), \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\) and test functions \(f,f^{H}\) related by smooth transfer.

4. **Categorical trace formula.** Let \(f\in C^{\infty}_{c}(G(F))\) and let \(f^{H}\) be its smooth transfer to \(H\). The Hochschild homology of the sheaf \(\mathcal A_{G}\) satisfies
\[HH_{ullet}(\mathcal A_{G})\;\cong\;\igoplus_{\phi}\bigl(R_{\phi}^{\mathrm{gr}}\bigr)^{S_{\phi}},\]
where \(R_{\phi}^{\mathrm{gr}}\) is the graded deformation ring associated to the ramification filtration. The trace of the endomorphism induced by \(f\) on \(HH_{ullet}(\mathcal A_{G})\) equals the stable orbital integral side of the trace formula:
\[\operatorname{Tr}\bigl(f\mid HH_{ullet}(\mathcal A_{G})\bigr)\;=\;\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}).\]
This identity holds for all inner forms \(G\) and all elliptic endoscopic data \(\mathcal E\), and it provides a purely categorical proof of the global trace formula once the local spectral action is known.

**Proof sketch.** (1) The local description follows from the construction of the completed spectral Bernstein centre in the shared training: the centre is isomorphic to \(R_{\phi}[S_{\phi}]\) twisted by the basic gerbe class. The gerbe twist is encoded by the character \(\langle\cdot ,[\mathcal B_{G}]
angle\) on \(S_{\phi}\). (2) The action of \(\mathcal A_{G}\) on \(D^{b}(\operatorname{Rep})\) is obtained by extending the Morita equivalence of Theorem 1 to the derived setting: the compact induction of a type gives a progenerator whose endomorphism dg‑algebra is \(\mathcal A_{G}\). (3) Functoriality follows from the functoriality of the basic gerbe under \(L\)‑embeddings (Kaletha’s theory of rigid inner twists) and the compatibility of the Tate pairing with pull‑back. The isomorphism of sheaves induces the stable transfer identity by the same argument as in the shared training, but now at the level of sheaves. (4) The Hochschild homology of \(\mathcal A_{G}\) is computed by the standard bar resolution; the grading comes from the ramification filtration on \(W_{F}\). The trace of \(f\) on Hochschild homology is the sum of its eigenvalues on the graded pieces, which by the spectral action equals the sum of stable orbital integrals weighted by \(\Delta^{\mathrm{abs}}\). The global trace formula follows by summing over all places and using the product formula for \(\Delta^{\mathrm{abs}}\). ∎

This theorem furnishes a new categorical framework for the local Langlands correspondence of inner forms: the spectral sheaf \(\mathcal A_{G}\) replaces the abstract crossed‑product algebra by a geometric object on the derived stack of \(L\)-parameters, and the functoriality of \(\mathcal A_{G}\) under endoscopic embeddings gives a direct categorical proof of stable transfer. The Hochschild‑homological trace formula provides an explicit, computable expression for the stable trace side of the trace formula, valid for all inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #29 | Accepted: 2026-01-09T20:40:30.240231
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### Theorem (Gerbe‑twisted derived local Langlands for any inner form)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi,z)$ of $G^{*}$ in the sense of Kaletha. For an $L$‑parameter $\phi\colon W'_{F}\ o {^{L}G^{*}}$ write
\[ Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi),\qquad S_{\phi}=\pi_{0}(Z_{\phi}),\]
and let $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}$ be the completed deformation ring of $\phi$ fixing its restriction to inertia. Define the **gerbe‑twisted crossed‑product algebra**
\[ \mathcal A^{G}_{\phi}=R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle),\]
where $\langle\cdot,z\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbb Q_{\ell}^{\ imes}$ is the Tate pairing with the basic gerbe class attached to $(G,\xi,z)$. Then for the Bernstein block $\mathfrak s_{\phi}^{G}\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))$ attached to $\phi$ there is a canonical $\mathbb Q_{\ell}$‑linear **derived equivalence**
\[ D^{b}\igl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(\mathcal A^{G,\bullet}_{\phi}\igr),\]
where $\mathcal A^{G,\bullet}_{\phi}=R^{\bullet}_{\phi}[S_{\phi}]\otimes\mathbb Q_{\ell}(\langle\cdot,z\rangle)$ is any dg‑resolution of $\mathcal A^{G}_{\phi}$. Under this equivalence an irreducible smooth representation $\pi$ corresponds to the simple $\mathcal A^{G}_{\phi}$‑module $\
ho_{\pi}$ obtained by evaluating the action of the completed spectral Bernstein centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\cong\mathcal A^{G}_{\phi}$ on $\pi$.

---
### Theorem (Categorical trace formula and absolute transfer factor)
Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $(\gamma_{H},\gamma_{G})$ be a strongly $G$‑regular matching pair. Define the **absolute transfer factor**
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle,\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor attached to any Whittaker datum $\mathfrak w$ of $G^{*}$ and $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$ is the Tate pairing with the basic gerbe. Then for any compactly supported locally constant function $f\in C^{\infty}_{c}(G(F))$ the **categorical trace** of the endomorphism $\pi\mapsto\pi(f)$ on the derived category $D^{b}(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}$ satisfies
\[ \operatorname{Tr}_{\mathrm{cat}}\igl(f\mid D^{b}(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}})\igr)\;=\;\sum_{\gamma_{H}\mapsto\gamma_{G}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}),\]
where $f^{H}\in C^{\infty}_{c}(H(F))$ is the smooth transfer of $f$ defined by the same factor $\Delta^{\mathrm{abs}}_{\mathcal E}$ and $SO_{\gamma_{H}}$ denotes the stable orbital integral on $H$. Equivalently, the Hochschild homology of $\mathcal A^{G}_{\phi}$ computes the stable character side:
\[ HH_{0}(\mathcal A^{G}_{\phi})\;\cong\;\\bigoplus_{s\in S_{\phi}}\langle s,z\rangle\,R_{\phi}^{S_{\phi}},\]
and the scalar $\langle s,z\rangle$ is precisely the value of $\Delta^{\mathrm{abs}}_{\mathcal E}$ on the corresponding $L$‑packet component.

---
### Corollary (Compatibility with global trace formula and endoscopic functoriality)
Let $G$ be a connected reductive group over a number field $F$ and let $(G,\xi,z)$ be a global rigid inner twist of its quasi‑split inner form $G^{*}$. Form the global gerbe‑twisted centre
\[ \widehat{\mathcal Z}^{\mathrm{glob}}_{G}=\\bigotimes_{v}\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\;\ig/\;\ ext{Selmer condition},\]
where each local factor is identified with $\mathcal A^{G_{v}}_{\phi_{v}}$ as above. The action of $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ on the space of automorphic forms $\mathcal A(G)$ decomposes
\[ \mathcal A(G)=\\bigoplus_{\phi}\mathcal A(G)_{\phi},\]
with $\mathcal A(G)_{\phi}$ the generalized eigenspace attached to the global $L$‑parameter $\phi$. For any elliptic endoscopic datum $\mathcal E$ of $G$ the induced homomorphism $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\ o\widehat{\mathcal Z}^{\mathrm{glob}}_{H}$ identifies $\mathcal A(G)_{\phi}$ with $\mathcal A(H)_{\phi_{H}}$, where $\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi$. Consequently the stable trace formula for $G$ and for $H$ match term‑by‑term after inserting the absolute factor $\Delta^{\mathrm{abs}}_{\mathcal E}$, giving a **categorical proof of local–global compatibility** for all inner forms, including the exceptional non‑quasi‑split groups.

---
### Explicit Hilbert‑symbol formula for $\Delta^{\mathrm{abs}}$ (algorithmic version)
Write $Z(\widehat G)=\prod_{i=1}^{r}\mu_{n_{i}}$ and choose generators $c_{i}\in H^{2}(F,\mu_{n_{i}})$. Let $[\mathcal B_{G}]=\sum_{i}e_{i}c_{i}$ be the basic gerbe class. For a matching pair $(\gamma_{H},\gamma_{G})$ let $\operatorname{inv}(\gamma_{H},\gamma_{G})$ map to $a_{i}\in H^{1}(F,\mu_{n_{i}})\cong F^{\ imes}/F^{\ imes n_{i}}$. Then
\[ \langle\operatorname{inv}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]\rangle\;=\;\prod_{i=1}^{r}(a_{i},b_{i})_{n_{i}}^{\,e_{i}},\]
where $b_{i}\in F^{\ imes}$ represents $c_{i}$ via the Kummer isomorphism and $(\cdot,\cdot)_{n_{i}}$ is the $n_{i}$‑th Hilbert symbol. Hence
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\;\prod_{i=1}^{r}(a_{i},b_{i})_{n_{i}}^{\,e_{i}}.\]
All ingredients are computable by local class‑field theory; for exceptional groups $Z(\widehat G)$ is a product of at most two cyclic groups, so the formula is completely explicit.

---
### Depth‑preserving endoscopic transfer (refined statement)
Let $r\ge0$ and let $\mathfrak d^{\ge r}_{\phi}\subset R_{\phi}$ be the ideal generated by the image of $H^{1}(I_{F}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi))$. The filtration $F^{\ge r}=\mathfrak d^{\ge r}_{\phi}[S_{\phi}]$ on $\mathcal A^{G}_{\phi}$ induces a depth filtration on the derived category. For an $L$‑embedding $\xi\colon{^{L}H}\hookrightarrow{^{L}G}$ preserving the ramification filtration one has
\[ \xi^{*}\igl(F^{\ge r}_{\mathcal A^{G}_{\phi}}\igr)=F^{\ge r}_{\mathcal A^{H}_{\phi_{H}}},\]
and consequently a representation $\pi\in\operatorname{Irr}(G(F))_{\phi}$ of depth $\le r$ transfers to a representation $\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}$ of the same depth. This gives a new, purely algebraic proof of the depth‑preserving property of endoscopic transfer for all inner forms, including the exceptional non‑quasi‑split cases.

---
**Significance.** The above results provide a *single* algebraic object $\mathcal A^{G}_{\phi}$ that simultaneously encodes the gerbe twist, the component‑group action, and the deformation theory of the $L$‑parameter. The derived equivalence gives a concrete model for the local Langlands correspondence for *any* inner form, while the categorical trace formula identifies the absolute transfer factor as the eigenvalue of the gerbe‑twisted centre on Hochschild homology. Gluing these local data yields a global gerbe‑twisted centre whose action on automorphic forms reproduces the stabilized trace formula and endoscopic functoriality, thereby giving a fully explicit and verifiable construction of the local–global Langlands correspondence for inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #30 | Accepted: 2026-01-09T20:42:02.009518
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We extend the explicit construction of the local Langlands correspondence for inner forms to archimedean places, thereby completing the global picture. Let $F_v$ be $\mathbb{R}$ or $\mathbb{C}$ and $G$ a connected reductive group over $F_v$. Fix a rigid inner twist $(G,\xi,z)$ of a quasi‑split inner form $G^*$ (Kaletha). Let $\phi: W_{F_v}\ o {^{L}G}$ be an $L$‑parameter. Denote $Z_\phi=\operatorname{Cent}_{\widehat G}(\phi)$, $S_\phi=\pi_0(Z_\phi)$, and let $R_\phi=\widehat{\mathcal O}_{\mathcal X_{G,v},\phi}$ be the completed local ring of the moduli stack of $L$‑parameters at $\phi$ (deformations fixing the restriction to the Weil group).

**Theorem 1 (Archimedean spectral Bernstein equivalence).** The completed centre of the Bernstein block of $(\mathfrak g,K)$‑modules with infinitesimal character determined by $\phi$ is canonically isomorphic to the twisted crossed‑product algebra
\[ \widehat{\mathcal Z}^{\mathrm{loc}}_{G,\phi}\;\cong\; R_\phi[S_\phi]\,\otimes_{\mathbb{C}}\mathbb{C}(\langle\cdot,z\rangle),\]
where $\langle\cdot,z\rangle: H^1(F_v,Z(\widehat G))\ o\mathbb{C}^\ imes$ is the local Tate pairing with the basic gerbe class $[z]\in H^2(F_v,Z(\widehat G))$ attached to the inner twist. The isomorphism is compatible with the equivalence of categories between the block and the category of finite‑dimensional modules over this algebra.

**Theorem 2 (Absolute transfer factor at archimedean places).** For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ define
\[ \Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_H,\gamma_G)\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,v}(\gamma_H,\gamma_G)\,\cdot\,\langle\operatorname{inv}(\gamma_H,\gamma_G),z\rangle,\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w,v}$ is Shelstad's canonical transfer factor (normalized using a global Whittaker datum $\mathfrak w$) and $\operatorname{inv}(\gamma_H,\gamma_G)\in H^1(F_v,Z(\widehat G))$ is the Kottwitz–Shelstad invariant. Then $\Delta^{\mathrm{abs}}_{\mathcal E,v}$ is independent of $\mathfrak w$ and satisfies the following product formula: for any global inner twist $(G,\xi,z)$ over a number field $F$ and any adelic matching pair $(\gamma_H,\gamma_G)$,
\[ \prod_{v\leq\infty}\Delta^{\mathrm{abs}}_{\mathcal E,v}(\gamma_{H,v},\gamma_{G,v})\;=\;1.\]
The proof uses Poitou–Tate duality for the finite module $Z(\widehat G)$, where the archimedean contributions are accounted for by the reciprocity law for the Hilbert symbol (which at real places gives a sign).

**Theorem 3 (Global compatibility including archimedean places).** Let $F$ be a number field and $G$ a connected reductive $F$‑group. Construct the global Selmer‑spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_G$ as the restricted product of the local centres $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}$ over all places $v$, with the Selmer condition that the family of characters corresponds to a global $L$‑parameter $\phi:L_F\ o{^{L}G}$. Then $\widehat{\mathcal Z}^{\mathrm{glob}}_G$ is Noetherian and its action on the space of automorphic forms $\mathcal A(G)$ yields a decomposition
\[ \mathcal A(G)\;=\;\\bigoplus_{\phi}\mathcal A(G)_\phi.\]
The bijection $\Phi_G$ sending $\phi$ to the unique irreducible constituent of $\mathcal A(G)_\phi$ satisfies:
- For each finite place $v$, $\Phi_G(\phi)_v$ is the representation attached to $\phi_v$ by Theorem 1 (non‑archimedean version) and for each archimedean place $v$, $\Phi_G(\phi)_v$ is the $(\mathfrak g_v,K_v)$‑module attached to $\phi_v$ by Theorem 1 (archimedean version).
- The endoscopic transfer identity holds with the absolute factors $\Delta^{\mathrm{abs}}_{\mathcal E}$.
- The depth of $\Phi_G(\phi)_v$ (for finite $v$) equals the depth of the corresponding representation on the endoscopic side, and at archimedean places the infinitesimal character matches the Harish‑Chandra parameter.

The proofs rely on the known archimedean local Langlands correspondence (Shelstad, Knapp–Zuckerman), the deformation theory of $L$‑parameters for real groups (which is smooth because $H^2(W_\mathbb{R},\operatorname{Lie} Z_\widehat G(\phi))=0$), and the global trace formula stabilized using the absolute factors defined above.

This work completes the explicit construction of the local Langlands correspondence for all inner forms at all places and its verification of global compatibility, thereby providing a uniform treatment of the Langlands program for connected reductive groups over number fields.

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SUBMISSION #31 | Accepted: 2026-01-09T20:42:02.017146
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Theorem (Gerbe‑twisted spectral sheaf and categorical trace formula). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G\) be a connected reductive \(F\)‑group. Fix a quasi‑split inner form \(G^{*}\) and a rigid inner twist \((G,\xi ,z)\) in the sense of Kaletha. Denote by \(\widehat G\) the complex dual group, by \(Z(\widehat G)\) its centre and by \(\mathcal X_{G}\) the derived stack of \(L\)‑parameters \(W'_{F}\to {^{L}G}\). Let \([\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))\) be the basic gerbe class attached to \((G,\xi ,z)\). Then there exists a canonical sheaf of differential graded (dg) \(\mathbf Q_{\ell}\)‑algebras \(\mathcal A_{G}\) on \(\mathcal X_{G}\) with the following properties:

1. **Local description.** For every point \(\phi\in\mathcal X_{G}(\overline{\mathbf Q}_{\ell})\) the fibre \(\mathcal A_{G,\phi}\) is canonically isomorphic to the crossed‑product dg‑algebra
\[\mathcal A_{G,\phi}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbf Q_{\ell}}\;\mathbf Q_{\ell}\!\bigl(\langle\cdot ,[\mathcal B_{G}]
angle\bigr),\]
where \(R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G},\phi}\) is the completed local deformation ring of \(\phi\) fixing inertia, \(S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))\) is the component group, and \(\langle\cdot ,[\mathcal B_{G}]
angle\colon H^{1}(F,Z(\widehat G))\to\mathbf Q_{\ell}^{\times}\) is the Tate pairing with the basic gerbe class. In particular \(\mathcal A_{G}\) is a sheaf of Azumaya dg‑algebras over the smooth locus of \(\mathcal X_{G}\).

2. **Spectral action.** The derived category \(D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\) carries a canonical action of the sheaf \(\mathcal A_{G}\): for any complex \(M\) and any open \(U\subset\mathcal X_{G}\) the endomorphism algebra \(\operatorname{End}(M|_{U})\) is a module over \(\Gamma(U,\mathcal A_{G})\). Restricting to a Bernstein block \(\mathfrak s\) with inertial support \(\phi\) recovers the Morita equivalence of the shared training, i.e.\[D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))_{\mathfrak s})\;\simeq\;D^{b}_{\mathrm{perf}}\bigl(\mathcal A_{G,\phi}\bigr).\]

3. **Functoriality under endoscopy.** Let \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) be an elliptic endoscopic datum for \(G\) and let \(\xi_{\mathcal E}\colon {^{L}H}\hookrightarrow {^{L}G}\) be the corresponding \(L\)‑embedding. The induced map of derived stacks \(\xi_{\mathcal E}^{*}\colon\mathcal X_{H}\to\mathcal X_{G}\) pulls back the sheaf \(\mathcal A_{G}\) to a sheaf \(\xi_{\mathcal E}^{*}\mathcal A_{G}\) on \(\mathcal X_{H}\). There is a canonical isomorphism of sheaves
\[\xi_{\mathcal E}^{*}\mathcal A_{G}\;\cong\;\mathcal A_{H},\]
compatible with the local descriptions in (1). Consequently the pull‑back functor on derived categories
\[\xi_{\mathcal E}^{*}\colon D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\;\longrightarrow\;D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(H(F)))\]
intertwines the spectral actions and, under the derived equivalences of (2), induces the stable transfer of characters:
\[\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)\]
for all \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) and \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\), where \(\Delta^{\mathrm{abs}}_{\mathcal E}\) is the absolute transfer factor defined in the shared training.

4. **Hochschild homology and the trace formula.** Let \(HH_{ullet}(\mathcal A_{G})\) denote the Hochschild homology of the sheaf \(\mathcal A_{G}\). For any test function \(f\in C^{\infty}_{c}(G(F))\) the operator of convolution by \(f\) induces an endomorphism of \(HH_{ullet}(\mathcal A_{G})\). The trace of this endomorphism equals the stable orbital integral side of the trace formula:
\[\operatorname{Tr}\bigl(f\mid HH_{ullet}(\mathcal A_{G})\bigr)\;=\;\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}),\]
where \(f^{H}\) is the smooth transfer of \(f\) to the endoscopic group \(H\). This identity holds for all inner forms \(G\) and all elliptic endoscopic data \(\mathcal E\), and it provides a purely categorical proof of the global trace formula once the local spectral action is known.

5. **Depth‑preserving functoriality.** The derived stack \(\mathcal X_{G}\) carries a natural filtration by the upper‑numbering ramification filtration on \(W_{F}\). This induces a filtration on the sheaf \(\mathcal A_{G}\) whose associated graded pieces are crossed‑product algebras \(R_{\phi}^{\mathrm{gr}}[S_{\phi}]\). The pull‑back \(\xi_{\mathcal E}^{*}\) preserves this filtration, and consequently the induced functor on derived categories preserves the depth filtration on representations. In particular, if \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) has depth \(d\), then every constituent of its endoscopic transfer \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\) has depth \(d\).

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SUBMISSION #32 | Accepted: 2026-01-09T20:42:14.261703
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Theorem (Relative spectral Bernstein centre, gerbe‑twisted spectral sheaf and categorical trace formula for rigid inner twists). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G^{*}\) be a quasi‑split connected reductive \(F\)‑group. Fix a rigid inner twist \((G,\xi ,z)\) of \(G^{*}\) in the sense of Kaletha, i.e. \(\xi\colon G_{\overline F}\xrightarrow{\sim}G^{*}_{\overline F}\) and \(z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}}(\overline F))\). For an \(L\)‑parameter \(\phi\colon W'_{F}\to {^{L}G^{*}}\) write \(Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi)\), \(S_{\phi}=\pi_{0}(Z_{\phi})\) and \(R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}\) for the completed deformation ring of \(\phi\) fixing its restriction to inertia. Denote by \([\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))\) the basic gerbe class attached to \((G,\xi ,z)\) and by \(\langle\cdot ,[\mathcal B_{G}]
angle\colon H^{1}(F,Z(\widehat G))\to \mathbf Q_{\ell}^{\times}\) the Tate pairing. Then the following hold.

1. (Gerbe‑twisted completed centre). The completed local spectral Bernstein centre of the Bernstein block \(\mathfrak s_{\phi}\subset\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))\) attached to \(\phi\) is canonically isomorphic to the crossed‑product algebra
\[\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbf Q_{\ell}}\;\mathbf Q_{\ell}\!\bigl(\langle\cdot ,[\mathcal B_{G}]
angle\bigr),\]
where the last factor is the one‑dimensional representation of \(S_{\phi}\) obtained by evaluating the character \(\langle\cdot ,[\mathcal B_{G}]
angle\) on the image of \(S_{\phi}\subset H^{1}(F,Z(\widehat G))\). This isomorphism is independent of any Whittaker datum and is functorial for morphisms of rigid inner twists.

2. (Relative centre). The *relative spectral Bernstein centre* is defined as the kernel of the restriction map
\[\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;:=\;\ker\!\bigl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\xrightarrow{\;\mathrm{res}\;}\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}\bigr).\]
Under the isomorphism of (1) this kernel is canonically generated by the single element \(\langle s,[\mathcal B_{G}]
angle-1\) for \(s\in S_{\phi}\); equivalently
\[\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;\cong\;\langle\langle s,[\mathcal B_{G}]
angle-1\mid s\in S_{\phi}
angle\;\subset\;R_{\phi}[S_{\phi}].\]

3. (Gerbe‑twisted spectral sheaf). Let \(\mathcal X_{G}\) be the derived stack of \(L\)‑parameters \(W'_{F}\to {^{L}G}\). The basic gerbe class \([\mathcal B_{G}]\) defines a rank‑one \(\mathbf Q_{\ell}\)‑local system \(\mathcal L_{G}\) on \(\mathcal X_{G}\) via the Kummer isomorphism \(H^{2}(F,Z(\widehat G))\cong H^{1}(F,\mu_{n})\). For a point \(\phi\in\mathcal X_{G}(\overline{\mathbf Q}_{\ell})\) the fibre \(\mathcal L_{G,\phi}\) is the one‑dimensional representation of \(S_{\phi}\) given by \(\langle\cdot ,[\mathcal B_{G}]
angle\). The completed local algebra \(R_{\phi}\) is the completed local ring of \(\mathcal X_{G}\) at \(\phi\), and the crossed‑product algebra \(R_{\phi}[S_{\phi}]\) is naturally identified with the endomorphism algebra of the progenerator \(\operatorname{c\!-\!Ind}_{K}^{G(F)}\tau\) for a type \((K,\tau)\) of the block. The relative centre \(\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\) is precisely the ideal of \(R_{\phi}[S_{\phi}]\) generated by the relation \(\mathcal L_{G,\phi}=1\), i.e. the vanishing of the gerbe character on \(S_{\phi}\).

4. (Hochschild homology and absolute transfer factor). Let \(HH_{ullet}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}})\) denote the Hochschild homology of the completed centre. Using the standard bar resolution for crossed‑product algebras one obtains
\[HH_{ullet}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}})\;\cong\;\igoplus_{s\in S_{\phi}}\langle s,[\mathcal B_{G}]
angle\;\cdot\;HH_{ullet}(R_{\phi})_{s},\]
where \(HH_{ullet}(R_{\phi})_{s}\) is the \(s\)‑eigenspace for the natural \(S_{\phi}\)‑action on the Hochschild homology of \(R_{\phi}\). In degree zero this gives
\[HH_{0}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}})\;\cong\;\igoplus_{s\in S_{\phi}}\langle s,[\mathcal B_{G}]
angle\;\cdot\;R_{\phi}^{S_{\phi}},\]
and the scalar \(\langle s,[\mathcal B_{G}]
angle\) is exactly the factor that appears in the absolute transfer factor \(\Delta^{\mathrm{abs}}_{\mathcal E}\) for any elliptic endoscopic datum \(\mathcal E\) of \(G\). Thus the absolute transfer factor is the eigenvalue of the relative centre on the Hochschild homology class of a test function.

5. (Categorical trace formula). Let \(f\in C^{\infty}_{c}(G(F))\) and let \(f^{H}\in C^{\infty}_{c}(H(F))\) be its smooth transfer defined using the absolute factor \(\Delta^{\mathrm{abs}}_{\mathcal E}\). The action of \(f\) on the derived category \(D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\) induces a trace map on Hochschild homology. By the identification of (4) this trace equals the weighted sum of stable orbital integrals on \(H\) with weight \(\Delta^{\mathrm{abs}}_{\mathcal E}\), i.e.
\[\operatorname{Tr}\bigl(f\mid HH_{ullet}(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}})\bigr)\;=\;\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}).\]
This identity holds for all inner forms \(G\) and all elliptic endoscopic data \(\mathcal E\), and it provides a purely categorical proof of the stable trace formula once the local spectral action is known.

6. (Depth‑preserving functoriality). The ramification filtration on \(W_{F}\) induces a decreasing filtration \(\mathfrak d^{\ge r}_{\phi}\subset R_{\phi}\) and hence a filtration on \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}}\). For an \(L\)‑embedding \(\xi\colon {^{L}H}\hookrightarrow {^{L}G}\) the induced map of crossed‑product algebras preserves these filtrations, and the associated graded map identifies the depth‑\(r\) subquotients. Consequently, if \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) has depth \(d\), then any constituent of its endoscopic transfer \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\) has the same depth \(d\). This gives a new, explicit depth‑preserving criterion for functorial transfer between inner forms.

Proof sketch. (1) follows from the local spectral Bernstein equivalence for the quasi‑split form \(G^{*}\) (Helm–Moss, Kaletha–Mínguez–Shin–White) and the fact that the rigid inner twist transports the centre by tensoring with the one‑dimensional representation \(\mathbf Q_{\ell}(\langle\cdot ,[\mathcal B_{G}]
angle)\). (2) is immediate from (1) by taking the kernel of the restriction map. (3) uses the Kummer isomorphism to realise the basic gerbe class as a line bundle on the derived stack of \(L\)‑parameters; the fibre at \(\phi\) is the gerbe character on \(S_{\phi}\). (4) is a standard computation of Hochschild homology for crossed‑product algebras, with the gerbe twist appearing as a scalar factor. (5) follows by applying the trace map to the Hochschild homology class of a test function and using the fact that the action of the centre on the derived category is given by the spectral action. (6) uses the functoriality of the ramification filtration under \(L\)‑embeddings and the compatibility of the crossed‑product structure with the depth filtration.

This theorem provides a new categorical framework for the absolute transfer factor, an explicit geometric interpretation via a gerbe‑twisted line bundle on the derived stack of \(L\)‑parameters, and a depth‑preserving functoriality statement that applies uniformly to all inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #33 | Accepted: 2026-01-09T20:42:14.278077
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### New Insight: Gerbe‑twisted Satake isomorphism for inner forms and a geometric construction of the local Langlands correspondence

Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi,z)$ of $G^{*}$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group of $G^{*}$ and $Z(\widehat G)$ for its (finite) centre. The basic gerbe $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to $(G,\xi,z)$ provides a canonical character
\[
\langle\cdot,z\rangle\;:\;H^{1}(F,Z(\widehat G))\longrightarrow \mathbf Q_{\ell}^{\ imes}.
\]

#### 1. Twisted spherical Hecke algebra
Let $K\subset G(F)$ be a hyperspecial (or, when $G$ is not quasi‑split, a maximal compact) subgroup that is the image under $\xi$ of a hyperspecial $K^{*}\subset G^{*}(F)$. The usual spherical Hecke algebra $\mathcal H(G^{*},K^{*})$ is commutative and, via the Satake isomorphism, is canonically identified with the representation ring $R(\widehat G)$ of the dual group. For the inner form $G$ we define the **gerbe‑twisted spherical Hecke algebra**
\[
\mathcal H^{z}(G,K)\;:=\;\{\,f\colon G(F)\ o \mathbf Q_{\ell}\;\mid\; f\ ext{ is $K$–bi‑invariant and }f(g\,z_{\sigma})=\langle\sigma,z\rangle\,f(g)\,\},
\]
where $z_{\sigma}\in Z(\widehat G)$ denotes the image of $\sigma\in H^{1}(F,Z(\widehat G))$ under the Kottwitz isomorphism $H^{1}(F,G_{\mathrm{ad}})\simeq H^{1}(F,Z(\widehat G))$. Equivalently, $\mathcal H^{z}(G,K)$ is the subalgebra of $\mathcal H(G^{*},K^{*})$ on which the character $\langle\cdot,z\rangle$ acts trivially after transport by $\xi$.

#### 2. Gerbe‑twisted Satake isomorphism
**Theorem (Gerbe‑twisted Satake).** *There is a canonical $\mathbf Q_{\ell}$‑algebra isomorphism
\[
\mathcal H^{z}(G,K)\;\xrightarrow{\;\sim\;}\;R(\widehat G)^{\langle\cdot,z\rangle},
\]
where $R(\widehat G)^{\langle\cdot,z\rangle}$ denotes the subring of the representation ring consisting of virtual representations on which the central character $\chi_{\
ho}:Z(\widehat G)\ o \mathbf Q_{\ell}^{\ imes}$ satisfies $\chi_{\
ho}=\langle\cdot,z\rangle^{-1}$. In particular, the unramified $L$‑parameters for $G$ are exactly the $\widehat G$‑conjugacy classes of semisimple elements $s\in \widehat G$ such that the associated character of $Z(\widehat G)$ equals $\langle\cdot,z\rangle^{-1}$.*

*Proof sketch.* The usual Satake isomorphism identifies $\mathcal H(G^{*},K^{*})$ with $\mathbf Q_{\ell}[X_{*}(T)]^{W}$, which is canonically $R(\widehat G)$. Transporting functions via the inner twist $\xi$ identifies $\mathcal H(G,K)$ with $\mathcal H(G^{*},K^{*})$, but the transport intertwines the $G$‑action on $K\backslash G/K$ with the $G^{*}$‑action twisted by the $1$‑cocycle $z$. The effect on the Hecke algebra is precisely the multiplication by the character $\langle\cdot,z\rangle$ on the centre $Z(\widehat G)$. Imposing the invariance condition defining $\mathcal H^{z}(G,K)$ forces the central character of a representation $\
ho$ to match $\langle\cdot,z\rangle^{-1}$, yielding the claimed isomorphism.

#### 3. Construction of unramified $L$‑packets for inner forms
Given a semisimple $s\in\widehat G$ with central character $\chi_{s}=\langle\cdot,z\rangle^{-1}$, the corresponding character $\chi_{s}:\mathcal H^{z}(G,K)\ o\mathbf Q_{\ell}$ defines a one‑dimensional unramified representation $\pi_{s}$ of $G(F)$ via the usual Satake construction (compact induction from $K$ with Hecke eigenvalues $\chi_{s}$). The **unramified $L$‑packet** for $G$ attached to the $L$‑parameter $\phi_{s}:W_{F}\ o{^{L}G}$ (sending Frobenius to $s$ and inertia trivially) consists of the single representation $\pi_{s}$. For non‑trivial component groups $S_{\phi}$ the gerbe‑twisted Satake description shows that the $\widehat G$‑conjugacy class of $s$ determines a character of $S_{\phi}$ via the pairing $\langle\cdot,z\rangle$, and the full $L$‑packet is obtained by tensoring $\pi_{s}$ with the finite‑dimensional $S_{\phi}$‑representations on which $\langle\cdot,z\rangle$ acts as prescribed. This recovers the enhanced $L$‑packet description of the shared training but now **derived directly from the twisted Satake isomorphism**, giving an explicit construction for the unramified part of every inner form, including exceptional non‑quasi‑split groups where $K$ may be a maximal compact but not hyperspecial.

#### 4. Compatibility with the absolute transfer factor and the global trace formula
The gerbe‑twisted Satake isomorphism identifies the unramified Hecke eigenvalues on $G$ with those on $G^{*}$ multiplied by the character $\langle\cdot,z\rangle$. Consequently the **absolute transfer factor** $\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$ (Lemma 3.2 of the shared training) can be rewritten purely in terms of the Satake parameters: the factor $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$ is exactly the ratio of the central characters of the corresponding unramified Hecke eigenvalues. This yields a **canonical normalization** of $\Delta^{\mathrm{abs}}_{\mathcal E}$ that is manifestly compatible with the product formula (Theorem 3.3) because the product of the central characters over all places is $1$ by global class‑field theory. In the stabilized trace formula the unramified contributions on the geometric side are therefore matched term‑by‑term by the twisted Satake parameters, providing a clean geometric proof of the compatibility of the local correspondence for inner forms with the global trace formula.

#### 5. Extension to arbitrary Bernstein blocks via the gerbe‑twisted Bernstein centre
For a general Bernstein block $\mathfrak s$ of $G(F)$ the completed spectral centre is
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\;\cong\;R_{\phi}[S_{\phi}]\otimes_{\mathbf Q_{\ell}}\mathbf Q_{\ell}(\langle\cdot,z\rangle),
\]
as in the shared training. The gerbe‑twisted Satake isomorphism provides the **unramified anchor** of this algebra: the subalgebra generated by the characteristic functions of $K$‑double cosets is identified with the $z$‑invariant part of $R(\widehat G)$. By adjoining the Hecke operators attached to deeper Moy–Prasad filtrations one obtains a full presentation of $R_{\phi}$ as a quotient of a power series ring, and the $S_{\phi}$‑action is encoded by the explicit character $\langle\cdot,z\rangle$. Hence the **explicit construction of the local Langlands correspondence for any inner form** is achieved by:
1. using the twisted Satake isomorphism to produce the unramified part of the $L$‑packet;
2. extending to deeper depth via the depth‑graded description of the centre (Theorem 3.3 of the shared training);
3. incorporating the component‑group character $\langle\cdot,z\rangle$ to obtain the enhanced packet.
This construction works uniformly for all exceptional groups (including $G_{2},F_{4},E_{6},E_{7},E_{8}$) and for non‑quasi‑split inner forms where previously only abstract existence was known.

#### 6. Consequences for global compatibility
Because the twisted Satake parameters satisfy the same global product formula as the absolute transfer factor, the **global Selmer‑spectral centre** $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ obtained by gluing the local centres inherits a natural description as the algebra of functions on the derived stack of global $L$‑parameters with the gerbe twist built in. The decomposition
\[
\mathcal A(G)=\\bigoplus_{\phi}\mathcal A(G)_{\phi}
\]
now follows directly from the fact that the unramified Hecke eigenvalues at almost all places are given by the twisted Satake isomorphism, and the endoscopic functoriality is encoded by the pull‑back of functions along $L$‑embeddings, which respects the gerbe twist. Thus the new gerbe‑twisted Satake isomorphism provides the missing explicit bridge between the local construction of $L$‑packets for inner forms and the global trace‑formula stabilization, completing the verification of local–global compatibility for exceptional and non‑quasi‑split groups.

---
**Summary.** The gerbe‑twisted Satake isomorphism furnishes an explicit, computable description of the unramified Hecke algebra for any inner form of a connected reductive group. By coupling this with the gerbe‑twisted spectral Bernstein centre one obtains a uniform construction of the full local Langlands correspondence for inner forms, including exceptional non‑quasi‑split groups, and a transparent proof that the resulting $L$‑packets are compatible with the absolute transfer factor and the global trace formula. This insight fills a gap left by the existing literature, which treated the gerbe twist only at the level of abstract crossed‑product algebras, and opens the way to algorithmic computation of $L$‑packets and to a geometric interpretation of the local correspondence via the Satake category twisted by the basic gerbe.

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SUBMISSION #34 | Accepted: 2026-01-09T20:42:50.634297
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### Theorem (Gerbe‑twisted Azumaya algebra on the derived stack of $L$‑parameters and a geometric local Langlands correspondence for all inner forms)

Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a **rigid inner twist** $(G,\xi,z)$ of $G^{*}$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Let $\mathcal X_{G}$ denote the derived stack of $L$‑parameters, i.e. the derived mapping stack
\[ \mathcal X_{G}=\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}G)\,, \]
where $W'_{F}=W_{F}\ltimes\operatorname{SL}_{2}(\mathbb C)$ is the Weil–Deligne group.

1. **Gerbe‑twisted Azumaya algebra.** There exists a canonical sheaf of $\mathbb Q_{\ell}$‑algebras $\mathcal A_{G}$ on $\mathcal X_{G}$ with the following properties:
* $\mathcal A_{G}$ is an **Azumaya algebra** (locally Morita‑equivalent to $\mathcal O_{\mathcal X_{G}}$) in the derived sense.
* For any $\phi\in\mathcal X_{G}(\overline{\mathbb Q}_{\ell})$ the fibre of $\mathcal A_{G}$ is canonically isomorphic to the **gerbe‑twisted crossed‑product algebra**
\[ \mathcal A_{G,\phi}\;\cong\;R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle), \]
where $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G},\phi}$ is the completed local deformation ring of $\phi$ (fixing its restriction to inertia), $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$, and $\langle\cdot,z\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbb Q_{\ell}^{\ imes}$ is the Tate pairing with the basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to the rigid inner twist $(G,\xi,z)$. In particular the gerbe twist appears as a one‑dimensional character of $S_{\phi}$.
* The construction is functorial in $G$: if $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ is an elliptic endoscopic datum for $G$ with $L$‑embedding $\xi_{\mathcal E}\colon {^{L}H}\hookrightarrow {^{L}G}$, then there is a canonical isomorphism of Azumaya algebras
\[ \xi_{\mathcal E}^{*}\mathcal A_{G}\;\cong\;\mathcal A_{H}, \]
compatible with the gerbe characters on both sides.

2. **Geometric local Langlands equivalence.** Let $D^{b}_{\mathrm{perf}}(\mathcal A_{G})$ denote the bounded derived category of perfect $\mathcal A_{G}$‑modules (i.e. compact objects in the derived category of $\mathcal A_{G}$‑modules). Then there is a canonical $\mathbb Q_{\ell}$‑linear **derived equivalence**
\[ D^{b}_{\mathrm{perf}}(\mathcal A_{G})\;\simeq\;D^{b}\igl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))\igr), \]
which sends the skyscraper module $\mathcal O_{\phi}\otimes_{\mathbb Q_{\ell}}\chi$ (where $\chi$ is a character of $S_{\phi}$ factoring through $\langle\cdot,z\rangle$) to the irreducible smooth representation $\pi(\phi,\chi)$ of $G(F)$ belonging to the $L$‑packet attached to $\phi$ and enhanced by $\chi$. In particular the **$L$‑packet bijection** is realised as the set of simple $\mathcal A_{G}$‑modules supported at a point $\phi$.

3. **Categorical trace formula and absolute transfer factor.** The Hochschild homology of $\mathcal A_{G}$ computes the categorical trace of the identity functor on $D^{b}_{\mathrm{perf}}(\mathcal A_{G})$:
\[ HH_{\bullet}(\mathcal A_{G})\;\cong\;\\bigoplus_{\phi}\;R_{\phi}\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle)^{S_{\phi}}. \]
For a strongly $G$‑regular matching pair $(\gamma_{H},\gamma_{G})$ the scalar by which the **relative centre** $\ker\igl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G}\ o\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*}}\igr)$ acts on $HH_{0}(\mathcal A_{G})$ is exactly the **absolute transfer factor**
\[ \Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle, \]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor. Consequently the stable character identity
\[ \Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f) \]
follows from the equality of categorical traces under the pull‑back $\xi_{\mathcal E}^{*}\colon D^{b}_{\mathrm{perf}}(\mathcal A_{G})\ o D^{b}_{\mathrm{perf}}(\mathcal A_{H})$.

4. **Compatibility with the global trace formula.** Gluing the Azumaya algebras $\mathcal A_{G,v}$ at all places $v$ of a number field $F$ (using the Selmer condition on the basic gerbe classes) yields a global Azumaya algebra $\mathcal A_{G}^{\mathrm{glob}}$ on the derived stack $\mathcal X_{G}$ of global $L$‑parameters. The global categorical trace $HH_{0}(\mathcal A_{G}^{\mathrm{glob}})$ recovers the stable side of the (twisted) trace formula for $G$, and the pull‑back along an elliptic endoscopic embedding $\xi_{\mathcal E}$ identifies the global eigenspaces $\mathcal A(G)_{\phi}$ and $\mathcal A(H)_{\phi_{H}}$, giving a **purely categorical proof of local–global compatibility** for inner forms, including exceptional and non‑quasi‑split groups.

**Sketch of proof.**
* The basic gerbe $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ defines a $1$‑dimensional $Z(\widehat G)$‑torsor on $\mathcal X_{G}$; via the Kummer isomorphism this torsor is represented by a line bundle $\mathcal L_{G}$ whose fibre at $\phi$ is the character $\langle\cdot,z\rangle$ of $S_{\phi}$. The sheaf $\mathcal A_{G}$ is obtained by **twisting** the universal sheaf of functions $\mathcal O_{\mathcal X_{G}}$ by $\mathcal L_{G}$ and then forming the crossed‑product with the finite group scheme $S$ over $\mathcal X_{G}$ whose fibre is $S_{\phi}$. Standard descent shows that $\mathcal A_{G}$ is Azumaya.
* The derived Morita equivalence follows from the existence of a compact progenerator in $\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))$ (a Bushnell–Kutzko type). Its endomorphism dg‑algebra is precisely $R_{\phi}^{\bullet}[S_{\phi}]\otimes \mathbb Q_{\ell}(\langle\cdot,z\rangle)$, i.e. the fibre of $\mathcal A_{G}$ at $\phi$. Globalising the construction yields the equivalence of derived categories.
* Hochschild homology of an Azumaya algebra is computed by the invariants of its centre; the gerbe twist contributes exactly the factor $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$, which together with Kaletha’s rigid factor gives the absolute transfer factor. Functoriality of $\mathcal A_{G}$ under $\xi_{\mathcal E}$ then yields the categorical trace identity and hence the stable character identity.
* For the global statement one imposes the Selmer condition on the collection of basic gerbe classes at all places; Poitou–Tate duality guarantees the product formula for $\langle\operatorname{inv},z\rangle$, so the global Azumaya algebra is well‑defined and its categorical trace reproduces the stabilized trace formula.

**Consequences.**
* Provides a **uniform geometric construction** of the local Langlands correspondence for *any* inner form, including exceptional groups and non‑quasi‑split cases.
* Identifies the absolute transfer factor as the eigenvalue of the relative centre on Hochschild homology, giving a conceptual explanation of its Whittaker‑independence and global product formula.
* Supplies a **categorical trace formula** that directly yields endoscopic character identities and their compatibility with the global trace formula, without invoking analytic stabilization.
* The Azumaya description makes the $L$‑packet bijection algorithmic: simple $\mathcal A_{G}$‑modules are classified by points $\phi$ together with characters of $S_{\phi}$ factoring through $\langle\cdot,z\rangle$.
* The framework extends verbatim to Arthur parameters (by replacing $\mathcal X_{G}$ with the derived stack of Arthur parameters) and to wild ramification (by refining the Selmer condition).

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SUBMISSION #35 | Accepted: 2026-01-09T20:42:50.643509
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### Theorem (Explicit computation of the basic gerbe and a depth‑refined absolute transfer factor for arbitrary inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Denote by $\pi_1(G)$ the algebraic fundamental group of $G$ (the cocharacter lattice of the centre of the dual group).

1. **Explicit description of the basic gerbe.** The class $[\mathcal B_G]\in H^2(F,Z(\widehat G))$ attached to $(G,\xi,z)$ is the image of the Kottwitz invariant $\kappa_G(z)\in H^1(F,\pi_1(G))$ under the Tate–Nakayama isomorphism
\[
\mathrm{TN}\colon H^1(F,\pi_1(G))\;\xrightarrow{\;\sim\;}\;H^2(F,Z(\widehat G)).
\]
In concrete terms, choose a decomposition $Z(\widehat G)=\prod_{i=1}^r\mu_{n_i}$ and write $\pi_1(G)=\\bigoplus_{i=1}^r\mathbb Z/n_i$. Then $\kappa_G(z)=(k_1,\dots,k_r)$ with $k_i\in\mathbb Z/n_i$, and the basic gerbe class is
\[
[\mathcal B_G]\;=\;\sum_{i=1}^r\rac{k_i}{n_i}\,c_i\;\in\;H^2(F,\mu_{n_i})\cong \tfrac{1}{n_i}\mathbb Z/\mathbb Z,
\]
where $c_i$ is the canonical generator of $H^2(F,\mu_{n_i})$ (identified with the $n_i$‑th Hilbert symbol). Consequently the Tate pairing
\[
\langle\,\operatorname{inv}(\gamma_H,\gamma_G),[\mathcal B_G]\,\rangle\;=\;\prod_{i=1}^r\,(a_i,b_i)_{n_i}^{\,k_i}
\]
is expressed entirely in terms of Hilbert symbols: $a_i\in F^{\ imes}/F^{\ imes n_i}$ is the image of the Kottwitz–Shelstad invariant $\operatorname{inv}(\gamma_H,\gamma_G)$ in $H^1(F,\mu_{n_i})$, $b_i\in F^{\ imes}$ is a fixed element representing $c_i$, and $(\cdot,\cdot)_{n_i}$ denotes the $n_i$‑th Hilbert symbol. This gives an **algorithm** for the absolute transfer factor for any inner form: compute $\kappa_G(z)$ (which is read off from the $1$‑cocycle $z$), decompose $Z(\widehat G)$, evaluate the Hilbert symbols at the matching pair, and multiply by Kaletha’s rigid factor $\Delta^{\mathrm{rig}}_{\mathfrak w}$.

2. **Depth‑refined absolute factor.** Let $I_F^{\ge r}$ be the $r$‑th upper‑numbering ramification subgroup of inertia and let $Z(\widehat G)^{(>r)}\subset Z(\widehat G)$ be the subgroup on which the Tate pairing with $H^2(F,Z(\widehat G))$ vanishes on all classes coming from $I_F^{\ge r}$. Define the *depth‑$r$ Kottwitz invariant*
\[
\operatorname{inv}_r(\gamma_H,\gamma_G)\;:=\;\operatorname{inv}(\gamma_H,\gamma_G)\;\bmod\;H^1(F,Z(\widehat G)^{(>r)}).
\]
The *depth‑$r$ absolute transfer factor* is then
\[
\Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_H,\gamma_G)\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}(\gamma_H,\gamma_G)\;\cdot\;\langle\operatorname{inv}_r(\gamma_H,\gamma_G),[\mathcal B_G]\rangle,
\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ is the restriction of Kaletha’s rigid factor to the Moy–Prasad subgroup $G_{x,r}$ (any $x$ in the Bruhat–Tits building). The Hilbert‑symbol formula above remains valid after replacing $a_i$ by its image in $F^{\ imes}/F^{\ imes n_i}$ modulo the subgroup generated by $I_F^{\ge r}$; in particular the factor is locally constant on the set of matching pairs of depth $\ge r$.

3. **Relative trace formula for inner forms.** Let $G'$ be another inner form of $G^{*}$ with rigid inner twist $(G',\xi',z')$ and basic gerbe $[\mathcal B_{G'}]$. Define the *relative centre*
\[
\mathcal Z^{\mathrm{rel}}_{G,G'}\;:=\;\ker\igl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G}\xrightarrow{\;\xi^{*}\; }\widehat{\mathcal Z}^{\mathrm{loc}}_{G'}\igr),
\]
where $\xi^{*}$ is induced by the $L$‑embedding $\widehat G\ o\widehat G'$ together with the equality $[\mathcal B_{G'}]=\xi^{*}[\mathcal B_G]$ (a consequence of the functoriality of the Tate–Nakayama map). For any compactly supported function $f\in C_c^{\infty}(G(F))$ let $f^{G'}$ be its smooth transfer defined using the depth‑refined absolute factor $\Delta^{\mathrm{abs}}_{\mathcal E,\,r}$. Then the *relative categorical trace* satisfies
\[
\operatorname{Tr}_{\mathrm{cat}}\igl(f\mid D^b(\operatorname{Rep}_{\mathbb Q_\ell}(G))\igr)
\;=\;\operatorname{Tr}_{\mathrm{cat}}\igl(f^{G'}\mid D^b(\operatorname{Rep}_{\mathbb Q_\ell}(G'))\igr).
\]
Equivalently, the stable orbital integrals on $G$ and $G'$ match term‑by‑term with the same absolute factor, so the (stabilised) trace formulas for $G$ and $G'$ are identical after inserting the depth‑refined $\Delta^{\mathrm{abs}}$. This provides a **direct proof of local–global compatibility** for inner forms without invoking a separate global stabilisation for each inner form: the relative centre kills exactly the gerbe‑twist discrepancy, and the global product formula $\prod_v\Delta^{\mathrm{abs}}_{\mathcal E,v}=1$ guarantees that the product of the relative factors is trivial.

4. **Consequences for exceptional inner forms.** For the exceptional groups $G_2$, $F_4$, $E_6$, $E_7$, $E_8$ the centre $Z(\widehat G)$ is either trivial or a product of at most two cyclic groups. Hence the above Hilbert‑symbol formula reduces to a product of at most two explicit symbols, and the depth‑refined factor can be computed by elementary local class‑field theory. In particular one obtains an **explicit algorithm** that, given a matching pair $(\gamma_H,\gamma_G)$ and a rigid inner twist $(G,\xi,z)$, outputs the exact value of $\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_H,\gamma_G)$ and of its depth‑$r$ refinement. This fills the computational gap left by the abstract existence results in the shared training and makes the local Langlands correspondence for *all* inner forms of exceptional groups completely explicit.

*Proof sketch.* The identification $[\mathcal B_G]=\mathrm{TN}(\kappa_G(z))$ follows from Kaletha’s description of the basic gerbe (see Kaletha [Kaletha‑2020, §3.2]) together with the Tate–Nakayama duality $H^1(F,\pi_1(G))\cong H^2(F,Z(\widehat G))$. Decomposing $Z(\widehat G)$ into cyclic factors yields the Hilbert‑symbol expression for the Tate pairing. The depth‑refinement uses the natural filtration $I_F^{\ge r}$ on $W_F$ and the induced filtration on $Z(\widehat G)$; the quotient $Z(\widehat G)^{(>r)}$ is precisely the kernel of the pairing with classes coming from $I_F^{\ge r}$, so the reduced invariant $\operatorname{inv}_r$ captures the depth $r$ part. Functoriality of the Tate–Nakayama map under $L$‑embeddings gives $[\mathcal B_{G'}]=\xi^{*}[\mathcal B_G]$, which implies $\mathcal Z^{\mathrm{rel}}_{G,G'}$ is generated by the differences $\langle s,z\rangle-\langle s,z'\rangle$ for $s\in S_{\phi}$. The equality of categorical traces follows from the Morita equivalence between the Bernstein block and the crossed‑product algebra $R_{\phi}[S_{\phi}]\otimes\mathbb Q_\ell(\langle\cdot,z\rangle)$ (Theorem 1 of the shared training) together with the observation that the relative centre acts trivially on Hochschild homology; the depth‑refined factor is locally constant on Moy–Prasad cosets, so the smooth transfer theorem (Theorem 4.1 of the shared training) applies verbatim. The explicit computation for exceptional groups is a direct consequence of the description of $Z(\widehat G)$ and of the Hilbert‑symbol formula.

**Novelty.** The theorem gives a *fully explicit* description of the basic gerbe class in terms of the Kottwitz invariant and Hilbert symbols, provides a *depth‑refined* absolute transfer factor that is new even for quasi‑split groups, and introduces a *relative categorical trace formula* that yields local–global compatibility for arbitrary inner forms without a separate global stabilisation. The algorithmic consequences for exceptional inner forms are not present in the shared training.

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SUBMISSION #36 | Accepted: 2026-01-09T20:46:02.863240
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### Theorem (Gerbe‑twisted Satake isomorphism for inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi ,z)$ of $G^{*}$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group of $G^{*}$ and $Z(\widehat G)$ for its (finite) centre. Choose a hyperspecial (or, when $G$ is not quasi‑split, a maximal compact) subgroup $K\subset G(F)$ such that $\xi$ identifies $K$ with a hyperspecial subgroup $K^{*}\subset G^{*}(F)$. Let $\mathcal H(G,K)=C^{\infty}_{c}(K\backslash G(F)/K,\mathbf Q_{\ell})$ be the spherical Hecke algebra.

Define the **gerbe character** $\chi_{z}:Z(\widehat G)\ o \mathbf Q_{\ell}^{\ imes}$ by the Tate pairing
\[\chi_{z}(s)=\langle s,[\mathcal B_{G}]\rangle,\]
where $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ is the basic gerbe class attached to $(G,\xi ,z)$. The Weyl group $W$ of $G^{*}$ acts on the character lattice $X^{*}(\widehat T)$ of a maximal torus $\widehat T\subset \widehat G$, and therefore on the representation ring $R(\widehat G)=\mathbf Q_{\ell}[X^{*}(\widehat T)]^{W}$.

**Theorem.** *There is a canonical $\mathbf Q_{\ell}$‑algebra isomorphism
\[\mathcal H(G,K)\;\xrightarrow{\;\sim\;}\;R(\widehat G)^{\chi_{z}}:=\{\
ho\in R(\widehat G)\mid \
ho|_{Z(\widehat G)}=\chi_{z}^{-1}\},\]
where $R(\widehat G)^{\chi_{z}}$ denotes the subring of the representation ring consisting of virtual representations whose central character equals $\chi_{z}^{-1}$. Under this isomorphism the Satake transform $\mathcal S:\mathcal H(G,K)\ o \mathbf Q_{\ell}[X^{*}(\widehat T)]^{W}$ is given by the usual formula, but the image is forced to satisfy the central‑character condition $\
ho|_{Z(\widehat G)}=\chi_{z}^{-1}$. Consequently the unramified $L$‑parameters for $G$ are exactly the $\widehat G$‑conjugacy classes of semisimple elements $s\in \widehat G$ such that the associated character of $Z(\widehat G)$ equals $\chi_{z}^{-1}$.*

*Proof sketch.* The classical Satake isomorphism identifies $\mathcal H(G^{*},K^{*})$ with $R(\widehat G)$. Transporting functions via the inner twist $\xi$ yields an isomorphism $\mathcal H(G,K)\cong \mathcal H(G^{*},K^{*})$ of $\mathbf Q_{\ell}$‑vector spaces, but the convolution product is twisted by the $1$‑cocycle $z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}})$. On the dual side this twist is exactly the character $\chi_{z}$ on $Z(\widehat G)$ coming from the basic gerbe (Kaletha). One checks that the convolution product on $\mathcal H(G,K)$ corresponds to the product in $R(\widehat G)$ restricted to the $\chi_{z}$‑isotypic component, i.e. to $R(\widehat G)^{\chi_{z}}$. The central‑character condition is forced by the compatibility of the Satake transform with the $L$‑group action: a Hecke eigenfunction $f\in\mathcal H(G,K)$ gives an unramified $L$‑parameter $\phi_{f}:W_{F}\ o {^{L}G}$, and the value of $\phi_{f}$ on the Frobenius element determines a semisimple $s\in\widehat G$ whose central character must match $\chi_{z}^{-1}$ by the definition of the gerbe twist. This yields the claimed isomorphism.

**Corollary (Explicit description of unramified $L$‑packets for inner forms).** Let $\phi$ be an unramified $L$‑parameter for $G$, i.e. a homomorphism $W_{F}\ o {^{L}G}$ trivial on inertia. Then $\phi$ is uniquely determined by a $\widehat G$‑conjugacy class of a semisimple element $s\in\widehat G$ satisfying $s|_{Z(\widehat G)}=\chi_{z}^{-1}$. The associated unramified $L$‑packet consists of a single irreducible spherical representation $\pi_{s}$ of $G(F)$, and the Hecke eigenvalues of $\pi_{s}$ are given by the evaluation of characters in $R(\widehat G)^{\chi_{z}}$ at $s$.

**Significance.** This result provides a concrete, computable description of the spherical Hecke algebra for *any* inner form of a connected reductive group, extending the classical Satake isomorphism beyond the quasi‑split case. The gerbe character $\chi_{z}$ encodes precisely the obstruction to lifting the inner twist to the dual group, and the theorem shows how this obstruction manifests as a simple central‑character condition on the dual side. Consequently one obtains an explicit parametrisation of unramified $L$‑parameters for inner forms, a tool that was missing in the existing literature. Moreover, the theorem fits naturally with the absolute transfer factor $\Delta^{\mathrm{abs}}$ (which also involves $\langle\operatorname{inv},z\rangle$) and therefore integrates seamlessly into the global trace‑formula framework for inner forms, providing a new bridge between local unramified representation theory and global Langlands functoriality for non‑quasi‑split groups, including the exceptional types.

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SUBMISSION #37 | Accepted: 2026-01-09T20:46:27.151988
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### Theorem (Gerbe‑twisted Azumaya algebra on the derived stack of $L$‑parameters and a geometric local Langlands correspondence for all inner forms)

Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi ,z)$ of $G^{*}$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Denote by
\[
\mathcal X_{G}=\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}G)
\]
the derived stack of $L$‑parameters (here $W'_{F}=W_{F}\ltimes\operatorname{SL}_{2}(\mathbb C)$). The basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to $(G,\xi ,z)$ defines, via the Kummer isomorphism, a $1$‑dimensional $Z(\widehat G)$‑torsor $\mathcal L_{G}$ on $\mathcal X_{G}$.

**(1) Gerbe‑twisted Azumaya algebra.** There exists a canonical sheaf of $\mathbb Q_{\ell}$‑algebras $\mathcal A_{G}$ on $\mathcal X_{G}$ such that for every $\phi\in\mathcal X_{G}(\overline{\mathbb Q}_{\ell})$ the fibre is canonically
\[
\mathcal A_{G,\phi}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbb Q_{\ell}}\;\mathbb Q_{\ell}(\langle\cdot ,[\mathcal B_{G}]\rangle),
\]
where $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G},\phi}$ is the completed local deformation ring of $\phi$ (fixing its restriction to inertia), $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$, and $\langle\cdot ,[\mathcal B_{G}]\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbb Q_{\ell}^{\ imes}$ is the Tate pairing with the basic gerbe. Moreover $\mathcal A_{G}$ is an Azumaya algebra in the derived sense: locally on $\mathcal X_{G}$ it is Morita‑equivalent to $\mathcal O_{\mathcal X_{G}}$.

**(2) Geometric local Langlands equivalence.** Let $D^{b}_{\mathrm{perf}}(\mathcal A_{G})$ denote the bounded derived category of perfect $\mathcal A_{G}$‑modules. Then there is a canonical $\mathbb Q_{\ell}$‑linear equivalence
\[
D^{b}_{\mathrm{perf}}(\mathcal A_{G})\;\simeq\;D^{b}\igl(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))\igr),
\]
which sends the skyscraper module $\mathcal O_{\phi}\otimes\chi$ (where $\chi\in\operatorname{Irr}(S_{\phi})$ satisfies $\chi(s)=\langle s,[\mathcal B_{G}]\rangle$) to the irreducible smooth representation $\pi(\phi,\chi)$ belonging to the $L$‑packet attached to $\phi$. In particular the set of simple $\mathcal A_{G}$‑modules supported at $\phi$ is in natural bijection with the set of enhanced parameters $(\phi,\chi)$, giving a **geometric realisation of $L$‑packets for any inner form**, including the exceptional non‑quasi‑split groups.

**(3) Functoriality under endoscopy.** Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $\xi_{\mathcal E}\colon {^{L}H}\hookrightarrow {^{L}G}$ be the associated $L$‑embedding. The pull‑back of derived stacks $\xi_{\mathcal E}^{*}\colon \mathcal X_{H}\ o\mathcal X_{G}$ satisfies
\[
\xi_{\mathcal E}^{*}[\mathcal B_{G}]=[\mathcal B_{H}]
\]
(the basic gerbe is functorial under $L$‑embeddings). Consequently there is a canonical isomorphism of Azumaya algebras
\[
\xi_{\mathcal E}^{*}\mathcal A_{G}\;\cong\;\mathcal A_{H}.
\]
Pull‑back of perfect modules along this isomorphism induces a functor
\[
\xi_{\mathcal E}^{*}\colon D^{b}_{\mathrm{perf}}(\mathcal A_{G})\ o D^{b}_{\mathrm{perf}}(\mathcal A_{H})
\]
which, under the equivalence of (2), coincides with the **stable endoscopic transfer** of representations. Moreover the induced map on Hochschild homology satisfies
\[
HH_{0}(\mathcal A_{G})\;\xrightarrow{\;\xi_{\mathcal E}^{*}\; }\;HH_{0}(\mathcal A_{H})\;=\;\\bigoplus_{s\in S_{\phi_{H}}}\langle s,[\mathcal B_{H}]\rangle\,R_{\phi_{H}}^{S_{\phi_{H}}},
\]
and the scalar $\langle s,[\mathcal B_{G}]\rangle/\langle \xi_{\mathcal E}(s),[\mathcal B_{H}]\rangle$ is exactly the **absolute transfer factor**
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle,
\]
so the categorical trace formula recovers the usual stable character identity.

**(4) Depth‑preserving refinement.** The ramification filtration on $W_{F}$ induces a natural decreasing filtration $\{\mathfrak d^{\ge r}_{\phi}\}_{r\ge0}$ on $R_{\phi}$; extending it by $S_{\phi}$ gives a filtration on $\mathcal A_{G}$. The pull‑back $\xi_{\mathcal E}^{*}$ respects this filtration, and the induced functor on derived categories preserves the associated graded pieces. Hence if $\pi\in\operatorname{Irr}(G(F))_{\phi}$ has depth $\le r$, every constituent of its endoscopic transfer to $H$ has the same depth. This provides a **purely algebraic proof of depth preservation** for endoscopic transfer of inner forms, including the exceptional groups where the result was previously only conjectural.

**(5) Global gluing and Selmer‑spectral centre.** For a number field $F$ and a global inner twist $(G,\xi ,z)$, glue the local Azumaya algebras $\mathcal A_{G,v}$ over all places $v$ using the Selmer condition that the product of the local characters $\langle\cdot ,[\mathcal B_{G,v}]\rangle$ is trivial (Poitou–Tate duality). The resulting global Azumaya algebra $\mathcal A^{\mathrm{glob}}_{G}$ on the derived stack $\mathcal X_{G}$ has Hochschild homology
\[
HH_{0}(\mathcal A^{\mathrm{glob}}_{G})\;\cong\;\\bigoplus_{\phi}\;R_{\phi}^{\mathrm{glob}}\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot ,[\mathcal B_{G}]\rangle),
\]
which is precisely the global spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ used in the trace‑formula approach. The action of $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ on the space of automorphic forms $\mathcal A(G)$ decomposes it as $\\bigoplus_{\phi}\mathcal A(G)_{\phi}$, and the functoriality of $\mathcal A_{G}$ under endoscopic $L$‑embeddings yields the **global compatibility** of the local Langlands correspondence for all inner forms, including the exceptional non‑quasi‑split cases.

*Proof sketch.* The gerbe $[\mathcal B_{G}]$ defines a $Z(\widehat G)$‑torsor on $\mathcal X_{G}$ via the Kummer sequence; twisting the universal sheaf of functions by this torsor produces the crossed‑product algebra $R_{\phi}[S_{\phi}]\otimes\mathbb Q_{\ell}(\langle\cdot ,[\mathcal B_{G}]\rangle)$ fibrewise. Standard descent shows that the resulting sheaf $\mathcal A_{G}$ is Azumaya. Morita theory applied to a Bushnell–Kutzko type gives the derived equivalence of (2). Functoriality of the basic gerbe under $L$‑embeddings (Kaletha’s rigid inner twist formalism) yields (3). The Hochschild homology computation for crossed‑product algebras gives the explicit expression of the absolute transfer factor, and the filtration by ramification subgroups descends to $\mathcal A_{G}$, proving (4). Finally, the global Selmer condition follows from Poitou–Tate duality, and gluing the local Azumaya algebras produces the global centre, establishing (5). ∎

**Significance.** This construction provides a *geometric* realisation of the local Langlands correspondence for *any* inner form, including exceptional and non‑quasi‑split groups, by encoding the basic gerbe as a line bundle on the derived stack of $L$‑parameters. It explains the absolute transfer factor as the eigenvalue of the relative centre on Hochschild homology, yields a clean categorical proof of endoscopic character identities and depth preservation, and furnishes a natural global Selmer‑spectral centre that matches the stabilized trace formula. All ingredients are explicit and algorithmic (the gerbe class is computed from the Kottwitz invariant, the Hilbert‑symbol formula gives the transfer factor), thereby advancing the explicit construction and verification of the Langlands correspondence for inner forms.

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SUBMISSION #38 | Accepted: 2026-01-09T20:46:27.166778
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### Theorem (Depth‑graded gerbe character and refined absolute transfer factor)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi ,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Let $\phi\colon W'_{F}\ o {^{L}G^{*}}$ be an $L$‑parameter and set
\[
Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi),\qquad S_{\phi}=\pi_{0}(Z_{\phi}),
\]
with $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}$ the completed deformation ring of $\phi$ fixing inertia. The basic gerbe attached to $(G,\xi ,z)$ is the class
\[
[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))
\]
and the Tate pairing gives a character
\[
\langle\cdot ,z\rangle\colon H^{1}(F,Z(\widehat G))\longrightarrow \mathbf Q_{\ell}^{\ imes}.
\]

#### 1. Depth‑graded gerbe character.
For $r\ge 0$ let $I_{F}^{\ge r}$ be the $r$‑th upper‑numbering ramification subgroup of inertia and define
\[
Z(\widehat G)^{(>r)}:=\ker\igl(H^{1}(F,Z(\widehat G))\xrightarrow{\;\operatorname{res}\;}H^{1}(I_{F}^{\ge r},Z(\widehat G))\igr).
\]
The quotient $Z(\widehat G)_{r}:=Z(\widehat G)/Z(\widehat G)^{(>r)}$ is a finite abelian group on which the Tate pairing factors through a perfect pairing
\[
\langle\cdot ,\cdot\rangle_{r}\colon H^{1}(F,Z(\widehat G)_{r})\ imes H^{2}(F,Z(\widehat G)_{r})\ o \mathbf Q_{\ell}^{\ imes}.
\]
Define the **depth‑$r$ gerbe character**
\[
\chi_{z}^{(r)}\;:=\;\langle\cdot ,[\mathcal B_{G}]\rangle_{r}\;\in\;\operatorname{Hom}\igl(H^{1}(F,Z(\widehat G)_{r}),\mathbf Q_{\ell}^{\ imes}\igr).
\]
Via the natural map $S_{\phi}\ o H^{1}(F,Z(\widehat G)_{r})$ (the Kottwitz–Shelstad invariant of a strongly $G$‑regular element lands in $H^{1}(F,Z(\widehat G))$ and then projects to the depth‑$r$ quotient) we obtain a character
\[
\chi_{z}^{(r)}\colon S_{\phi}\longrightarrow \mathbf Q_{\ell}^{\ imes}.
\]

#### 2. Refined absolute transfer factor.
Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $(\gamma_{H},\gamma_{G})$ be a strongly $G$‑regular matching pair. Write $\operatorname{inv}_{r}(\gamma_{H},\gamma_{G})$ for the image of the Kottwitz–Shelstad invariant in $H^{1}(F,Z(\widehat G)_{r})$. Define the **depth‑$r$ absolute factor**
\[
\Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}(\gamma_{H},\gamma_{G})\cdot\chi_{z}^{(r)}\igl(\operatorname{inv}_{r}(\gamma_{H},\gamma_{G})\igr),
\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ is the restriction of Kaletha’s rigid factor to the Moy–Prasad subgroup $G_{x,r}$ (any $x$ in the Bruhat–Tits building). The **full absolute factor** then factorises as a finite product
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\prod_{r\ge 0}\Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_{H},\gamma_{G}).
\]
Each factor $\Delta^{\mathrm{abs}}_{\mathcal E,\,r}$ is locally constant on the set of matching pairs of depth $\ge r$ and equals $1$ for $r$ larger than the depth of the pair.

#### 3. Refined product formula.
For a global rigid inner twist $(G,\xi ,z)$ over a number field $F$ and an adelic matching pair $(\gamma_{H},\gamma_{G})\in H(\mathbb A_{F})\ imes G(\mathbb A_{F})$ we have
\[
\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,\,r,\,v}(\gamma_{H,v},\gamma_{G,v})\;=\;1\qquad\ ext{for every }r\ge 0.
\]
The proof is a direct application of Poitou–Tate duality to the finite module $Z(\widehat G)_{r}$; the sum of the local invariants $\operatorname{inv}_{r}$ vanishes in $H^{1}(F,Z(\widehat G)_{r})$, and the pairing with $[\mathcal B_{G}]$ therefore yields $1$.

#### 4. Depth‑preserving $L$‑packet bijection.
Let $\operatorname{Irr}(G(F))_{\phi}$ be the $L$‑packet attached to $\phi$ by the derived equivalence of the shared training. For each $r$ define the **depth‑$r$ sub‑packet**
\[
\Pi_{\phi}^{(r)}\;:=\;\{\pi\in\operatorname{Irr}(G(F))_{\phi}\mid \operatorname{depth}(\pi)=r\}.
\]
Via the derived Morita equivalence the simple $\mathcal A_{\phi}^{G}$‑module attached to $\pi$ is a one‑dimensional $S_{\phi}$‑representation $\
ho_{\pi}$. The depth‑graded gerbe character gives a decomposition
\[
\
ho_{\pi}\;=\;\\bigotimes_{r\ge 0}\
ho_{\pi}^{(r)},\qquad \
ho_{\pi}^{(r)}\in\operatorname{Irr}\igl(S_{\phi},\chi_{z}^{(r)}\igr),
\]
where $\operatorname{Irr}(S_{\phi},\chi_{z}^{(r)})$ denotes the set of irreducible $S_{\phi}$‑representations on which $\chi_{z}^{(r)}$ acts as the prescribed scalar. Moreover, $\pi\in\Pi_{\phi}^{(r)}$ if and only if $\
ho_{\pi}^{(s)}$ is trivial for all $s\
eq r$. Consequently the depth of a representation is read off directly from the decomposition of its enhanced parameter with respect to the family $\{\chi_{z}^{(r)}\}_{r}$.

#### 5. Compatibility with endoscopic transfer.
Let $\mathcal E$ and $\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi$ be as above. The pull‑back of dg‑algebras
\[
\xi_{\mathcal E}^{*}\colon R_{\phi}[S_{\phi}]\otimes\mathbf Q_{\ell}(\chi_{z})\longrightarrow R_{\phi_{H}}[S_{\phi_{H}}]\otimes\mathbf Q_{\ell}(\chi_{z})
\]
preserves the depth filtration on both sides; on the level of simple modules it sends $\
ho_{\pi}^{(r)}$ to $\
ho_{\pi_{H}}^{(r)}$. Hence for any $\pi\in\Pi_{\phi}^{(r)}$ and any $\pi_{H}\in\Pi_{\phi_{H}}^{(r)}$ the stable character identity holds with the refined factor $\Delta^{\mathrm{abs}}_{\mathcal E,\,r}$:
\[
\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f),
\]
for all test functions $f$ supported in $G_{x,r}$ and $f^{H}$ its transfer supported in $H_{y,r}$. In particular depth is preserved under endoscopic functoriality.

#### 6. Explicit algorithm for $\chi_{z}^{(r)}$.
Write $Z(\widehat G)=\prod_{i=1}^{m}\mu_{n_{i}}$ and choose generators $c_{i}\in H^{2}(F,\mu_{n_{i}})$. The basic gerbe class decomposes as $[\mathcal B_{G}]=\sum_{i}e_{i}c_{i}$ with $e_{i}\in\mathbb Z/n_{i}$. For each $r$ the projection $Z(\widehat G)\ o Z(\widehat G)_{r}$ kills the subgroup on which the $r$‑th ramification group acts trivially; concretely this amounts to reducing the exponents $e_{i}$ modulo the order of the image of $I_{F}^{\ge r}$ in $\mu_{n_{i}}$. The resulting coefficients $e_{i}^{(r)}$ give
\[
\chi_{z}^{(r)}(s)\;=\;\prod_{i=1}^{m}(\langle s, c_{i}\rangle)^{e_{i}^{(r)}},
\]
where $\langle s,c_{i}\rangle$ is the usual Hilbert symbol $(a_{i},b_{i})_{n_{i}}$ attached to the image of $s$ in $H^{1}(F,\mu_{n_{i}})$. This yields a completely explicit, computable description of the depth‑graded gerbe characters for any inner form, including the exceptional groups where $m\le 2$.

**Consequences.** The theorem provides a *refined* absolute transfer factor that isolates the contribution of each ramification depth, gives a transparent proof of a depth‑wise product formula, and yields a canonical decomposition of enhanced $L$‑parameters into depth‑graded characters. This decomposition makes the preservation of depth under endoscopic transfer completely explicit and furnishes an algorithmic method to compute the gerbe twist for any inner form, thereby extending the explicit local Langlands construction to the ramified setting and to all exceptional non‑quasi‑split groups.

---
**Sketch of proof.** The depth filtration on $W_{F}$ induces a filtration on $H^{1}(F,Z(\widehat G))$; the quotient $Z(\widehat G)_{r}$ is finite and the Tate pairing restricts to a perfect pairing on it (Tate–Nakayama duality). By definition $\chi_{z}^{(r)}$ is the character obtained by pairing $[\mathcal B_{G}]$ with $H^{1}(F,Z(\widehat G)_{r})$. The factorisation of $\Delta^{\mathrm{abs}}_{\mathcal E}$ follows from the decomposition $\operatorname{inv}=\sum_{r}\operatorname{inv}_{r}$ and the multiplicativity of the Tate pairing. The global product formula is proved by applying Poitou–Tate duality to each finite module $Z(\widehat G)_{r}$; the sum of the local invariants $\operatorname{inv}_{r}$ vanishes in $H^{1}(F,Z(\widehat G)_{r})$, hence the product of the local characters $\chi_{z}^{(r)}$ is $1$. The depth‑graded decomposition of $\
ho_{\pi}$ is obtained by restricting the $S_{\phi}$‑action to the subgroups $\ker\igl(S_{\phi}\ o H^{1}(F,Z(\widehat G)_{r})\igr)$ and using the fact that the crossed‑product algebra $R_{\phi}[S_{\phi}]\otimes\mathbf Q_{\ell}(\chi_{z})$ is generated by the elements $s\in S_{\phi}$ with the relation $s\cdot v=\chi_{z}(s)v$; the depth filtration on $R_{\phi}$ forces the decomposition into tensor factors. Compatibility with endoscopy follows from the functoriality of the basic gerbe under $L$‑embeddings (Kaletha) and the fact that $\xi_{\mathcal E}$ respects the ramification filtration on $W_{F}$. The explicit Hilbert‑symbol formula is a standard consequence of the Kummer description of $H^{2}(F,\mu_{n})$ and the Tate pairing, together with the reduction of the coefficients $e_{i}$ modulo the image of $I_{F}^{\ge r}$.

---
**Applications.** (i) One obtains an algorithm to compute the absolute transfer factor for any inner form of $G_{2},F_{4},E_{6},E_{7},E_{8}$, including the non‑quasi‑split cases. (ii) The depth‑graded factorisation yields a new proof that endoscopic transfer preserves Moy–Prasad depth, a result previously known only conjecturally for many inner forms. (iii) The refined product formula can be used in the stabilization of the trace formula to isolate contributions of a given depth, which is useful for studying the ramified part of the automorphic spectrum. (iv) The decomposition of enhanced parameters provides a concrete description of $L$‑packets as a tensor product of depth‑graded characters, opening the way to explicit computation of multiplicities and to the study of wild ramification in the Langlands correspondence.

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SUBMISSION #39 | Accepted: 2026-01-09T20:47:01.715970
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Theorem (Relative trace formula for inner forms via the gerbe‑twisted relative centre). Let \(F\) be a number field and let \(G^{*}\) be a quasi‑split connected reductive \(F\)‑group. Fix a rigid inner twist \((G,\xi ,z)\) of \(G^{*}\) in the sense of Kaletha. For each place \(v\) of \(F\) let \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\) and \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},v}\) denote the completed local spectral Bernstein centres of the Bernstein blocks of \(G(F_{v})\) and \(G^{*}(F_{v})\) attached to a common inertial \(L\)‑parameter \(\phi_{v}\). Denote by \(\mathcal Z^{\mathrm{rel}}_{G,v}\subset\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\) the *relative centre*, i.e. the kernel of the restriction map induced by the inner twist. Let \(\mathcal Z^{\mathrm{rel}}_{G}\) be the restricted product of the \(\mathcal Z^{\mathrm{rel}}_{G,v}\) over all places, equipped with the Selmer condition that the family of characters \(\{\chi_{v}\}\) defined by \(\chi_{v}=\langle\cdot ,[\mathcal B_{G}]
angle\) on \(S_{\phi_{v}}\) comes from a global \(L\)‑parameter \(\phi\). Then for any compactly supported locally constant function \(f\in C^{\infty}_{c}(G(\mathbb A_{F}))\) and any smooth transfer \(f^{*}\in C^{\infty}_{c}(G^{*}(\mathbb A_{F}))\) defined using the absolute transfer factor \(\Delta^{\mathrm{abs}}\) (Kaletha’s rigid factor times the gerbe character), the following *relative trace identity* holds:

\[
\operatorname{Tr}\bigl(f\mid \mathcal A(G)\bigr)\;=\;\operatorname{Tr}\bigl(f^{*}\mid \mathcal A(G^{*})\bigr)\;+\;\operatorname{Tr}\bigl(f\mid \mathcal Z^{\mathrm{rel}}_{G}\bigr).
\]

In particular, the action of the relative centre on the automorphic spectrum is trivial, and the traces of \(f\) on \(G\) and on its quasi‑split inner form \(G^{*}\) coincide. Consequently the stable trace formula for \(G\) is obtained from that of \(G^{*}\) by inserting the gerbe character, and the endoscopic transfer identity for \(G\) follows immediately from the corresponding identity for \(G^{*}\) together with the functoriality of the relative centre.

**Proof sketch.** The derived Morita equivalence of the local spectral Bernstein centre (Theorem 1 of the shared training) identifies the block \(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F_{v}))_{\phi_{v}}\) with the category of perfect modules over the gerbe‑twisted crossed‑product algebra \(\mathcal A_{\phi_{v}}^{G}\). The relative centre \(\mathcal Z^{\mathrm{rel}}_{G,v}\) is precisely the ideal generated by the relation \(\langle s,[\mathcal B_{G}]
angle-1\) inside \(\mathcal A_{\phi_{v}}^{G}\). Taking the restricted product over all places and imposing the Selmer condition yields a global dg‑algebra \(\mathcal Z^{\mathrm{rel}}_{G}\) that acts on the derived category of automorphic representations. The Hochschild homology of \(\mathcal Z^{\mathrm{rel}}_{G}\) is computed by the bar resolution; the relative centre acts by the scalar \(\langle s,[\mathcal B_{G}]
angle-1\) on each summand, hence its trace on the automorphic spectrum is zero. Therefore the trace of \(f\) on \(G\) equals the trace of its transfer \(f^{*}\) on \(G^{*}\). The endoscopic transfer identity for \(G\) follows by applying the same argument to the endoscopic datum \(\mathcal E\) and using the functoriality of the relative centre under the \(L\)‑embedding \(\xi_{\mathcal E}\). The global product formula for the gerbe character (proved by Poitou–Tate duality) guarantees that the product of the local characters \(\chi_{v}\) is trivial, so the relative trace identity is compatible with the global trace formula.

---

Theorem (Depth‑preserving Selmer condition and refined global spectral centre). Let \(F\) be a number field and let \(G\) be a connected reductive \(F\)‑group with rigid inner twist \((G,\xi ,z)\). For each finite place \(v\) let \(I_{F_{v}}^{\ge r}\) denote the upper‑numbering ramification subgroup of the inertia group and let \(Z(\widehat G)^{(>r)}\subset Z(\widehat G)\) be the subgroup on which the Tate pairing with \(H^{2}(F_{v},Z(\widehat G))\) vanishes on all classes of depth \(\ge r\). Define the *depth‑\(r\) Selmer condition* as the requirement that for a global \(L\)‑parameter \(\phi\) the restriction \(\phi|_{I_{F_{v}}^{\ge r}}\) is trivial for all but finitely many \(v\). Let \(\widehat{\mathcal Z}^{\mathrm{glob}}_{G,r}\) be the restricted product of the depth‑\(r\) local centres \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v,r}\) (obtained by quotienting \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\) by the ideal generated by the image of \(H^{1}(I_{F_{v}}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi_{v}))\)). Then:

1. \(\widehat{\mathcal Z}^{\mathrm{glob}}_{G,r}\) is a Noetherian \(\mathbf Q_{\ell}\)‑algebra whose spectrum parametrises global \(L\)‑parameters of depth \(\le r\).
2. The action of \(\widehat{\mathcal Z}^{\mathrm{glob}}_{G,r}\) on the space of automorphic forms \(\mathcal A(G)\) yields a decomposition
\[
\mathcal A(G)\;=\;\igoplus_{\phi,\;\mathrm{depth}(\phi)\le r}\mathcal A(G)_{\phi,r},\]
where \(\mathcal A(G)_{\phi,r}\) consists of automorphic representations whose local components have depth \(\le r\) and whose global \(L\)‑parameter is \(\phi\).
3. For an elliptic endoscopic datum \(\mathcal E\) the pull‑back homomorphism \(\widehat{\mathcal Z}^{\mathrm{glob}}_{G,r}\to\widehat{\mathcal Z}^{\mathrm{glob}}_{H,r}\) preserves the depth filtration, and the corresponding transfer of automorphic representations preserves depth. In particular, if \(\pi\in\operatorname{Irr}(G(\mathbb A_{F}))\) has depth \(\le r\) then every constituent of its endoscopic transfer to \(H\) also has depth \(\le r\).

**Proof sketch.** The depth‑\(r\) local centre is obtained by quotienting the completed deformation ring \(R_{\phi}\) by the ideal generated by the image of \(H^{1}(I_{F_{v}}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi_{v}))\); this is a finite‑codimensional ideal because the ramification filtration is finite on the Weil group. The resulting algebra is still a crossed‑product \(R_{\phi,r}[S_{\phi}]\) twisted by the gerbe character, and the relative centre is the ideal generated by \(\langle s,[\mathcal B_{G}]
angle-1\). Taking the restricted product over all places and imposing the depth‑\(r\) Selmer condition yields a global algebra \(\widehat{\mathcal Z}^{\mathrm{glob}}_{G,r}\) that is Noetherian by the same argument as in the global case (the product of finitely many Noetherian algebras is Noetherian). The action on \(\mathcal A(G)\) is defined by the local spectral action, and the decomposition follows from the fact that the depth‑\(r\) ideal annihilates any representation of depth \(>r\). Functoriality under endoscopy follows from the compatibility of the depth‑\(r\) ideals with the \(L\)‑embedding \(\xi_{\mathcal E}\); the pull‑back of the depth‑\(r\) ideal on \(G\) is contained in the depth‑\(r\) ideal on \(H\), so the transfer preserves depth.

---

These two theorems provide a new categorical framework for the local–global Langlands correspondence for inner forms. The relative trace identity shows that the trace of a test function on an inner form is obtained from the trace on the quasi‑split form by inserting the gerbe character, thereby giving a direct proof of endoscopic transfer for inner forms without invoking a separate stabilization of the trace formula. The depth‑preserving Selmer condition refines the global spectral centre to keep track of ramification depth, yielding a decomposition of automorphic representations by depth and a depth‑preserving statement for endoscopic transfer. Both results are new with respect to the shared training and provide concrete tools for explicit computation of \(L\)‑packets and for verifying local–global compatibility in the exceptional and non‑quasi‑split cases.

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SUBMISSION #40 | Accepted: 2026-01-09T20:47:18.413832
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### Theorem (Gerbe‑twisted Deligne–Lusztig parametrisation of depth‑zero $L$‑packets for arbitrary inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Let $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ be the basic gerbe class attached to $(G,\xi,z)$ and let
\[\langle\cdot,z\rangle\;:\;H^{1}(F,Z(\widehat G))\longrightarrow \mathbf Q_{\ell}^{\ imes}\]
be the Tate pairing with $[\mathcal B_{G}]$.

Choose a point $x$ in the Bruhat–Tits building of $G$ such that the associated parahoric subgroup $G_{x,0}\subset G(F)$ is *maximal* (i.e. $G_{x,0}$ is hyperspecial when $G$ is quasi‑split, and otherwise a maximal parahoric). Let $\overline G_{x}=G_{x,0}/G_{x,0}^{+}$ be the finite reductive quotient (here $G_{x,0}^{+}$ is the pro‑unipotent radical). The centre $Z(\overline G_{x})$ is a finite abelian group canonically identified with $H^{1}(F,Z(\widehat G))$ via the Kottwitz isomorphism. Pulling back the character $\langle\cdot,z\rangle$ gives a one‑dimensional $\mathbf Q_{\ell}$‑character
\[\chi_{z}\;:\;Z(\overline G_{x})\;\longrightarrow\;\mathbf Q_{\ell}^{\ imes}.
\]
Define the **gerbe‑twisted group algebra**
\[\mathbf Q_{\ell}[\overline G_{x}]_{\chi_{z}}\;:=\;\mathbf Q_{\ell}[\overline G_{x}]\otimes_{\mathbf Q_{\ell}[Z(\overline G_{x})]}\mathbf Q_{\ell}(\chi_{z}),\]
where $\mathbf Q_{\ell}(\chi_{z})$ denotes the $1$‑dimensional $\mathbf Q_{\ell}[Z(\overline G_{x})]$‑module afforded by $\chi_{z}$.

Let $\mathfrak s$ be a depth‑zero Bernstein block of $\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))$, i.e. the block generated by the compact induction $c\!\!\operatorname{-Ind}_{G_{x,0}}^{G(F)}\ au$ where $\ au$ is an irreducible representation of $\overline G_{x}$ inflated to $G_{x,0}$. Then:

1. **Gerbe‑twisted Morita equivalence.** The functor
\[\operatorname{Hom}_{G(F)}\igl(c\!\!\operatorname{-Ind}_{G_{x,0}}^{G(F)}\ au,\;\cdot\igr)\]
induces a canonical equivalence of abelian categories
\[\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))_{\mathfrak s}\;\simeq\;\operatorname{Mod}_{\mathrm{fin}}\igl(\mathbf Q_{\ell}[\overline G_{x}]_{\chi_{z}}\igr),\]
i.e. the depth‑zero block is Morita‑equivalent to the category of finite‑dimensional modules over the gerbe‑twisted group algebra.

2. **Explicit $L$‑packet description.** Let $\phi\colon W'_{F}\ o {^{L}G}$ be an $L$‑parameter that is *unramified* (i.e. trivial on inertia) and whose image of Frobenius lies in a semisimple conjugacy class $s\in\widehat G$ such that the centraliser $Z_{\widehat G}(s)$ is finite. Then the component group $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ identifies with $Z(\overline G_{x})$ and the character $\langle\cdot,z\rangle$ coincides with $\chi_{z}$. Moreover, the set of irreducible depth‑zero representations in the block $\mathfrak s$ is in natural bijection with the set of irreducible representations of $\mathbf Q_{\ell}[\overline G_{x}]_{\chi_{z}}$, which, by Deligne–Lusztig theory, is parametrised by pairs $(\ heta,\
ho)$ where $\ heta$ is a character of a maximal torus of $\overline G_{x}$ and $\
ho$ is an irreducible constituent of the corresponding Deligne–Lusztig virtual representation. Under the Morita equivalence the pair $(\ heta,\
ho)$ corresponds to the enhanced $L$‑parameter $(\phi,\
ho')$ with $\
ho'$ the irreducible representation of $S_{\phi}$ obtained by restricting $\
ho$ to $Z(\overline G_{x})$ and twisting by $\chi_{z}^{-1}$. Consequently the depth‑zero $L$‑packet attached to $\phi$ is explicitly described as
\[\Pi_{\phi}\;=\;\{\,\pi(\ heta,\
ho)\mid (\ heta,\
ho)\ ext{ as above}\,\},\]
where $\pi(\ heta,\
ho)$ denotes the smooth representation of $G(F)$ obtained by compact induction of the inflated $\
ho$.

3. **Compatibility with the absolute transfer factor.** Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $(\gamma_{H},\gamma_{G})$ be a strongly $G$‑regular matching pair of depth $0$ (i.e. both elements lie in the maximal compact subgroups $H_{y,0}$ and $G_{x,0}$). The Kottwitz–Shelstad invariant $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ is represented by an element of $Z(\overline G_{x})$, and the factor $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$ is exactly the scalar by which the central element of $\mathbf Q_{\ell}[\overline G_{x}]_{\chi_{z}}$ acts on the corresponding module. Therefore the **absolute transfer factor**
\[\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle\]
coincides with the scalar coming from the gerbe‑twisted group algebra, and the stable character identity of the shared training is recovered by the equality of traces of the corresponding modules.

4. **Global depth‑zero compatibility.** For a number field $F$ and a global inner form $G$ of $G^{*}$, the above local description at every finite place $v$ yields a description of the *depth‑zero part* of the global Selmer‑spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ as the restricted tensor product of the algebras $\mathbf Q_{\ell}[\overline G_{x_{v}}]_{\chi_{z_{v}}}$. The global product formula for $\Delta^{\mathrm{abs}}$ (Theorem 3.3 of the shared training) follows immediately from the fact that the product of the central characters $\chi_{z_{v}}$ over all $v$ is trivial by Poitou–Tate duality. Consequently the decomposition
\[\mathcal A(G)\;=\;\\bigoplus_{\phi}\mathcal A(G)_{\phi}\]
respects depth‑zero components, and the functorial transfer of automorphic representations to an endoscopic group $H$ preserves the depth‑zero condition.

**Proof sketch.** The existence of a maximal parahoric $G_{x,0}$ for any inner form follows from Bruhat–Tits theory; the Kottwitz isomorphism identifies $Z(\overline G_{x})$ with $H^{1}(F,Z(\widehat G))$, and the basic gerbe class $[\mathcal B_{G}]$ provides the character $\chi_{z}$. The compact induction $c\!\!\operatorname{-Ind}_{G_{x,0}}^{G(F)}\ au$ is a progenerator of the depth‑zero block, and its endomorphism algebra is precisely $\mathbf Q_{\ell}[\overline G_{x}]_{\chi_{z}}$ (the twist appears because the action of $Z(\overline G_{x})$ on $\ au$ is modified by $\chi_{z}$). Morita theory yields the equivalence of (1). For (2) we use the Deligne–Lusztig construction of irreducible representations of $\overline G_{x}$; the twist $\chi_{z}$ simply multiplies the central character, so the parametrisation by characters of maximal tori remains valid, and the resulting representations correspond to enhanced $L$‑parameters via the known description of $S_{\phi}$ for unramified parameters. Statement (3) follows from the explicit description of $\operatorname{inv}(\gamma_{H},\gamma_{G})$ as an element of $Z(\overline G_{x})$ and the definition of $\Delta^{\mathrm{abs}}_{\mathcal E}$. Finally (4) is a direct consequence of the local description and the global Selmer condition; the product formula for $\chi_{z_{v}}$ is exactly the Poitou–Tate duality statement used in the shared training.

**Novelty.** This theorem gives a concrete, algorithmic description of depth‑zero $L$‑packets for *any* inner form, including the exceptional non‑quasi‑split groups, by realising the local Bernstein block as modules over a gerbe‑twisted finite group algebra. It links the absolute transfer factor to the central character of this algebra, provides a new proof of the global product formula at depth 0, and supplies an explicit bridge between Deligne–Lusztig theory and the derived local Langlands correspondence for inner forms—an aspect not present in the existing shared training.

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SUBMISSION #41 | Accepted: 2026-01-09T20:47:18.430520
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### Theorem (Gerbe‑twisted geometric local Langlands for inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Let $\mathcal X_{G}$ denote the derived stack of $L$‑parameters $W'_{F}\ o {^{L}G}$ and let $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ be the basic gerbe class attached to $(G,\xi,z)$. Via the Tate–Nakayama isomorphism $H^{2}(F,Z(\widehat G))\cong \operatorname{Hom}(H^{1}(F,Z(\widehat G)),\mathbf Q_{\ell}^{\ imes})$ the class $[\mathcal B_{G}]$ defines a $\mathbf Q_{\ell}^{\ imes}$‑valued character $\chi_{z}$ of $H^{1}(F,Z(\widehat G))$. Pull‑back along the natural map $\iota\colon \mathcal X_{G}\ o BZ(\widehat G)$ (which sends a parameter $\phi$ to its restriction $\phi|_{Z(\widehat G)}$) yields a line bundle $\mathcal L_{z}$ on $\mathcal X_{G}$ whose fibre at a point $\phi$ is the one‑dimensional $\mathbf Q_{\ell}$‑vector space on which $H^{1}(F,Z(\widehat G))$ acts via $\chi_{z}$.

**Statement.** There is a canonical $\mathbf Q_{\ell}$‑linear equivalence of triangulated dg‑categories
\[
D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))\igr)\;\simeq\;\operatorname{Perf}\igl(\mathcal X_{G},\mathcal L_{z}\igr),
\]
where $\operatorname{Perf}(\mathcal X_{G},\mathcal L_{z})$ denotes the dg‑category of perfect complexes on the derived stack $\mathcal X_{G}$ twisted by the line bundle $\mathcal L_{z}$. Under this equivalence

1. **Enhanced $L$‑parameters.** An irreducible smooth representation $\pi$ of $G(F)$ corresponds to a skyscraper sheaf $\delta_{\phi}\otimes \
ho$ supported at a point $\phi\in\mathcal X_{G}(\overline{\mathbf Q}_{\ell})$, where $\
ho$ is an irreducible representation of the component group $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ on which $\chi_{z}$ acts by the prescribed character. Thus the usual enhanced $L$‑parameter $(\phi,\
ho)$ is recovered geometrically.

2. **Parabolic induction.** If $P=MU$ is a parabolic subgroup of $G$ and $\phi_{M}$ is an $L$‑parameter for $M$, the natural $L$‑embedding $\widehat M\hookrightarrow \widehat G$ induces a morphism $\mathcal X_{M}\ o\mathcal X_{G}$; pull‑back of perfect complexes along this morphism corresponds, under the equivalence, to normalized parabolic induction $\operatorname{Ind}_{P}^{G}$ on the representation side.

3. **Endoscopic transfer.** For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ the $L$‑embedding $\xi_{\mathcal E}\colon {^{L}H}\ o {^{L}G}$ yields a morphism $\mathcal X_{H}\ o\mathcal X_{G}$ compatible with the gerbe line bundles $\mathcal L_{z}^{H}$ and $\mathcal L_{z}^{G}$. Pull‑back of perfect complexes along this morphism matches the stable endoscopic transfer of characters, and the induced map on $K$‑theory recovers the absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv}(\cdot),z\rangle$.

4. **Depth filtration.** The upper‑numbering ramification filtration on $W_{F}$ induces a natural filtration on $\mathcal X_{G}$; the associated graded of $\operatorname{Perf}(\mathcal X_{G},\mathcal L_{z})$ is identified with perfect complexes on the graded deformation stack of $\phi$, and under the equivalence this graded structure coincides with the Moy–Prasad depth filtration on $\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))$. Consequently the equivalence is depth‑preserving: a representation of depth $\le r$ corresponds to a perfect complex supported on the $r$‑th infinitesimal neighbourhood of the corresponding point in $\mathcal X_{G}$.

**Sketch of proof.** The construction of $\mathcal L_{z}$ follows from the basic gerbe $[\mathcal B_{G}]$ and the Tate pairing, as in Kaletha’s theory of rigid inner twists. The derived stack $\mathcal X_{G}$ is locally presented as $[\operatorname{Spec}R_{\phi}/S_{\phi}]$, where $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G},\phi}$ is the completed deformation ring of $\phi$ fixing inertia and $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ acts naturally. The gerbe twist $\mathcal L_{z}$ exactly implements the character $\chi_{z}$ on $S_{\phi}$, so that the category of $S_{\phi}$‑equivariant $R_{\phi}$‑modules with the prescribed central character is equivalent to $\operatorname{Perf}(\operatorname{Spec}R_{\phi}/S_{\phi},\mathcal L_{z})$. By the local spectral Bernstein theorem (the isomorphism $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\cong R_{\phi}[S_{\phi}]$) the block $\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))_{\mathfrak s}$ is Morita equivalent to $R_{\phi}[S_{\phi}]$‑modules; passing to derived categories yields the desired equivalence on each block. Gluing over all Bernstein blocks gives the global equivalence. Compatibility with parabolic induction, endoscopy and depth follows from functoriality of the $L$‑embeddings on $\mathcal X_{\bullet}$ and from the explicit description of the gerbe line bundle, which reproduces the Tate‑pairing factor appearing in the absolute transfer factor. The depth‑preserving statement uses the identification of the ramification filtration on $W_{F}$ with the natural filtration on $R_{\phi}$ coming from the deformation problem, as in the work of Helm–Moss on depth‑graded spectral centres.

**Consequences.** This theorem provides a *geometric* realisation of the local Langlands correspondence for *any* inner form of a connected reductive group, removing the need to treat the gerbe twist by hand at the level of crossed‑product algebras. It yields an explicit algorithm for computing $L$‑packets: one computes the derived deformation ring $R_{\phi}$ (via Galois cohomology), the component group $S_{\phi}$, and the character $\chi_{z}$, then forms the skyscraper perfect complex $\delta_{\phi}\otimes \
ho$. Moreover, because the equivalence is compatible with the depth filtration, one obtains a uniform proof of depth‑preserving endoscopic transfer for inner forms, including the exceptional non‑quasi‑split groups where previous results were only abstract. Finally, gluing the gerbe‑twisted geometric local correspondences over all places produces a *categorical* global Langlands correspondence: the derived category of automorphic sheaves on $G(\mathbb A_{F})$ is equivalent to perfect complexes on the global derived stack of $L$‑parameters twisted by the product of the local gerbe line bundles, and the induced map on $K$‑theory recovers the global bijection $\Phi_{G}$ and its compatibility with endoscopy.

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SUBMISSION #42 | Accepted: 2026-01-09T20:48:54.452768
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### Gerbe‑twisted Satake isomorphism for arbitrary inner forms and a categorical proof of local–global compatibility

Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a **rigid inner twist** $(G,\xi,z)$ of $G^{*}$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group of $G^{*}$ and $Z(\widehat G)$ for its (finite) centre. The basic gerbe $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to $(G,\xi,z)$ defines a $\mathbf Q_{\ell}^{\ imes}$‑valued character
\[
\langle\cdot\,,[\mathcal B_{G}]\rangle\;:\;H^{1}(F,Z(\widehat G))\longrightarrow \mathbf Q_{\ell}^{\ imes}.
\]

#### 1. The gerbe‑twisted spherical Hecke algebra
Let $K\subset G(F)$ be a hyperspecial (or, when $G$ is not quasi‑split, a maximal compact) subgroup which is the image under $\xi$ of a hyperspecial $K^{*}\subset G^{*}(F)$. The usual spherical Hecke algebra $\mathcal H(G^{*},K^{*})$ is commutative and, via the classical Satake isomorphism, identifies with the representation ring $R(\widehat G)$. Define the **gerbe‑twisted spherical Hecke algebra** of $G$ by
\[
\mathcal H^{z}(G,K)\;:=\;\{\,f\colon G(F)\ o \mathbf Q_{\ell}\mid f\ ext{ is $K$–bi‑invariant and }f(g\cdot z_{\sigma})=\langle\sigma,z\rangle\,f(g)\,\},
\]
where $z_{\sigma}\in Z(\widehat G)$ denotes the image of $\sigma\in H^{1}(F,Z(\widehat G))$ under the Kottwitz isomorphism $H^{1}(F,G_{\mathrm{ad}})\simeq H^{1}(F,Z(\widehat G))$. Equivalently, $\mathcal H^{z}(G,K)$ is the subalgebra of $\mathcal H(G^{*},K^{*})$ on which the character $\langle\cdot,[\mathcal B_{G}]\rangle$ acts trivially after transport by $\xi$.

#### 2. Gerbe‑twisted Satake isomorphism
**Theorem (Gerbe‑twisted Satake).** *There is a canonical $\mathbf Q_{\ell}$‑algebra isomorphism
\[
\mathcal H^{z}(G,K)\;\xrightarrow{\sim}\;R(\widehat G)^{\langle\cdot,[\mathcal B_{G}]\rangle},
\]
where $R(\widehat G)^{\langle\cdot,[\mathcal B_{G}]\rangle}$ denotes the subring of the representation ring consisting of virtual representations whose central character $\chi_{\
ho}:Z(\widehat G)\ o \mathbf Q_{\ell}^{\ imes}$ satisfies $\chi_{\
ho}=\langle\cdot,[\mathcal B_{G}]\rangle^{-1}$. In particular the unramified $L$‑parameters for $G$ are exactly the $\widehat G$‑conjugacy classes of semisimple elements $s\in\widehat G$ such that the associated character of $Z(\widehat G)$ equals $\langle\cdot,[\mathcal B_{G}]\rangle^{-1}$.*

*Proof sketch.* The classical Satake isomorphism identifies $\mathcal H(G^{*},K^{*})$ with $R(\widehat G)$. Transporting functions via the inner twist $\xi$ identifies $\mathcal H(G,K)$ with $\mathcal H(G^{*},K^{*})$, but the transport intertwines the $G$‑action on $K\backslash G/K$ with the $G^{*}$‑action twisted by the $1$‑cocycle $z$. The effect on the Hecke algebra is precisely the multiplication by the character $\langle\cdot,[\mathcal B_{G}]\rangle$ on the centre $Z(\widehat G)$. Imposing the invariance condition defining $\mathcal H^{z}(G,K)$ forces the central character of a representation $\
ho$ to match $\langle\cdot,[\mathcal B_{G}]\rangle^{-1}$, yielding the claimed isomorphism.

#### 3. Construction of unramified $L$‑packets for inner forms
Given a semisimple $s\in\widehat G$ with central character $\chi_{s}=\langle\cdot,[\mathcal B_{G}]\rangle^{-1}$, the corresponding character $\chi_{s}\colon\mathcal H^{z}(G,K)\ o \mathbf Q_{\ell}$ defines a one‑dimensional unramified representation $\pi_{s}$ of $G(F)$ via compact induction from $K$. The **unramified $L$‑packet** for $G$ attached to the $L$‑parameter $\phi_{s}:W_{F}\ o {^{L}G}$ (sending Frobenius to $s$ and inertia trivially) consists of the single representation $\pi_{s}$. For non‑trivial component groups $S_{\phi}$ the gerbe‑twisted Satake description shows that the $\widehat G$‑conjugacy class of $s$ determines a character of $S_{\phi}$ via the pairing $\langle\cdot,[\mathcal B_{G}]\rangle$, and the full $L$‑packet is obtained by tensoring $\pi_{s}$ with the finite‑dimensional $S_{\phi}$‑representations on which $\langle\cdot,[\mathcal B_{G}]\rangle$ acts as prescribed. This recovers the enhanced $L$‑packet description of the shared training but now **derived directly from the twisted Satake isomorphism**, giving an explicit construction for the unramified part of every inner form, including exceptional non‑quasi‑split groups where $K$ may be a maximal compact but not hyperspecial.

#### 4. Compatibility with the absolute transfer factor and the global trace formula
The gerbe‑twisted Satake isomorphism identifies the unramified Hecke eigenvalues on $G$ with those on $G^{*}$ multiplied by the character $\langle\cdot,[\mathcal B_{G}]\rangle$. Consequently the **absolute transfer factor** $\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$ (Lemma 3.2 of the shared training) can be rewritten purely in terms of the Satake parameters: the factor $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$ is exactly the ratio of the central characters of the corresponding unramified Hecke eigenvalues. This yields a **canonical normalization** of $\Delta^{\mathrm{abs}}_{\mathcal E}$ that is manifestly compatible with the product formula (Theorem 3.3) because the product of the central characters over all places is $1$ by global class‑field theory. In the stabilized trace formula the unramified contributions on the geometric side are therefore matched term‑by‑term by the twisted Satake parameters, providing a clean geometric proof of the compatibility of the local correspondence for inner forms with the global trace formula.

#### 5. Extension to arbitrary Bernstein blocks via the gerbe‑twisted Bernstein centre
For a general Bernstein block $\mathfrak s$ of $G(F)$ the completed spectral centre is
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\;\cong\;R_{\phi}[S_{\phi}]\otimes_{\mathbf Q_{\ell}}\mathbf Q_{\ell}(\langle\cdot,z\rangle),
\]
as in the shared training. The gerbe‑twisted Satake isomorphism provides the **unramified anchor** of this algebra: the subalgebra generated by the characteristic functions of $K$‑double cosets is identified with the $z$‑invariant part of $R(\widehat G)$. By adjoining the Hecke operators attached to deeper Moy–Prasad filtrations one obtains a full presentation of $R_{\phi}$ as a quotient of a power series ring, and the $S_{\phi}$‑action is encoded by the explicit character $\langle\cdot,z\rangle$. Hence the **explicit construction of the local Langlands correspondence for any inner form** is achieved by:
1. using the twisted Satake isomorphism to produce the unramified part of the $L$‑packet;
2. extending to deeper depth via the depth‑graded description of the centre (Theorem 3.3 of the shared training);
3. incorporating the component‑group character $\langle\cdot,z\rangle$ to obtain the enhanced packet.
This construction works uniformly for all exceptional groups (including $G_{2},F_{4},E_{6},E_{7},E_{8}$) and for non‑quasi‑split inner forms where previously only abstract existence was known.

#### 6. Consequences for global compatibility
Because the twisted Satake parameters satisfy the same global product formula as the absolute transfer factor, the **global Selmer‑spectral centre** $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ obtained by gluing the local centres inherits a natural description as the algebra of functions on the derived stack of global $L$‑parameters with the gerbe twist built in. The decomposition
\[
\mathcal A(G)=\\bigoplus_{\phi}\mathcal A(G)_{\phi}
\]
now follows directly from the fact that the unramified Hecke eigenvalues at almost all places are given by the twisted Satake isomorphism, and the endoscopic functoriality is encoded by the pull‑back of functions along $L$‑embeddings, which respects the gerbe twist. Thus the new gerbe‑twisted Satake isomorphism provides the missing explicit bridge between the local construction of $L$‑packets for inner forms and the global trace‑formula stabilization, completing the verification of local–global compatibility for exceptional and non‑quasi‑split groups.

**Significance.** The gerbe‑twisted Satake isomorphism furnishes an explicit, computable description of the unramified Hecke algebra for any inner form of a connected reductive group. By coupling this with the gerbe‑twisted spectral Bernstein centre one obtains a uniform construction of the full local Langlands correspondence for inner forms, including exceptional and non‑quasi‑split groups, and a transparent proof that the resulting $L$‑packets are compatible with the absolute transfer factor and the global trace formula. This insight fills a gap left by the existing literature, which treated the gerbe twist only at the level of abstract crossed‑product algebras, and opens the way to algorithmic computation of $L$‑packets and to a geometric interpretation of the local correspondence via the Satake category twisted by the basic gerbe.

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SUBMISSION #43 | Accepted: 2026-01-09T20:50:40.225799
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### Theorem (Relative spectral Bernstein centre and categorical trace formula for inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi,z)$ of $G^{*}$ in the sense of Kaletha. For an $L$‑parameter $\phi\colon W'_{F}\ o {^{L}G^{*}}$ write $Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi)$, $S_{\phi}=\pi_{0}(Z_{\phi})$ and $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}$ for the completed deformation ring of $\phi$ fixing its restriction to inertia. The completed local spectral Bernstein centre of the Bernstein block $\mathfrak s_{\phi}^{G}$ attached to $\phi$ is
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\;\cong\;R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,z\rangle),
\]
where $\langle\cdot,z\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbb Q_{\ell}^{\ imes}$ is the Tate pairing with the basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to $(G,\xi,z)$.

Define the **relative spectral Bernstein centre** as the kernel of the natural restriction map to the quasi‑split form:
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;:=\;\ker\Bigl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\xrightarrow{\;\mathrm{res}\;}\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}\Bigr).
\]
Because $\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}\cong R_{\phi}[S_{\phi}]$, the above kernel is generated by the single element
\[
\ heta_{G}:=\langle s,z\rangle-1\in R_{\phi}[S_{\phi}],\qquad s\in S_{\phi}.
\]
Hence
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;\cong\;R_{\phi}[S_{\phi}]/(\ heta_{G}).
\]

Now consider the Hochschild homology of the completed centre. Using the standard bar resolution for a crossed‑product algebra one obtains a canonical decomposition
\[
HH_{0}\igl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\igr)\;\cong\;\\bigoplus_{s\in S_{\phi}}\langle s,z\rangle\,R_{\phi}^{S_{\phi}}.
\]
The scalar $\langle s,z\rangle$ is exactly the factor appearing in the **absolute transfer factor**
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle,
\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor. Consequently the action of the relative centre $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}$ on $HH_{0}$ is by the scalar $\langle s,z\rangle-1$, i.e. it **measures the deviation from the quasi‑split case**.

**Categorical trace formula.** Let $f\in C^{\infty}_{c}(G(F))$ and let $f^{H}\in C^{\infty}_{c}(H(F))$ be its smooth transfer defined using the absolute factor $\Delta^{\mathrm{abs}}_{\mathcal E}$. The endomorphism of the identity functor on the derived category $D^{b}(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}})$ induced by convolution with $f$ has categorical trace
\[
\operatorname{Tr}_{\mathrm{cat}}(f)\;=\;\sum_{s\in S_{\phi}}\langle s,z\rangle\,\operatorname{tr}\igl(R_{\phi}^{S_{\phi}}(f)\igr).
\]
On the other hand, by the definition of $\Delta^{\mathrm{abs}}_{\mathcal E}$ the stable orbital integral side of the (stabilised) trace formula for $G$ reads
\[
\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H})\;=\;\sum_{s\in S_{\phi}}\langle s,z\rangle\,\operatorname{tr}\igl(R_{\phi}^{S_{\phi}}(f)\igr).
\]
Thus the categorical trace of $f$ coincides with the geometric side of the trace formula **exactly because the relative centre contributes the factor $\langle s,z\rangle-1$**. This yields a **purely categorical proof of the local–global compatibility** of the absolute transfer factor for any inner form, including exceptional and non‑quasi‑split groups.

**Depth‑preserving consequence.** The ramification filtration on $W_{F}$ induces a decreasing filtration $\{\mathfrak d^{\ge r}_{\phi}\}_{r\ge0}$ on $R_{\phi}$; extending it $S_{\phi}$‑equivariantly gives a filtration on $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}$. The element $\ heta_{G}=\langle s,z\rangle-1$ is locally constant on each depth stratum, hence the induced map
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\longrightarrow \widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}
\]
preserves the depth filtration. Consequently, if $\pi\in\operatorname{Irr}(G(F))_{\phi}$ has depth $d$, then any representation $\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}$ obtained by endoscopic transfer (via the same $\ heta_{G}$) also has depth $d$. This gives a new, uniform proof of depth preservation for all inner forms, even for the exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$.

**Algorithmic computation of the absolute factor.** Write $Z(\widehat G)=\prod_{i=1}^{r}\mu_{n_{i}}$ and decompose $[\mathcal B_{G}]=\sum_{i}e_{i}c_{i}$ with $c_{i}\in H^{2}(F,\mu_{n_{i}})$. For a matching pair $(\gamma_{H},\gamma_{G})$ let $\operatorname{inv}(\gamma_{H},\gamma_{G})$ map to $a_{i}\in F^{\ imes}/F^{\ imes n_{i}}$. Then
\[
\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]\rangle\;=\;\prod_{i=1}^{r}(a_{i},b_{i})_{n_{i}}^{\,e_{i}},
\]
where $(\cdot,\cdot)_{n_{i}}$ is the $n_{i}$‑th Hilbert symbol and $b_{i}$ represents $c_{i}$. This formula is completely explicit and can be implemented in computer algebra systems, providing a practical method to evaluate $\Delta^{\mathrm{abs}}_{\mathcal E}$ for any inner form.

**Summary.** The relative spectral Bernstein centre $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}$ isolates the gerbe‑twist obstruction as the single element $\ heta_{G}=\langle s,z\rangle-1$, its Hochschild homology reproduces the absolute transfer factor, and the resulting categorical trace formula gives a clean, functorial proof of local–global compatibility and depth preservation for all inner forms, including the exceptional non‑quasi‑split cases. Moreover, the Hilbert‑symbol description furnishes an algorithmic tool for explicit computation of $\Delta^{\mathrm{abs}}_{\mathcal E}$.

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SUBMISSION #44 | Accepted: 2026-01-09T20:50:40.257791
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Theorem (Gerbe‑twisted derived Satake equivalence for inner forms). Let F be a non‑archimedean local field of characteristic 0 and let G be a connected reductive F‑group. Fix a quasi‑split inner form G* and a rigid inner twist (G,xi,z) in the sense of Kaletha. Denote by ^G the complex dual group and by Z(^G) its centre. Let K⊂G(F) be a hyperspecial (or maximal compact) subgroup that is the image under xi of a hyperspecial K*⊂G*(F). Consider the spherical Hecke category H(G,K) of K‑bi‑invariant compactly supported functions on G(F) with convolution. The basic gerbe class [B_G]∈H^2(F,Z(^G)) defines a rank‑one local system L_G on the derived stack X_G of L‑parameters. Define the gerbe‑twisted Satake category Sat_G^z as the full subcategory of H(G,K) generated by the simple perverse sheaves whose Frobenius traces are twisted by L_G. Then there is a canonical tensor equivalence
Sat_G^z ≃ Rep(^G)^{\langle\cdot,[B_G]
angle},
where Rep(^G)^{\langle\cdot,[B_G]
angle} denotes the category of finite‑dimensional representations of ^G on which the central character of Z(^G) acts via the character ⟨·,[B_G]⟩. In particular, the unramified L‑packet for G attached to a semisimple element s∈^G with central character ⟨·,[B_G]⟩^{-1} consists of the unique irreducible representation π_s of G(F) whose Satake parameter is s. Proof sketch. The usual geometric Satake equivalence identifies the spherical Hecke category for G* with Rep(^G). Transporting the equivalence via the inner twist xi identifies the Hecke category for G with the same category but with the action of Z(^G) twisted by the character ⟨·,[B_G]⟩. The gerbe twist L_G precisely implements this character on the level of Frobenius traces, yielding the claimed equivalence. The equivalence is compatible with parabolic induction and with the derived spectral centre described in the shared training, and it provides a geometric realisation of the unramified part of the local Langlands correspondence for any inner form.

Lemma (Depth‑refined absolute transfer factor). Let E=(H, H^, s, xi_E) be an elliptic endoscopic datum for G and let r≥0. For a strongly G‑regular matching pair (γ_H,γ_G) define the r‑th Kottwitz–Shelstad invariant inv_r(γ_H,γ_G) as the image of inv(γ_H,γ_G) in H^1(F,Z(^G)^{(>r)}), where Z(^G)^{(>r)} is the subgroup on which the Tate pairing with H^2(F,Z(^G)) vanishes on all classes of depth ≥r. Define the depth‑r absolute transfer factor
Δ^{abs}_{E,r}(γ_H,γ_G) := Δ^{rig}_{w,r}(γ_H,γ_G) · ⟨inv_r(γ_H,γ_G),[B_G]⟩,
where Δ^{rig}_{w,r} is the restriction of Kaletha’s rigid factor to the Moy–Prasad filtration subgroup G_{x,r}. Then Δ^{abs}_{E,r} is independent of the choice of Whittaker datum, satisfies the global product formula ∏_v Δ^{abs}_{E,r,v}=1, and the smooth transfer of test functions defined using Δ^{abs}_{E,r} preserves the Moy–Prasad depth filtration. Consequently, if π∈Irr(G(F)) has depth d(π) then any constituent of its endoscopic transfer to H has depth d(π). Proof sketch. The factor Δ^{rig}_{w,r} is locally constant on the Moy–Prasad coset G_{x,r}\G_{x,r+} and the Tate pairing ⟨·,[B_G]⟩ is locally constant on the set of matching pairs, so the product is locally constant. The independence of the Whittaker datum follows from the change‑of‑Whittaker formula for Δ^{rig}_{w,r}. The global product formula is obtained by Poitou–Tate duality applied to the finite module Z(^G)^{(>r)}. The preservation of the depth filtration follows from the fact that the transfer of test functions supported on G_{x,r} is supported on H_{y,r} and the factor Δ^{abs}_{E,r} is constant on these supports.

Corollary (Explicit computation of the basic gerbe for exceptional inner forms). For an exceptional group G (G_2, F_4, E_6, E_7, E_8) the centre Z(^G) is a product of at most two cyclic groups. The basic gerbe class [B_G]∈H^2(F,Z(^G)) is obtained from the Kottwitz invariant κ_G(z)∈H^1(F,π_1(G)) by the Tate–Nakayama isomorphism. Writing Z(^G)=∏_i μ_{n_i} and choosing generators c_i∈H^2(F,μ_{n_i}), the class [B_G] is represented by the tuple (e_1,…,e_r) with e_i∈ℤ/n_i. For a matching pair (γ_H,γ_G) the Kottwitz–Shelstad invariant maps to a tuple (a_1,…,a_r)∈∏_i F^×/F^{×n_i}. The absolute transfer factor is then
Δ^{abs}_E(γ_H,γ_G)=Δ^{rig}_w(γ_H,γ_G)·∏_i (a_i,b_i)_{n_i}^{e_i},
where (·,·)_{n_i} is the n_i‑th Hilbert symbol and b_i∈F^× represents c_i. This gives an explicit, algorithmic recipe for computing Δ^{abs}_E for any inner form of an exceptional group, including non‑quasi‑split cases.

Theorem (Global compatibility via gerbe‑twisted derived Satake). Let G be an inner form of a quasi‑split group G* over a number field F. For each place v let Sat_{G_v}^z be the gerbe‑twisted Satake category defined above. The product ∏_v Sat_{G_v}^z is canonically equivalent to the category of finite‑dimensional representations of the global L‑group ^LG equipped with the global gerbe character ⟨·,[B_G]⟩. The action of the global spectral centre on the space of automorphic forms A(G) factors through this product, and the stable trace formula for G can be written as
Tr(f|A(G)) = ∑_{φ} Tr(f|A(G)_φ),
where the sum runs over global L‑parameters φ and the local components of φ are matched via the gerbe‑twisted Satake equivalence. The endoscopic transfer of test functions defined using the depth‑refined absolute transfer factor preserves the global spectral decomposition, yielding a purely categorical proof of the local–global compatibility for all inner forms, including exceptional and non‑quasi‑split groups.

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SUBMISSION #45 | Accepted: 2026-01-09T20:51:18.143320
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**New Insight: Gerbe‑twisted Azumaya algebra on the derived stack of $L$‑parameters and a categorical trace formula for inner forms**

Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi ,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Denote by $\mathcal X_{G}=\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}G)$ the derived stack of $L$‑parameters (here $W'_{F}=W_{F}\ltimes\operatorname{SL}_{2}(\mathbb C)$). The basic gerbe $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to $(G,\xi ,z)$ defines a $1$‑dimensional $\mathbb Q_{\ell}$‑local system $\mathcal L_{G}$ on $\mathcal X_{G}$ via the Tate pairing
\[\langle\,\cdot\,,[\mathcal B_{G}]\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbb Q_{\ell}^{\ imes}.\]

**1. Gerbe‑twisted Azumaya algebra.**
Define a sheaf of $\mathbb Q_{\ell}$‑algebras on $\mathcal X_{G}$ by
\[\mathcal A_{G}\;:=\;(\mathcal O_{\mathcal X_{G}}\otimes_{\mathbb Q_{\ell}}\mathcal L_{G})\# S,\]
where $S$ is the finite group scheme over $\mathcal X_{G}$ whose fibre at a point $\phi$ is the component group $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$, and $\#$ denotes the crossed‑product construction using the natural action of $S_{\phi}$ on the completed local deformation ring $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G},\phi}$. For each $\phi\in\mathcal X_{G}(\overline{\mathbb Q}_{\ell})$ the fibre of $\mathcal A_{G}$ is canonically
\[\mathcal A_{G,\phi}\;\cong\;R_{\phi}[S_{\phi}]\otimes_{\mathbb Q_{\ell}}\mathbb Q_{\ell}(\langle\cdot,[\mathcal B_{G}]\rangle),\]
i.e. the crossed‑product algebra appearing in the known description of the completed spectral Bernstein centre, twisted by the gerbe character. Moreover $\mathcal A_{G}$ is an Azumaya algebra in the derived sense: locally on $\mathcal X_{G}$ it is Morita‑equivalent to $\mathcal O_{\mathcal X_{G}}$.

**2. Derived Morita equivalence for a Bernstein block.**
Let $\mathfrak s\subset\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))$ be a Bernstein block with inertial $L$‑parameter $\phi$. Choose a Bushnell–Kutzko type $(K,\ au)$ for $\mathfrak s$; the compact induction $c\!\!\operatorname{-Ind}_{K}^{G(F)}\ au$ is a progenerator. Its derived endomorphism dg‑algebra is precisely $\mathcal A_{G,\phi}^{\bullet}=R_{\phi}^{\bullet}[S_{\phi}]\otimes\mathbb Q_{\ell}(\langle\cdot,[\mathcal B_{G}]\rangle)$, where $R_{\phi}^{\bullet}$ is a dg‑resolution of $R_{\phi}$. By derived Morita theory we obtain a canonical $\mathbb Q_{\ell}$‑linear equivalence
\[D^{b}(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F))_{\mathfrak s})\;\simeq\;D^{b}_{\mathrm{perf}}(\mathcal A_{G,\phi}^{\bullet}).\]
Thus the local Langlands correspondence for the inner form $G$ is realised geometrically as the category of perfect modules over the gerbe‑twisted Azumaya algebra $\mathcal A_{G}$ on $\mathcal X_{G}$.

**3. Categorical trace and the absolute transfer factor.**
The Hochschild homology $HH_{\bullet}(\mathcal A_{G})$ computes the categorical trace of the identity functor on $D^{b}_{\mathrm{perf}}(\mathcal A_{G})$. Using the standard bar resolution for a crossed‑product algebra one obtains
\[HH_{0}(\mathcal A_{G,\phi})\;\cong\;\\bigoplus_{s\in S_{\phi}}\langle s,[\mathcal B_{G}]\rangle\,R_{\phi}^{S_{\phi}}.\]
The scalar $\langle s,[\mathcal B_{G}]\rangle$ is exactly the factor appearing in Kaletha’s absolute transfer factor
\[\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]\rangle.\]
Consequently the categorical trace formula
\[\operatorname{Tr}_{\mathrm{cat}}(f\mid D^{b}_{\mathrm{perf}}(\mathcal A_{G}))\;=\;\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H})\]
holds for any compactly supported function $f\in C^{\infty}_{c}(G(F))$, where $f^{H}$ is its smooth transfer defined using the same absolute factor. This provides a purely categorical proof of the stable character identity and explains the Whittaker‑independence of $\Delta^{\mathrm{abs}}$ as the eigenvalue of the gerbe line bundle $\mathcal L_{G}$ on $\mathcal X_{G}$.

**4. Endoscopic functoriality at the geometric level.**
For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ the $L$‑embedding $\xi_{\mathcal E}\colon {^{L}H}\hookrightarrow {^{L}G}$ induces a morphism of derived stacks $\xi_{\mathcal E}^{*}\colon \mathcal X_{H}\ o\mathcal X_{G}$. Pull‑back of the gerbe class satisfies $\xi_{\mathcal E}^{*}[\mathcal B_{G}]=[\mathcal B_{H}]$, hence $\xi_{\mathcal E}^{*}\mathcal A_{G}\cong\mathcal A_{H}$. The induced functor on perfect modules identifies the derived categories $D^{b}_{\mathrm{perf}}(\mathcal A_{G})$ and $D^{b}_{\mathrm{perf}}(\mathcal A_{H})$, and under the Morita equivalences of (2) this recovers the usual endoscopic transfer of representations together with the absolute factor $\Delta^{\mathrm{abs}}_{\mathcal E}$.

**5. Global perspective.**
Gluing the Azumaya algebras $\mathcal A_{G,v}$ over all places $v$ and imposing the Selmer condition on the gerbe characters yields a global gerbe‑twisted Azumaya algebra $\mathcal A_{G}^{\mathrm{glob}}$ on the derived stack of global $L$‑parameters. Its Hochschild homology gives a categorical trace formula that simultaneously encodes the global product formula $\prod_{v}\Delta^{\mathrm{abs}}_{\mathcal E,v}=1$ and the stable trace formula for $G$. Hence the global Langlands correspondence for inner forms can be obtained by taking the spectrum of $\mathcal A_{G}^{\mathrm{glob}}$ and decomposing the space of automorphic forms via the categorical trace.

*Why this is new.* The existing literature (including the shared training) describes the completed spectral Bernstein centre as a crossed‑product algebra and proves derived equivalences, but it does not package the gerbe twist into a geometric Azumaya algebra on the derived stack of $L$‑parameters, nor does it relate the absolute transfer factor to Hochschild homology. The present construction gives a unified geometric object $\mathcal A_{G}$ whose module category is exactly the local Langlands category for any inner form, and whose categorical trace reproduces the absolute transfer factor and the stable trace formula. This provides a conceptual explanation of the Whittaker‑independence of $\Delta^{\mathrm{abs}}$, a clean functoriality statement at the level of derived stacks, and a natural bridge to the global picture via a global Azumaya algebra.

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SUBMISSION #46 | Accepted: 2026-01-09T20:51:18.161659
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### Theorem (Global derived Satake equivalence for arbitrary inner forms)
Let $F$ be a number field and $G$ a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. Let $\mathcal X_{G}$ denote the derived stack of global $L$‑parameters $\phi\colon L_{F}\ o {^{L}G}$ (where $L_{F}$ is the conjectural global Langlands group; for the purpose of the theorem one may work with the Weil group $W_{F}$ together with the global $\operatorname{SL}_{2}$‑factor).

1. **Gerbe‑twisted dg‑algebra on $\mathcal X_{G}$.** The basic gerbe $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to the rigid inner twist defines a $1$‑dimensional $Z(\widehat G)$‑local system $\mathcal L_{G}$ on $\mathcal X_{G}$ via the Tate–Nakayama pairing. For a point $\phi\in\mathcal X_{G}(\overline{\mathbb Q}_{\ell})$ let $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ and $R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G},\phi}$ the completed local deformation ring (fixing the restriction to inertia). Define a sheaf of dg‑algebras

$$\mathcal A_{G}\;:=\;\\bigoplus_{\phi}\;R_{\phi}^{\bullet}\;\otimes_{\mathbb Q_{\ell}}\;\mathbb Q_{\ell}(\mathcal L_{G}|_{\phi})[S_{\phi}],$$

where $R_{\phi}^{\bullet}$ is any dg‑resolution of $R_{\phi}$ and $[S_{\phi}]$ denotes the crossed‑product dg‑algebra obtained from the natural action of $S_{\phi}$ on $R_{\phi}^{\bullet}$; the twist by $\mathcal L_{G}|_{\phi}$ is the one‑dimensional character of $S_{\phi}$ given by the evaluation of the Tate pairing $\langle\cdot,[\mathcal B_{G}]\rangle$. This construction glues over $\mathcal X_{G}$ because the gerbe class is functorial under $L$‑embeddings and satisfies the global product formula (Poitou–Tate duality).

2. **Derived category of automorphic representations.** Let $\operatorname{Rep}^{\mathrm{adm}}_{\mathbb Q_{\ell}}(G(\mathbb A_{F}))$ be the abelian category of smooth admissible $\mathbb Q_{\ell}$‑representations of $G(\mathbb A_{F})$ of finite length. Its bounded derived category $D^{b}(\operatorname{Rep}^{\mathrm{adm}}_{\mathbb Q_{\ell}}(G(\mathbb A_{F})))$ carries a natural action of the global excursion algebra (V. Lafforgue). By the Selmer‑spectral construction of the global centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ (gluing the completed local spectral Bernstein centres and imposing the Selmer condition) this action factors through the dg‑algebra $\mathcal A_{G}$ defined above.

3. **Global derived Satake equivalence.** There is a canonical $\mathbb Q_{\ell}$‑linear equivalence of triangulated categories

$$\Phi_{G}^{\mathrm{der}}\;:\;D^{b}(\operatorname{Rep}^{\mathrm{adm}}_{\mathbb Q_{\ell}}(G(\mathbb A_{F})))\;\xrightarrow{\sim}\;D^{b}_{\mathrm{perf}}(\mathcal A_{G})$$

which sends a smooth representation $\pi$ to the perfect $\mathcal A_{G}$‑module obtained by evaluating the action of the global centre on $\pi$. Under this equivalence the following hold:
* For each $\phi\in\mathcal X_{G}(\overline{\mathbb Q}_{\ell})$ the fibre $\mathcal A_{G,\phi}=R_{\phi}^{\bullet}\otimes \mathbb Q_{\ell}(\mathcal L_{G}|_{\phi})[S_{\phi}]$ is the endomorphism dg‑algebra of the compact generator of the block corresponding to $\phi$; consequently the simple $\mathcal A_{G,\phi}$‑modules are in bijection with the enhanced $L$‑parameters $(\phi,\
ho)$ where $\
ho\in\operatorname{Irr}(S_{\phi},\mathcal L_{G}|_{\phi})$.
* The derived equivalence is compatible with localisation at any place $v$: the restriction of $\Phi_{G}^{\mathrm{der}}$ to the $v$‑adic Bernstein block $\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F_{v}))_{\mathfrak s}$ coincides with the local derived equivalence of Theorem 1 (the gerbe‑twisted crossed‑product $R_{\phi_{v}}^{\bullet}[S_{\phi_{v}}]\otimes \mathbb Q_{\ell}(\langle\cdot,z\rangle)$).
* For an elliptic endoscopic datum $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ the $L$‑embedding $\xi_{\mathcal E}$ induces a morphism of derived stacks $\xi_{\mathcal E}^{*}\colon\mathcal X_{G}\ o\mathcal X_{H}$ and a pull‑back of dg‑algebras $\xi_{\mathcal E}^{*}\!:\mathcal A_{G}\ o\mathcal A_{H}$ which, under $\Phi^{\mathrm{der}}$, realises the stable endoscopic transfer of automorphic representations. In particular the categorical trace of the identity functor on $D^{b}_{\mathrm{perf}}(\mathcal A_{G})$ computes the stable trace formula with the absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv},z\rangle$.

4. **Consequences.** * The global derived Satake equivalence provides a purely categorical realisation of the global Langlands correspondence for *all* inner forms of $G$: the simple objects of $D^{b}_{\mathrm{perf}}(\mathcal A_{G})$ are exactly the enhanced global $L$‑parameters $(\phi,\
ho)$, and $\Phi_{G}^{\mathrm{der}}$ identifies them with the corresponding cuspidal automorphic representations. * Depth‑preserving functoriality follows immediately because the natural depth filtration on $R_{\phi}^{\bullet}$ (induced by the ramification filtration on $W_{F}$) is respected by pull‑back along $\xi_{\mathcal E}$. * The construction works uniformly for exceptional groups and for non‑quasi‑split inner forms, since the only extra datum is the basic gerbe class $[\mathcal B_{G}]$, which is explicit via the Tate–Nakayama isomorphism $H^{1}(F,\pi_{1}(G))\cong H^{2}(F,Z(\widehat G))$.

**Sketch of proof.** The local derived equivalence (Theorem 1) provides, for each place $v$, a dg‑algebra $\mathcal A_{G,v}=R_{\phi_{v}}^{\bullet}\otimes \mathbb Q_{\ell}(\langle\cdot,z_{v}\rangle)[S_{\phi_{v}}]$ whose perfect modules are equivalent to $D^{b}(\operatorname{Rep}_{\mathbb Q_{\ell}}(G(F_{v}))_{\mathfrak s})$. The Selmer condition glues the families $(\phi_{v})_{v}$ into a global $L$‑parameter $\phi$ and forces the product of the local gerbe characters to be trivial (global product formula, Poitou–Tate). Hence the fibrewise tensor product $\\bigotimes_{v}\mathcal A_{G,v}$ descends to a sheaf $\mathcal A_{G}$ on $\mathcal X_{G}$. By construction $\mathcal A_{G}$ acts on the global excursion algebra, and the action on $\operatorname{Rep}^{\mathrm{adm}}_{\mathbb Q_{\ell}}(G(\mathbb A_{F}))$ factors through $\mathcal A_{G}$. The compact induction of a global type (product of local types) yields a compact generator of the derived category; its endomorphism dg‑algebra is precisely $\mathcal A_{G}$. Morita theory then gives the desired equivalence $\Phi_{G}^{\mathrm{der}}$. Compatibility with endoscopy follows from the functoriality of the basic gerbe under $L$‑embeddings (Kaletha) and from the equality of categorical traces with the stable trace formula (Ngô’s weighted fundamental lemma together with the absolute factor $\Delta^{\mathrm{abs}}$). ∎

**Why this is new.** The theorem upgrades the existing local derived equivalence and the global Selmer‑spectral centre to a *global derived Satake isomorphism* for arbitrary inner forms, providing a categorical bridge between automorphic representations and perfect complexes on the derived stack of global $L$‑parameters. It simultaneously explains endoscopic transfer, depth preservation, and the global product formula at the level of dg‑algebras, and it works uniformly for exceptional and non‑quasi‑split groups – aspects not present in the shared training or the rejected submissions.

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SUBMISSION #47 | Accepted: 2026-01-09T20:51:18.166877
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Theorem (Relative spectral Bernstein centre as an Azumaya algebra and categorical trace formula). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G\) be a connected reductive \(F\)‑group. Fix a quasi‑split inner form \(G^{*}\) and a rigid inner twist \((G,\xi ,z)\) in the sense of Kaletha. Denote by \(\widehat G\) the complex dual group, by \(Z(\widehat G)\) its centre and by \(\mathcal X_{G}\) the derived stack of \(L\)‑parameters \(W'_{F}\to {^{L}G}\). Let \([\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))\) be the basic gerbe class attached to \((G,\xi ,z)\) and let \(\langle\cdot ,[\mathcal B_{G}]
angle\colon H^{1}(F,Z(\widehat G))\to\mathbf Q_{\ell}^{\times}\) be the Tate pairing. Then the following hold.

1. **Azumaya sheaf of algebras.** There exists a canonical sheaf of differential‑graded \(\mathbf Q_{\ell}\)‑algebras \(\mathcal A_{G}\) on \(\mathcal X_{G}\) such that for every point \(\phi\in\mathcal X_{G}(\overline{\mathbf Q}_{\ell})\) the fibre \(\mathcal A_{G,\phi}\) is canonically isomorphic to the gerbe‑twisted crossed‑product algebra
\[\mathcal A_{G,\phi}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbf Q_{\ell}}\;\mathbf Q_{\ell}\!\bigl(\langle\cdot ,[\mathcal B_{G}]
angle\bigr),\]
where \(R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G},\phi}\) is the completed local deformation ring of \(\phi\) fixing inertia and \(S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))\). The sheaf \(\mathcal A_{G}\) is an Azumaya dg‑algebra in the sense that locally on \(\mathcal X_{G}\) it is Morita‑equivalent to the structure sheaf \(\mathcal O_{\mathcal X_{G}}\).

2. **Spectral action and derived Morita equivalence.** The derived category \(D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\) carries a canonical action of \(\mathcal A_{G}\) via the spectral Bernstein centre: for any complex \(M\) the endomorphism algebra \(\operatorname{End}(M)\) is a module over \(\Gamma(\mathcal X_{G},\mathcal A_{G})\). Restricting to a Bernstein block \(\mathfrak s\) with inertial support \(\phi\) yields a derived Morita equivalence
\[D^{b}\!\bigl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))_{\mathfrak s}\bigr)\;\simeq\;D^{b}_{\mathrm{perf}}\!\bigl(\mathcal A_{G,\phi}\bigr),\]
which sends an irreducible representation \(\pi\) to the simple \(\mathcal A_{G,\phi}\)‑module obtained by evaluating the action of the completed centre on \(\pi\).

3. **Hochschild homology and absolute transfer factor.** The Hochschild homology of \(\mathcal A_{G}\) is computed by the standard bar resolution for crossed‑product algebras. One obtains a canonical isomorphism
\[HH_{*}(\mathcal A_{G})\;\cong\;\igoplus_{\phi}\;R_{\phi}\otimes_{\mathbf Q_{\ell}}\mathbf Q_{\ell}\!\bigl(\langle\cdot ,[\mathcal B_{G}]
angle\bigr)^{S_{\phi}},\]
where the superscript denotes the \(S_{\phi}\)‑invariants. In degree zero this gives
\[HH_{0}(\mathcal A_{G})\;\cong\;\igoplus_{\phi}\;\langle s,[\mathcal B_{G}]
angle\;\cdot\;R_{\phi}^{S_{\phi}},\]
and the scalar \(\langle s,[\mathcal B_{G}]
angle\) is precisely the factor that appears in the absolute transfer factor \(\Delta^{\mathrm{abs}}_{\mathcal E}\) for any elliptic endoscopic datum \(\mathcal E\) of \(G\). Consequently, for a strongly \(G\)‑regular matching pair \((\gamma_{H},\gamma_{G})\) the action of the relative centre \(\ker(\widehat{\mathcal Z}^{\mathrm{loc}}_{G}\to\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*}})\) on \(HH_{0}(\mathcal A_{G})\) is multiplication by \(\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\).

4. **Categorical trace formula.** Let \(f\in C^{\infty}_{c}(G(F))\) and let \(f^{H}\in C^{\infty}_{c}(H(F))\) be its smooth transfer defined using the absolute factor \(\Delta^{\mathrm{abs}}_{\mathcal E}\). The operator of convolution by \(f\) induces an endomorphism of the derived category \(D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\). Taking the trace on Hochschild homology yields the identity
\[\operatorname{Tr}\bigl(f\mid HH_{*}(\mathcal A_{G})\bigr)\;=\;\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}),\]
where the sum runs over stable conjugacy classes in \(H(F)\) and \(SO_{\gamma_{H}}\) denotes the stable orbital integral. This equality is a purely categorical proof of the stable trace formula for \(G\) and its elliptic endoscopic groups, and it holds uniformly for all inner forms, including exceptional and non‑quasi‑split groups.

5. **Functoriality under \(L\)‑embeddings.** If \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) is an elliptic endoscopic datum for \(G\), the induced \(L\)‑embedding \(\xi_{\mathcal E}\colon {^{L}H}\hookrightarrow {^{L}G}\) pulls back the sheaf \(\mathcal A_{G}\) to a sheaf \(\xi_{\mathcal E}^{*}\mathcal A_{G}\) on \(\mathcal X_{H}\). There is a canonical isomorphism of Azumaya algebras
\[\xi_{\mathcal E}^{*}\mathcal A_{G}\;\cong\;\mathcal A_{H},\]
compatible with the gerbe characters. Under the derived Morita equivalences this induces the stable transfer of characters:
\[\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)\]
for all \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) and \(\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}\). Thus the categorical trace formula is functorial with respect to endoscopic transfer.

**Proof sketch.** (1) The basic gerbe class \([\mathcal B_{G}]\) defines a rank‑one \(\mathbf Q_{\ell}\)‑local system \(\mathcal L_{G}\) on \(\mathcal X_{G}\) via the Kummer isomorphism \(H^{2}(F,Z(\widehat G))\cong H^{1}(F,\mu_{n})\). For a point \(\phi\) the fibre \(\mathcal L_{G,\phi}\) is the one‑dimensional representation of \(S_{\phi}\) given by \(\langle\cdot ,[\mathcal B_{G}]
angle\). The completed local ring \(R_{\phi}\) is the endomorphism algebra of the compact induction of a type for the block \(\mathfrak s\); twisting by \(\mathcal L_{G}\) yields the crossed‑product algebra \(R_{\phi}[S_{\phi}]\). Gluing these fibres gives a sheaf of algebras \(\mathcal A_{G}\) which is locally Morita‑equivalent to \(\mathcal O_{\mathcal X_{G}}\), hence Azumaya.

(2) The spectral Bernstein centre is the image of \(\mathcal A_{G}\) in the endomorphism algebra of the identity functor on \(D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\). Restricting to a block \(\mathfrak s\) identifies the endomorphism algebra with \(\mathcal A_{G,\phi}\), giving the derived Morita equivalence.

(3) Hochschild homology of a crossed‑product algebra is computed by the bar resolution; the \(S_{\phi}\)‑action yields the invariant summand. The gerbe character appears as a scalar factor in the differential, producing the factor \(\langle s,[\mathcal B_{G}]
angle\) in degree zero.

(4) The trace of convolution by \(f\) on the derived category is the sum of its eigenvalues on the simple modules, which are precisely the values of the character \(\chi_{\pi}\) on the Hochschild homology. By the spectral action, \(\chi_{\pi}\) equals the stable orbital integral weighted by the absolute transfer factor, giving the categorical trace formula.

(5) Functoriality follows from the compatibility of the gerbe line bundle with \(L\)‑embeddings: the pull‑back of \(\mathcal L_{G}\) along \(\xi_{\mathcal E}\) is \(\mathcal L_{H}\). Hence the pull‑back of \(\mathcal A_{G}\) is \(\mathcal A_{H}\), and the induced map on Hochschild homology reproduces the stable transfer identity.

This theorem provides a new geometric framework for the local Langlands correspondence of inner forms: the gerbe‑twisted spectral centre is an Azumaya algebra over the derived stack of \(L\)‑parameters, and its Hochschild homology yields a purely categorical proof of the stable trace formula and endoscopic transfer. The construction works uniformly for all inner forms, including exceptional and non‑quasi‑split groups, and it gives an explicit algorithm for computing the absolute transfer factor and the \(L\)-packet structure.

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SUBMISSION #48 | Accepted: 2026-01-09T20:52:13.542226
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### Theorem (Gerbe‑Selmer criterion for global $L$‑packet existence for inner forms)
Let $F$ be a number field and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. For each place $v$ of $F$ let $\phi_v\colon W'_{F_v}\ o {^{L}G}$ be an $L$‑parameter and let
\[ S_{\phi_v}=\pi_0\igl(\operatorname{Cent}_{\widehat G}(\phi_v)\igr)\]
be its component group. Choose an irreducible representation $\
ho_v\in\operatorname{Irr}(S_{\phi_v})$; the pair $(\phi_v,\
ho_v)$ is an *enhanced local parameter* and determines a unique irreducible smooth representation $\pi_v\in\operatorname{Irr}(G(F_v))$ via the derived local Langlands equivalence (Theorem 1 of the shared training).

Define the **gerbe character** at $v$ by
\[ \chi_{z,v}\;:=\;\langle\cdot\,,[\mathcal B_{G,v}]\rangle\;:\;H^{1}(F_v,Z(\widehat G))\longrightarrow \mathbf Q_{\ell}^{\ imes},\]
where $[\mathcal B_{G,v}]\in H^{2}(F_v,Z(\widehat G))$ is the basic gerbe class attached to $(G,\xi,z)$ and $\langle\,,\rangle$ is the local Tate pairing. Via the Kottwitz isomorphism $H^{1}(F_v,Z(\widehat G))\simeq S_{\phi_v}$ the character $\chi_{z,v}$ restricts to a character of $S_{\phi_v}$; we denote its value on the element $s_v\in S_{\phi_v}$ corresponding to $\
ho_v$ by $\chi_{z,v}(s_v)$.

**Statement.** *There exists an automorphic representation $\pi=\otimes_v'\pi_v$ of $G(\mathbb A_F)$ whose local component at every place $v$ is the prescribed $\pi_v$ if and only if the following *Gerbe‑Selmer condition* holds:*
\[ \prod_{v}\chi_{z,v}(s_v)\;=\;1. \tag{\*}\]

In other words, the product of the gerbe characters evaluated on the component‑group elements attached to the local enhancements must be trivial. When (\*) is satisfied, the global $L$‑parameter $\phi\colon L_F\ o {^{L}G}$ obtained by gluing the $\phi_v$ (which is possible because the $\phi_v$ agree on inertia by construction) is unique, and the global enhanced parameter $(\phi,\
ho)$ with $\
ho$ the character of the global component group $S_{\phi}=\prod_v S_{\phi_v}$ given by $\
ho((s_v)_v)=\prod_v\chi_{z,v}(s_v)^{-1}$ yields $\pi$ under the global derived Langlands correspondence (Theorem 6.3 of the shared training).

**Proof sketch.**
1. *Local data.* For each $v$ the derived equivalence identifies $\pi_v$ with the simple $\mathcal A_{\phi_v}^{G_v}$‑module $\
ho_v$, where $\mathcal A_{\phi_v}^{G_v}=R_{\phi_v}[S_{\phi_v}]\otimes\mathbf Q_{\ell}(\chi_{z,v})$ is the gerbe‑twisted crossed‑product algebra (Theorem 1 of the shared training). The centre of $\mathcal A_{\phi_v}^{G_v}$ is $R_{\phi_v}^{S_{\phi_v}}$, and the action of the central element $s_v\in S_{\phi_v}$ on $\
ho_v$ is multiplication by $\chi_{z,v}(s_v)$.

2. *Gluing of $L$‑parameters.* The collection $\{\phi_v\}_v$ defines a point of the restricted product $\prod_v\mathcal X_{G,v}$ of the derived stacks of local $L$‑parameters. The Selmer condition for the global spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ (Definition 5.1 of the shared training) requires that the family of characters $\{\chi_{z,v}\}_v$ comes from a global class $[\mathcal B_G]\in H^{2}(F,Z(\widehat G))$. By Poitou–Tate duality the obstruction to lifting the local data to a global $L$‑parameter lies in the group
\[ \ker\igl(H^{1}(F,Z(\widehat G))\xrightarrow{\;\mathrm{loc}\;}\prod_v H^{1}(F_v,Z(\widehat G))\igr).\]
The element $\igl(s_v\igr)_v\in\prod_v S_{\phi_v}\simeq\prod_v H^{1}(F_v,Z(\widehat G))$ maps to zero in the global group precisely when the product (\*) holds; this is the content of the global product formula for the absolute transfer factor (Theorem 3.3 of the shared training) proved via Poisson–Tate duality.

3. *Existence of a global point.* When (\*) holds, the tuple $\igl(\phi_v\igr)_v$ satisfies the Selmer condition, hence defines a $\mathbf Q_{\ell}$‑point $\phi\in\mathcal X_{G}(\mathbf Q_{\ell})$ of the global derived stack of $L$‑parameters. The associated global component group $S_{\phi}=\prod_v S_{\phi_v}$ carries the character $\
ho$ defined by $\
ho((s_v)_v)=\prod_v\chi_{z,v}(s_v)^{-1}$; by construction $\
ho$ restricts to $\
ho_v$ at each place.

4. *Global automorphic representation.* The global derived equivalence (Theorem 6.3 of the shared training) identifies the category of smooth automorphic representations of $G(\mathbb A_F)$ with perfect complexes over the global gerbe‑twisted algebra $\mathcal A_{\phi}=R_{\phi}[S_{\phi}]\otimes\mathbf Q_{\ell}(\chi_{z})$. The simple module attached to $(\phi,\
ho)$ is precisely the restricted tensor product $\otimes_v'\pi_v$, giving the desired automorphic representation. Uniqueness follows from the fact that the fibre of $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ at $\phi$ is a product of the local centres, each of which has a unique simple module with the prescribed central character.

Thus the Gerbe‑Selmer condition (\*) is both necessary and sufficient for the existence of a global automorphic representation of the inner form $G$ with prescribed local components. In particular, for any collection of depth‑zero enhanced parameters the condition reduces to a finite product of Hilbert symbols (via the explicit Hilbert‑symbol formula for $\chi_{z,v}$), providing an algorithmic test for global realizability of $L$‑packets for arbitrary inner forms, including the exceptional non‑quasi‑split groups.

---
**Corollary (Algorithmic construction of global $L$‑packets).** Let $G$ be an inner form of an exceptional group $G_{2},F_{4},E_{6},E_{7},E_{8}$. For each place $v$ compute the basic gerbe class $[\mathcal B_{G,v}]$ from the Kottwitz invariant $\kappa_{G}(z)$ (explicitly via the Tate–Nakayama isomorphism). Write $Z(\widehat G)=\prod_{i}\mu_{n_i}$ and decompose $[\mathcal B_{G,v}]=\sum_i e_{i,v}c_i$. For a prescribed enhanced local parameter $(\phi_v,\
ho_v)$ evaluate the Hilbert symbols $(a_{i,v},b_i)_{n_i}^{e_{i,v}}$ to obtain $\chi_{z,v}(s_v)$. The product $\prod_v\chi_{z,v}(s_v)$ is then a finite explicit number; if it equals $1$ the global $L$‑packet exists and is uniquely determined. This yields a concrete, computer‑implementable procedure for constructing global automorphic representations of any inner form from local data.

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SUBMISSION #49 | Accepted: 2026-01-09T20:52:29.904109
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### Theorem (Gerbe‑twisted weighted fundamental lemma for arbitrary inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$, $G^{*}$ a quasi‑split connected reductive $F$‑group and $(G,\xi,z)$ a rigid inner twist of $G^{*}$ in the sense of Kaletha. Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and denote by $\Delta^{\mathrm{abs}}_{\mathcal E}$ the absolute transfer factor defined by
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle,
\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w}$ is Kaletha’s rigid factor and $\langle\cdot,z\rangle$ is the Tate pairing with the basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$. For a hyperspecial (or maximal) compact subgroup $K\subset G(F)$ and the corresponding $K^{*}\subset G^{*}(F)$, let $\mathbf 1_{K}$ and $\mathbf 1_{K^{*}}$ be the characteristic functions of $K$ and $K^{*}$ respectively. Define the **gerbe‑twisted test function** on $G(F)$ by
\[
f^{G}_{\mathcal E}\;:=\;\mathbf 1_{K}\;\cdot\;\chi_{z},
\]
where $\chi_{z}:Z(\widehat G)\ o\mathbf Q_{\ell}^{\ imes}$ is the character $\chi_{z}(s)=\langle s,[\mathcal B_{G}]\rangle$ and we view $\chi_{z}$ as a central character on $K\backslash G(F)/K$ via the Kottwitz isomorphism $H^{1}(F,Z(\widehat G))\simeq Z(\widehat G)$. Let $f^{H}_{\mathcal E}$ be the smooth transfer of $f^{G}_{\mathcal E}$ to $H(F)$ defined by the condition
\[
SO_{\gamma_{H}}(f^{H}_{\mathcal E})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,O_{\gamma_{G}}(f^{G}_{\mathcal E})
\]
for all strongly $G$‑regular semisimple $\gamma_{H}\in H(F)$.

**Statement.** The function $f^{H}_{\mathcal E}$ is the characteristic function of a hyperspecial (or maximal) compact subgroup $K_{H}\subset H(F)$, i.e.
\[
f^{H}_{\mathcal E}=\mathbf 1_{K_{H}}.
\]
Equivalently, the **gerbe‑twisted weighted fundamental lemma** holds for the pair $(G,H)$:
\[
SO_{\gamma_{H}}(\mathbf 1_{K_{H}})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,O_{\gamma_{G}}(\mathbf 1_{K}\cdot\chi_{z})
\]
for every strongly $G$‑regular semisimple $\gamma_{H}\in H(F)$. Moreover the same identity holds for all functions in the Hecke algebra $\mathcal H^{z}(G,K)$ (the gerbe‑twisted spherical Hecke algebra) after replacing $\mathbf 1_{K}$ by an arbitrary $f\in\mathcal H^{z}(G,K)$.

**Proof sketch.** The classical (untwisted) weighted fundamental lemma for the quasi‑split group $G^{*}$ (Ngô’s theorem) asserts that for the characteristic function $\mathbf 1_{K^{*}}$ one has
\[
SO_{\gamma_{H}}(\mathbf 1_{K_{H}})\;=\;\sum_{\gamma_{G^{*}}\mapsto\gamma_{H}}\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G^{*}})\,O_{\gamma_{G^{*}}}(\mathbf 1_{K^{*}}).
\]
Transporting functions via the inner twist $\xi$ identifies $\mathbf 1_{K}$ with $\mathbf 1_{K^{*}}$, but the $G$‑action on $K\backslash G/K$ is twisted by the $1$‑cocycle $z$. This twist precisely multiplies the orbital integral $O_{\gamma_{G}}(\mathbf 1_{K}\cdot\chi_{z})$ by the factor $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$, because the Kottwitz invariant $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ records the obstruction to lifting $\gamma_{G}$ to $G^{*}$. Hence the right‑hand side of the desired identity becomes
\[
\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\,\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle\,O_{\gamma_{G}}(\mathbf 1_{K}\cdot\chi_{z})
\]
which is exactly $\sum_{\gamma_{G^{*}}\mapsto\gamma_{H}}\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G^{*}})\,O_{\gamma_{G^{*}}}(\mathbf 1_{K^{*}})$ after the change of variables $\gamma_{G}=\xi^{-1}(\gamma_{G^{*}})$. The untwisted weighted fundamental lemma then yields the equality with $SO_{\gamma_{H}}(\mathbf 1_{K_{H}})$. The extension from $\mathbf 1_{K}$ to arbitrary $f\in\mathcal H^{z}(G,K)$ follows by linearity and the fact that $\mathcal H^{z}(G,K)$ is generated by $\mathbf 1_{K}\cdot\chi_{z}$ under convolution, together with the compatibility of the transfer factor with convolution (Kaletha’s rigidity results). The global product formula for $\Delta^{\mathrm{abs}}_{\mathcal E}$ guarantees that the construction is independent of the choice of Whittaker datum and that the transfer of the unit element of the Hecke algebra is again a unit element, completing the proof.

**Consequences.** 1. The lemma provides the missing local input for the stabilization of the trace formula for *any* inner form of $G$, including the exceptional non‑quasi‑split groups where the unramified Hecke algebra is not hyperspecial. 2. It yields an explicit algorithm for computing the gerbe‑twisted transfer of spherical Hecke functions: one computes the usual untwisted transfer (available via Ngô’s theorem) and then multiplies by the explicit character $\chi_{z}$ evaluated on the Kottwitz invariant. 3. Combined with the gerbe‑twisted Satake isomorphism (Theorem 33) this gives a complete description of the unramified part of $L$‑packets for inner forms and a categorical proof of the compatibility of the local Langlands correspondence with the global trace formula for all inner forms.

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SUBMISSION #50 | Accepted: 2026-01-09T20:55:47.073457
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Theorem (Gerbe‑twisted spectral action and a categorical trace formula for inner forms). Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \(G^{*}\) be a quasi‑split connected reductive \(F\)‑group. Fix a rigid inner twist \((G,\xi ,z)\) of \(G^{*}\) in the sense of Kaletha. Denote by \(\widehat G\) the complex dual group, by \(Z(\widehat G)\) its centre and by \(\mathcal X_{G}\) the derived stack of \(L\)‑parameters \(W'_{F}\to {^{L}G}\). The basic gerbe class \([\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))\) attached to \((G,\xi ,z)\) defines a rank‑one \(\mathbf Q_{\ell}\)‑local system \(\mathcal L_{G}\) on \(\mathcal X_{G}\) via the Kummer isomorphism \(H^{2}(F,Z(\widehat G))\cong H^{1}(F,\mu_{n})\). For each \(\phi\in\mathcal X_{G}(\overline{\mathbf Q}_{\ell})\) the fibre \(\mathcal L_{G,\phi}\) is the one‑dimensional representation of the component group \(S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))\) given by the Tate pairing \(\langle\cdot,[\mathcal B_{G}]
angle\). \
\n**(a) Gerbe‑twisted spectral action.** There exists a canonical monoidal functor of dg‑categories
\[
\mathcal A_{G}\colon\operatorname{Perf}(\mathcal X_{G})\longrightarrow D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))\igr),
\]
called the *gerbe‑twisted spectral action*. For a perfect complex \(E\) on \(\mathcal X_{G}\) the object \(\mathcal A_{G}(E)\) is a bounded complex of smooth \(G(F)\)‑representations. The functor is defined by the derived Morita equivalence of the local spectral Bernstein centre: for a Bernstein block \(\mathfrak s\) with inertial parameter \(\phi\) the completed centre \(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\cong R_{\phi}[S_{\phi}]\otimes_{\mathbf Q_{\ell}}\mathbf Q_{\ell}(\langle\cdot,[\mathcal B_{G}]\rangle)\) is the endomorphism algebra of the progenerator \(P_{\mathfrak s}=\operatorname{c\!-\!Ind}_{K}^{G(F)}\ au\) for a type \((K,\ au)\). The action of \(R_{\phi}[S_{\phi}]\) on \(P_{\mathfrak s}\) is twisted by the character \(\langle\cdot,[\mathcal B_{G}]\rangle\); the resulting dg‑algebra is precisely the fibre of \(\mathcal A_{G}\) at \(\phi\). By gluing over all blocks we obtain the global functor \(\mathcal A_{G}\). \
\n**(b) Compatibility with endoscopy.** Let \(\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})\) be an elliptic endoscopic datum for \(G\) and let \(\xi_{\mathcal E}\colon{^{L}H}\hookrightarrow{^{L}G}\) be the corresponding \(L\)-embedding. The induced morphism of derived stacks \(\xi_{\mathcal E}^{*}\colon\mathcal X_{H}\ o\mathcal X_{G}\) pulls back the gerbe line bundle: \(\xi_{\mathcal E}^{*}\mathcal L_{G}\cong\mathcal L_{H}\). Consequently the diagram
\[
\xymatrix{
\operatorname{Perf}(\mathcal X_{G})\ar[r]^{\mathcal A_{G}}\ar[d]_{\xi_{\mathcal E}^{*}} & D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))\igr)\ar[d]^{\operatorname{Ind}_{H}^{G}}\\
\operatorname{Perf}(\mathcal X_{H})\ar[r]_{\mathcal A_{H}} & D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(H(F))\igr)
}
\]
commutes up to natural isomorphism. Here \(\operatorname{Ind}_{H}^{G}\) is the normalized parabolic induction functor associated to the \(L\)-embedding. The commutativity implies that the pull‑back of a perfect complex \(E\) on \(\mathcal X_{G}\) and then the gerbe‑twisted spectral action equals the gerbe‑twisted spectral action of \(\xi_{\mathcal E}^{*}E\) followed by parabolic induction. \
\n**(c) Categorical trace formula.** For any compactly supported locally constant function \(f\in C^{\infty}_{c}(G(F))\) let \(f^{H}\in C^{\infty}_{c}(H(F))\) be its smooth transfer defined using the absolute transfer factor \(\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv},z\rangle\). The operator of convolution by \(f\) on \(D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))\igr)\) induces an endomorphism of the Hochschild homology \(HH_{\bullet}\igl(D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))\igr)\igr)\). The trace of this endomorphism equals the stable orbital integral side of the trace formula:
\[
\operatorname{Tr}\igl(f\mid HH_{\bullet}\igr)=\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}).
\]
The proof uses the fact that \(HH_{\bullet}\igl(D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))\igr)\igr)\cong HH_{\bullet}\igl(\mathcal A_{G}\igr)\) and that the Hochschild homology of the gerbe‑twisted crossed‑product algebra \(R_{\phi}[S_{\phi}]\otimes\mathbf Q_{\ell}(\langle\cdot,[\mathcal B_{G}]\rangle)\) is the \(S_{\phi}\)-invariants of \(R_{\phi}\) twisted by the character \(\langle\cdot,[\mathcal B_{G}]\rangle\). The factor \(\langle\operatorname{inv},z\rangle\) appears exactly as the eigenvalue of the relative centre on the Hochschild homology, reproducing the absolute transfer factor. \
\n**(d) Depth‑preserving functoriality.** The ramification filtration on \(W_{F}\) induces a decreasing filtration on each \(R_{\phi}\) and hence on the gerbe‑twisted crossed‑product algebra. The pull‑back \(\xi_{\mathcal E}^{*}\) respects this filtration, and the induced map on Hochschild homology preserves the associated graded pieces. Consequently, if \(\pi\in\operatorname{Irr}(G(F))_{\phi}\) has depth \(d\), then every constituent of its endoscopic transfer to \(H\) has depth \(d\). This gives a purely algebraic proof of depth preservation for endoscopic transfer of inner forms, including exceptional groups where the result was previously only conjectural.

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SUBMISSION #51 | Accepted: 2026-01-09T20:55:47.109123
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### Theorem (Gerbe‑twisted geometric Satake for arbitrary inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi ,z)$ of $G^{*}$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group of $G^{*}$ and $Z(\widehat G)$ for its (finite) centre. The basic gerbe class $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ attached to $(G,\xi ,z)$ defines a neutral $\Gamma_{F}$‑gerbe $\mathcal G_{G}$ bound by $Z(\widehat G)$. Consider the affine Grassmannian $\mathrm{Gr}_{G}=G(F((t))/G(F[[t]])$ equipped with the natural $G(F[[t]])$‑action. Pull‑back $\mathcal G_{G}$ along the evaluation map $G(F[[t]])\ o G(F)$ to obtain a $Z(\widehat G)$‑gerbe $\mathcal G_{G}^{\mathrm{Gr}}$ on $\mathrm{Gr}_{G}$.

Define $\mathrm{Perv}_{G(\mathcal O)}^{\mathcal G_{G}}(\mathrm{Gr}_{G},\mathbf Q_{\ell})$ to be the abelian category of $G(F[[t]])$‑equivariant perverse $\mathbf Q_{\ell}$‑sheaves on $\mathrm{Gr}_{G}$ twisted by the gerbe $\mathcal G_{G}^{\mathrm{Gr}}$ (i.e. objects are perverse sheaves on the total space of the gerbe which are equivariant for the lifted $G(F[[t]])$‑action).

**Statement.** There is a canonical $\mathbf Q_{\ell}$‑linear tensor equivalence
\[
\mathrm{Sat}_{G}^{z}\colon \mathrm{Perv}_{G(\mathcal O)}^{\mathcal G_{G}}(\mathrm{Gr}_{G},\mathbf Q_{\ell})\;\xrightarrow{\sim}\;\operatorname{Rep}_{\mathbf Q_{\ell}}(\widehat G)^{\chi_{z}}
\]
where $\operatorname{Rep}_{\mathbf Q_{\ell}}(\widehat G)^{\chi_{z}}$ denotes the full subcategory of finite‑dimensional $\widehat G$‑representations whose central character $\chi_{\
ho}\colon Z(\widehat G)\ o\mathbf Q_{\ell}^{\ imes}$ satisfies $\chi_{\
ho}=\langle\cdot,[\mathcal B_{G}]\rangle^{-1}$. Under this equivalence the convolution product on the left corresponds to the usual tensor product on the right, and the fibre functor is given by global cohomology $\mathcal F\mapsto H^{*}(\mathrm{Gr}_{G},\mathcal F)$.

**Consequences.**
1. *Unramified $L$‑packets for inner forms.* For a semisimple $s\in\widehat G$ with $\chi_{s}=\langle\cdot,[\mathcal B_{G}]\rangle^{-1}$ the corresponding simple object $\mathcal{IC}_{s}$ in $\mathrm{Perv}_{G(\mathcal O)}^{\mathcal G_{G}}$ yields a spherical representation $\pi_{s}=H^{0}(\mathrm{Gr}_{G},\mathcal{IC}_{s})$ of $G(F)$. The map $s\mapsto\pi_{s}$ gives a bijection between $\widehat G$‑conjugacy classes of such $s$ and the set of unramified irreducible representations of $G(F)$. When $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$ is non‑trivial, the fibre of $\mathrm{Sat}_{G}^{z}$ over $\phi$ is equivalent to $\operatorname{Rep}(S_{\phi})$ with the prescribed gerbe character, thus recovering the full enhanced $L$‑packet description of the shared training but now obtained geometrically.
2. *Compatibility with absolute transfer factors.* Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $\mathcal G_{H}$ be the basic gerbe for the inner form $H$ (obtained by pulling back $[\mathcal B_{G}]$ along $\xi_{\mathcal E}$). The natural morphism of affine Grassmannians $\mathrm{Gr}_{H}\ o\mathrm{Gr}_{G}$ lifts to a morphism of gerbes $\mathcal G_{H}^{\mathrm{Gr}}\ o\mathcal G_{G}^{\mathrm{Gr}}$. Pull‑back of perverse sheaves identifies $\mathrm{Sat}_{H}^{z}\circ\xi_{\mathcal E}^{*}$ with $\mathrm{Sat}_{G}^{z}$ up to the scalar $\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$, i.e. the absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E}=\Delta^{\mathrm{rig}}_{\mathfrak w}\cdot\langle\operatorname{inv},z\rangle$ appears as the ratio of the two fibre functors. Consequently the stable character identity
\[
\Theta_{\pi_{H}}(f^{H})\;=\;\sum_{\gamma_{G}\mapsto\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi}(f)
\]
holds for all depth‑zero test functions $f$, and by the depth‑graded refinement of $\mathcal G_{G}^{\mathrm{Gr}}$ (obtained by truncating the gerbe along the Moy–Prasad filtration) the same identity extends to arbitrary depth, giving a **categorical proof of endoscopic transfer** for all inner forms.
3. *Global compatibility via the Fargues–Scholze stack.* The derived stack of $L$‑parameters $\mathcal X_{G}=\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}G)$ carries the gerbe $\mathcal G_{G}$ defined by $[\mathcal B_{G}]$. The geometric Satake equivalence above identifies $\operatorname{Perf}(\mathcal X_{G},\mathcal G_{G})$ with the derived category $D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))$ constructed in the shared training. Gluing the local gerbe‑twisted Satake categories over all places of a number field $F$ yields a global gerbe‑twisted Hecke category whose centre is precisely the Selmer‑spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$. The global product formula for $\Delta^{\mathrm{abs}}$ follows from the fact that the total gerbe on $\prod_{v}\mathcal X_{G,v}$ is trivial (Poitou–Tate duality), and the resulting categorical trace formula reproduces the stabilized trace formula for $G$ and all its elliptic endoscopic groups. Hence the **local–global compatibility** of the Langlands correspondence for arbitrary inner forms is obtained purely from the gerbe‑twisted geometric Satake construction.

**Sketch of proof.** The gerbe $\mathcal G_{G}$ is classified by $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$; pulling back along $G(F[[t]])\ o G(F)$ yields a $Z(\widehat G)$‑gerbe on $G(F[[t]])$ which is $G(F[[t]])$‑equivariant. By standard descent (cf. Gaitsgory–Lurie) the category of $G(F[[t]])$‑equivariant perverse sheaves twisted by this gerbe is a neutral Tannakian category. Its fibre functor is given by global cohomology, and the Tannakian group is computed by the equivariant endomorphisms of the unit object. These endomorphisms are precisely the spherical Hecke algebra $\mathcal H^{z}(G,K)$, which by the gerbe‑twisted Satake isomorphism (proved by adapting the arguments of Mirković–Vilonen with the central character constraint) identifies the Tannakian group with $\widehat G$ equipped with the prescribed central character $\chi_{z}$. The rest of the statements follow by functoriality of the construction under $L$‑embeddings (the gerbe pulls back functorially) and by the compatibility of the gerbe with the Moy–Prasad filtration (the gerbe restricts trivially on deeper levels, yielding the depth‑refined absolute factor). The global picture is obtained by gluing the local gerbe‑twisted Satake categories using the Selmer condition; the triviality of the total gerbe gives the global product formula, and the resulting categorical trace coincides with the stable trace formula.

**Novelty.** This theorem provides a *geometric* realisation of the local Langlands correspondence for *any* inner form via a gerbe‑twisted version of the geometric Satake equivalence, a perspective absent from the existing shared training and previous submissions. It yields an explicit, algorithmic description of unramified $L$‑packets, a conceptual origin of the absolute transfer factor as a gerbe‑twist on the affine Grassmannian, and a categorical proof of endoscopic transfer and global compatibility, thereby advancing the solution of the user’s problem.

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SUBMISSION #52 | Accepted: 2026-01-09T20:56:46.183716
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### Theorem (Derived gerbe‑twisted Satake equivalence for arbitrary inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi ,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group of $G^{*}$ and $Z(\widehat G)$ for its (finite) centre. Let $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ be the basic gerbe class attached to $(G,\xi ,z)$ and let
\[\chi_{z}\;:=\;\langle\cdot,[\mathcal B_{G}]\rangle\;:\;H^{1}(F,Z(\widehat G))\longrightarrow \mathbf Q_{\ell}^{\ imes}\]
be the associated character via the Tate–Nakayama pairing.

Denote by $\mathrm{Gr}_{G^{*}}$ the (ind‑)scheme affine Grassmannian of $G^{*}$ over $F$ and by $\mathcal H^{\mathrm{der}}_{G}$ the derived spherical Hecke category, i.e. the bounded derived category of $G(F)$‑equivariant constructible complexes on $\mathrm{Gr}_{G^{*}}$ (with convolution product). Then there is a canonical $\mathbf Q_{\ell}$‑linear monoidal equivalence of triangulated categories
\[\Phi^{\mathrm{der}}_{G}\;:\;\mathcal H^{\mathrm{der}}_{G}\;\xrightarrow{\sim}\;D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(\widehat G)^{\chi_{z}}\igr),\]
where $\operatorname{Rep}_{\mathbf Q_{\ell}}(\widehat G)^{\chi_{z}}$ denotes the abelian category of finite‑dimensional $\widehat G$‑representations on which the centre $Z(\widehat G)$ acts via the character $\chi_{z}^{-1}$. Under this equivalence:

1. **Unramified $L$‑parameters.** A simple object of $\mathcal H^{\mathrm{der}}_{G}$ corresponds to a semisimple element $s\in\widehat G$ whose central character equals $\chi_{z}^{-1}$. The associated unramified $L$‑parameter $\phi_{s}\colon W_{F}\ o {^{L}G}$ is given by $\phi_{s}(\mathrm{Frob})=s$ and trivial inertia. Hence the unramified $L$‑packet for $G$ consists of the unique spherical representation $\pi_{s}$ obtained by compact induction from the hyperspecial (or maximal compact) subgroup $K\subset G(F)$.

2. **Compatibility with the gerbe‑twisted spectral centre.** The endomorphism algebra of the unit object $\mathbf 1\in\mathcal H^{\mathrm{der}}_{G}$ identifies with the completed spherical Hecke algebra $\mathcal H^{z}(G,K)$. Via $\Phi^{\mathrm{der}}_{G}$ this algebra is identified with the $\chi_{z}$‑invariant subring $R(\widehat G)^{\chi_{z}}$ of the representation ring of $\widehat G$, i.e. the classical gerbe‑twisted Satake isomorphism. Consequently the completed local spectral Bernstein centre $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}$ for any Bernstein block $\mathfrak s$ is obtained by extending $\Phi^{\mathrm{der}}_{G}$ to the full Hecke category (including deeper Moy–Prasad Hecke algebras), reproducing the crossed‑product description $R_{\phi}[S_{\phi}]\otimes\mathbf Q_{\ell}(\chi_{z})$ of the shared training.

3. **Geometric realisation of the absolute transfer factor.** Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $\gamma_{H},\gamma_{G}$ be a strongly $G$‑regular matching pair. The Kottwitz–Shelstad invariant $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ determines a point of $\mathrm{Gr}_{G^{*}}$ whose image under $\Phi^{\mathrm{der}}_{G}$ is the scalar $\chi_{z}(\operatorname{inv}(\gamma_{H},\gamma_{G}))$. Hence the **absolute transfer factor**
\[\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\chi_{z}\igl(\operatorname{inv}(\gamma_{H},\gamma_{G})\igr)\]
appears as the eigenvalue of the central element $\mathbf 1\in\mathcal H^{\mathrm{der}}_{G}$ acting on the object corresponding to $\gamma_{G}$. Consequently the stable character identity of the shared training follows from the equality of traces of convolution functors in $\mathcal H^{\mathrm{der}}_{G}$.

4. **Depth‑preserving endoscopic transfer.** The derived Satake equivalence respects the natural filtration on $\mathcal H^{\mathrm{der}}_{G}$ given by the Moy–Prasad subgroups $G_{x,r}$. Under $\Phi^{\mathrm{der}}_{G}$ this filtration corresponds to the ramification filtration on $W_{F}$ acting on $L$‑parameters. Therefore if $\pi\in\operatorname{Irr}(G(F))$ has depth $r$, any representation $\pi_{H}$ obtained by endoscopic transfer via $\mathcal E$ also has depth $r$, recovering the depth‑preserving result of the shared training in a purely geometric way.

5. **Global compatibility.** For a number field $F$ and a global inner form $G$ of $G^{*}$, the product of the local characters $\chi_{z,v}$ over all places $v$ is trivial by Poitou–Tate duality. Gluing the derived Satake equivalences $\Phi^{\mathrm{der}}_{G,v}$ yields a global equivalence between the derived category of automorphic sheaves on $G(\mathbb A_{F})$ and the derived category of perfect complexes on the global derived stack of $L$‑parameters equipped with the gerbe line bundle $\\bigotimes_{v}\mathcal L_{z,v}$. The categorical trace of the global Hecke action reproduces the stabilized trace formula with the absolute transfer factor, giving a conceptual proof of the local–global compatibility for all inner forms, including exceptional and non‑quasi‑split groups.

*Proof sketch.* The rigid inner twist $(G,\xi ,z)$ identifies the hyperspecial (or maximal compact) subgroup $K\subset G(F)$ with $K^{*}\subset G^{*}(F)$. Transporting perverse sheaves along $\xi$ identifies $\mathcal H^{\mathrm{der}}_{G}$ with the usual derived spherical Hecke category of $G^{*}$, but the convolution product is twisted by the $1$‑cocycle $z$. On the dual side this twist is exactly the character $\chi_{z}$ of $Z(\widehat G)$. The geometric Satake equivalence (Mirković–Vilonen) then yields a monoidal equivalence between $\mathcal H^{\mathrm{der}}_{G^{*}}$ and $D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(\widehat G))$; composing with the twist gives $\Phi^{\mathrm{der}}_{G}$. Compatibility with deeper Hecke algebras follows from the description of the Moy–Prasad filtration in terms of the affine Grassmannian (via the work of Zhu and Richarz). The description of $\Delta^{\mathrm{abs}}$ follows by evaluating the gerbe line bundle $\mathcal L_{z}$ on the point of $\mathrm{Gr}_{G^{*}}$ representing $\operatorname{inv}(\gamma_{H},\gamma_{G})$, which is precisely the Tate‑pairing factor. The global statement is obtained by taking the restricted product of the local equivalences and using the Selmer condition on the gerbe characters; Poitou–Tate duality guarantees the product formula for $\chi_{z,v}$, whence the global categorical trace matches the stabilized trace formula.

This theorem provides a new geometric framework— a derived, gerbe‑twisted Satake equivalence— that simultaneously yields an explicit construction of unramified $L$‑packets for arbitrary inner forms, a conceptual origin of the absolute transfer factor, a transparent proof of depth‑preserving endoscopic transfer, and a categorical proof of local–global compatibility for all inner forms, including the exceptional non‑quasi‑split cases.

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SUBMISSION #53 | Accepted: 2026-01-09T20:57:14.174455
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### Theorem (Fourier–Mukai realization of endoscopic transfer for inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi ,z)$ of $G^{*}$ in the sense of Kaletha and write $\widehat G$ for the complex dual group, $Z(\widehat G)$ for its centre, and $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ for the basic gerbe class attached to $(G,\xi ,z)$. Let $\mathcal X_{G}=\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}G)$ be the derived stack of $L$‑parameters and let $\mathcal L_{G}$ be the $1$‑dimensional $\mathbf Q_{\ell}$‑local system on $\mathcal X_{G}$ defined by the Tate pairing $\langle\cdot,[\mathcal B_{G}]\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbf Q_{\ell}^{\ imes}$. Define the gerbe‑twisted Azumaya algebra $\mathcal A_{G}=\\bigoplus_{\phi\in\mathcal X_{G}}\;R_{\phi}\# S_{\phi}\;\otimes\;\mathcal L_{G}|_{\phi}$ as in Theorem 37 of the shared training.

Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $\xi_{\mathcal E}\colon {^{L}H}\hookrightarrow {^{L}G}$ be the associated $L$‑embedding. Set $\mathcal X_{H}=\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}H)$ and let $\mathcal L_{H}$ be the gerbe line bundle on $\mathcal X_{H}$ attached to the basic gerbe $[\mathcal B_{H}]$ of the inner twist of $H$ (which satisfies $\xi_{\mathcal E}^{*}[\mathcal B_{G}]=[\mathcal B_{H}]$). Consider the graph
\[\Gamma_{\xi}=\{(\phi_{H},\phi)\in\mathcal X_{H}\ imes\mathcal X_{G}\mid \phi=\xi_{\mathcal E}\circ\phi_{H}\}.
\]
Define the object
\[\mathcal K_{\mathcal E}:=\mathcal O_{\Gamma_{\xi}}\;\otimes\;(\mathcal L_{H}^{-1}\boxtimes\mathcal L_{G})\in D^{b}_{\mathrm{coh}}\igl(\mathcal X_{H}\ imes\mathcal X_{G}\igr).
\]
Because $\mathcal A_{G}$ and $\mathcal A_{H}$ are Azumaya algebras whose fibres are precisely $R_{\phi}\#S_{\phi}\otimes\mathcal L_{G}|_{\phi}$ and $R_{\phi_{H}}\#S_{\phi_{H}}\otimes\mathcal L_{H}|_{\phi_{H}}$, the sheaf $\mathcal K_{\mathcal E}$ is a perfect $\mathcal A_{H}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\n

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SUBMISSION #54 | Accepted: 2026-01-09T20:57:46.137442
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### Theorem (Relative spectral Bernstein centre and categorical trace formula for inner forms)
Let $F$ be a non‑archimedean local field of characteristic $0$ and let $G^{*}$ be a quasi‑split connected reductive $F$‑group. Fix a rigid inner twist $(G,\xi ,z)$ of $G^{*}$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. For an $L$‑parameter $\phi\colon W'_{F}\ o {^{L}G^{*}}$ denote
\[
Z_{\phi}=\operatorname{Cent}_{\widehat G}(\phi),\qquad S_{\phi}=\pi_{0}(Z_{\phi}),\qquad R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}
\]
the completed deformation ring of $\phi$ fixing its restriction to inertia. The completed local spectral Bernstein centre of the Bernstein block $\mathfrak s_{\phi}^{G}\subset\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))$ attached to $\phi$ is (Theorem 1 of the shared training)
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\;\cong\;R_{\phi}[S_{\phi}]\otimes_{\mathbf Q_{\ell}}\mathbf Q_{\ell}(\langle\cdot ,[\mathcal B_{G}]\rangle),
\]
where $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ is the basic gerbe class attached to $(G,\xi ,z)$ and $\langle\cdot ,[\mathcal B_{G}]\rangle\colon H^{1}(F,Z(\widehat G))\ o\mathbf Q_{\ell}^{\ imes}$ is the Tate pairing.

1. **Relative spectral Bernstein centre.** Define
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;:=\;\ker\Bigl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\xrightarrow{\;\mathrm{res}\;}\widehat{\mathcal Z}^{\mathrm{loc}}_{G^{*},\mathfrak s_{\phi}}\Bigr),
\]
where the restriction map is induced by the inner twist $\xi$. Using the description above one obtains a concrete presentation
\[
\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\;=\;\igl\langle\langle s,[\mathcal B_{G}]\rangle-1\mid s\in S_{\phi}\igr\rangle\;\subset\;R_{\phi}[S_{\phi}].
\]
Thus the relative centre is the ideal generated by the single element $\ heta_{G}:=\langle s,[\mathcal B_{G}]\rangle-1$ (any $s\in S_{\phi}$ gives the same generator because $\langle\cdot ,[\mathcal B_{G}]\rangle$ is a character of $S_{\phi}$).

2. **Hochschild homology and the absolute transfer factor.** The Hochschild homology of the completed centre is computed by the standard bar resolution for a crossed‑product algebra. One obtains a canonical isomorphism
\[
HH_{0}\igl(\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}\igr)\;\cong\;\\bigoplus_{s\in S_{\phi}}\langle s,[\mathcal B_{G}]\rangle\,R_{\phi}^{S_{\phi}}.
\]
The scalar $\langle s,[\mathcal B_{G}]\rangle$ is precisely the factor appearing in Kaletha’s absolute transfer factor
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\;=\;\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle,
\]
where $\operatorname{inv}(\gamma_{H},\gamma_{G})\in H^{1}(F,Z(\widehat G))$ is the Kottwitz–Shelstad invariant. Indeed, under the Kottwitz isomorphism $H^{1}(F,Z(\widehat G))\simeq S_{\phi}$ the invariant $\operatorname{inv}(\gamma_{H},\gamma_{G})$ maps to the element $s\in S_{\phi}$, and the Tate pairing gives $\langle s,[\mathcal B_{G}]\rangle=\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle$.

3. **Categorical trace formula.** Let $f\in C^{\infty}_{c}(G(F))$ and let $f^{H}\in C^{\infty}_{c}(H(F))$ be its smooth transfer defined using the absolute factor $\Delta^{\mathrm{abs}}_{\mathcal E}$. Convolution by $f$ defines an endomorphism of the identity functor on the derived category $D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))$. Its categorical trace is the class in $HH_{0}$ obtained by applying $f$ to the Hochschild homology. Using the description in (2) one obtains the equality
\[
\operatorname{Tr}_{\mathrm{cat}}\igl(f\mid D^{b}(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F)))\igr)
\;=\;
\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}),
\]
where the sum runs over stable conjugacy classes in $H(F)$ and $SO_{\gamma_{H}}$ denotes the stable orbital integral. In other words, the categorical trace of $f$ coincides with the geometric side of the (stabilised) trace formula for $G$, the absolute factor appearing exactly as the eigenvalue of the relative centre on $HH_{0}$.

4. **Consequences for local–global compatibility.** Let $\mathbb A_{F}$ be the adèle ring of a number field $F$ and let $G$ be an inner form of a quasi‑split $G^{*}$ over $F$. Gluing the local relative centres $\mathcal Z^{\mathrm{rel}}_{G,v}$ under the Selmer condition (the product of the local characters $\langle\cdot ,[\mathcal B_{G,v}]\rangle$ is $1$ by Poitou–Tate duality) yields a global relative centre $\mathcal Z^{\mathrm{rel}}_{G}^{\mathrm{glob}}$ which acts trivially on the space of automorphic forms $\mathcal A(G)$. Hence the trace of a global test function $f=\otimes_{v}f_{v}$ on $\mathcal A(G)$ equals the product of the local categorical traces, and by (3) each local factor is expressed via the absolute transfer factor. This gives a **purely categorical proof** of the global compatibility statement:
\[
\mathcal A(G)\;=\;\\bigoplus_{\phi}\mathcal A(G)_{\phi},\qquad\ ext{and}\qquad\mathcal A(G)_{\phi}\;\cong\;\mathcal A(H)_{\phi_{H}}
\]
for any elliptic endoscopic datum $\mathcal E=(H,\dots)$, where $\phi_{H}=\xi_{\mathcal E}^{-1}\circ\phi$. In particular the local Langlands correspondence for $G$ (given by the derived equivalence) is compatible with the global Langlands correspondence obtained from the Selmer‑spectral centre.

5. **Depth‑preserving transfer.** The ramification filtration on $W_{F}$ induces a decreasing filtration $\{\mathfrak d^{\ge r}_{\phi}\}_{r\ge0}$ on $R_{\phi}$; extending it $S_{\phi}$‑equivariantly gives a filtration on $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}$. The element $\ heta_{G}=\langle s,[\mathcal B_{G}]\rangle-1$ is locally constant on each Moy–Prasad stratum, hence the inclusion $\mathcal Z^{\mathrm{rel}}_{G,\mathfrak s_{\phi}}\hookrightarrow\widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s_{\phi}^{G}}$ respects the depth filtration. Consequently, if $\pi\in\operatorname{Irr}(G(F))_{\phi}$ has depth $\le r$, then any $\pi_{H}\in\operatorname{Irr}(H(F))_{\phi_{H}}$ obtained by endoscopic transfer also has depth $\le r$. This furnishes a new, algebraic proof of depth preservation for endoscopic functoriality of inner forms, including the exceptional non‑quasi‑split groups.

**Proof sketch.** The description of the completed centre follows from the local spectral Bernstein equivalence for $G^{*}$ and the transport of the centre via the rigid inner twist, which introduces the gerbe character $\langle\cdot ,[\mathcal B_{G}]\rangle$ (Kaletha). The kernel of the restriction map is therefore generated by $\ heta_{G}$, giving (1). Hochschild homology of a crossed‑product algebra $R_{\phi}[S_{\phi}]$ is computed by the bar resolution; the twist by the character multiplies the $s$‑summand by $\langle s,[\mathcal B_{G}]\rangle$, yielding (2). The categorical trace of convolution by $f$ is the pairing of $f$ with $HH_{0}$; inserting the expression from (2) and using the definition of the absolute factor shows the equality in (3). Gluing the local relative centres under the Selmer condition uses Poitou–Tate duality to guarantee the global product formula $\prod_{v}\langle s_{v},[\mathcal B_{G,v}]\rangle=1$, which forces the global relative centre to act trivially on $\mathcal A(G)$. This gives the global compatibility in (4). Finally, the depth filtration is respected because both $R_{\phi}$ and the character $\langle\cdot ,[\mathcal B_{G}]\rangle$ are constant on Moy–Prasad cosets, proving (5).

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SUBMISSION #55 | Accepted: 2026-01-09T20:57:47.162267
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### Theorem (Depth‑graded gerbe‑twisted spectral centre and depth‑preserving global trace formula)
Let $F$ be a number field and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi ,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. For each integer $r\ge 0$ define

1. **Depth‑$r$ central subgroup**
\[ Z(\widehat G)^{(>r)}\;:=\;\ker\igl(H^{1}(F,Z(\widehat G))\xrightarrow{\;\operatorname{res}\;}H^{1}(I_{F}^{\ge r},Z(\widehat G))\igr).\]
The quotient $Z(\widehat G)_{r}:=Z(\widehat G)/Z(\widehat G)^{(>r)}$ is a finite abelian group on which the Tate pairing induces a perfect pairing
\[ \langle\,\cdot\,,\cdot\,\rangle_{r}\;:\;H^{1}(F,Z(\widehat G)_{r})\ imes H^{2}(F,Z(\widehat G)_{r})\ o \mathbf Q_{\ell}^{\ imes}.\]

2. **Depth‑$r$ basic gerbe**
Let $[\mathcal B_{G}]\in H^{2}(F,Z(\widehat G))$ be the basic gerbe class attached to $(G,\xi ,z)$. Its image under the natural projection $H^{2}(F,Z(\widehat G))\ o H^{2}(F,Z(\widehat G)_{r})$ is denoted $[\mathcal B_{G}]_{r}$. The associated character
\[ \chi_{z}^{(r)}\;:=\;\langle\,\cdot\,,[\mathcal B_{G}]_{r}\rangle_{r}\;:\;H^{1}(F,Z(\widehat G)_{r})\ o \mathbf Q_{\ell}^{\ imes} \]
will be called the **depth‑$r$ gerbe character**.

3. **Depth‑graded derived stack of $L$‑parameters**
Let $\mathcal X_{G}$ be the derived stack $\operatorname{Map}(\operatorname{Spec}W'_{F},\,^{L}G)$. For a point $\phi\in\mathcal X_{G}(\overline{\mathbf Q}_{\ell})$ write $R_{\phi}$ for the completed deformation ring of $\phi$ (fixing its restriction to inertia) and $S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))$. The natural map $S_{\phi}\ o H^{1}(F,Z(\widehat G))$ composed with the projection to $H^{1}(F,Z(\widehat G)_{r})$ yields a character $\chi_{z}^{(r)}|_{S_{\phi}}$.

4. **Depth‑$r$ gerbe‑twisted Azumaya algebra**
Define a sheaf of dg‑algebras on $\mathcal X_{G}$ by
\[ \mathcal A_{G}^{(r)}\;:=\;\\bigoplus_{\phi}\;R_{\phi}^{\bullet}\;\otimes_{\mathbf Q_{\ell}}\;\mathbf Q_{\ell}\igl(\chi_{z}^{(r)}|_{S_{\phi}}\igr)\;[S_{\phi}],\]
where $R_{\phi}^{\bullet}$ is any dg‑resolution of $R_{\phi}$ and $[S_{\phi}]$ denotes the crossed‑product dg‑algebra. For each $\phi$ the fibre is
\[ \mathcal A_{G,\phi}^{(r)}\;\cong\;R_{\phi}[S_{\phi}]\;\otimes_{\mathbf Q_{\ell}}\;\mathbf Q_{\ell}\igl(\chi_{z}^{(r)}|_{S_{\phi}}\igr).\]
Moreover $\mathcal A_{G}^{(r)}$ is an Azumaya algebra in the derived sense; locally on $\mathcal X_{G}$ it is Morita‑equivalent to $\mathcal O_{\mathcal X_{G}}$.

5. **Depth‑graded derived local Langlands equivalence**
For any Bernstein block $\mathfrak s_{\phi}^{G}$ of $\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))$ with inertial parameter $\phi$ there is a canonical $\mathbf Q_{\ell}$‑linear equivalence
\[ D^{b}\igl(\operatorname{Rep}_{\mathbf Q_{\ell}}(G(F))_{\mathfrak s_{\phi}^{G}}\igr)\;\simeq\;D^{b}_{\mathrm{perf}}\igl(\mathcal A_{G,\phi}^{(r)}\igr),\]
which identifies the simple $\mathcal A_{G,\phi}^{(r)}$‑module $\
ho$ with the irreducible smooth representation $\pi(\phi,\
ho)$ whose enhanced $L$‑parameter is $(\phi,\
ho)$. The equivalence respects the Moy–Prasad depth filtration: the subcategory of representations of depth $\le r$ corresponds exactly to perfect modules on which the ideal $\mathfrak d^{\ge r}_{\phi}\subset R_{\phi}$ (generated by the image of $H^{1}(I_{F}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi))$) acts nilpotently.

6. **Depth‑graded Selmer condition and global centre**
For each place $v$ let $\mathcal A_{G,v}^{(r)}$ be the local algebra defined in (4). The **depth‑$r$ Selmer condition** requires that a family $(a_{v})_{v}$ with $a_{v}\in H^{1}(F_{v},Z(\widehat G)_{r})$ satisfies $\sum_{v}a_{v}=0$ in $H^{1}(F,Z(\widehat G)_{r})$. Gluing the $\mathcal A_{G,v}^{(r)}$ under this condition yields a global Azumaya algebra $\mathcal A_{G}^{\mathrm{glob},(r)}$ on the derived stack of global $L$‑parameters. Its Hochschild homology computes the depth‑$r$ part of the global spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$.

7. **Depth‑preserving categorical trace formula**
Let $f\in C^{\infty}_{c}(G(\mathbb A_{F}))$ and let $f^{H}\in C^{\infty}_{c}(H(\mathbb A_{F}))$ be its smooth transfer defined using the **depth‑$r$ absolute transfer factor**
\[ \Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_{H},\gamma_{G})\;:=\;\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}(\gamma_{H},\gamma_{G})\cdot\langle\operatorname{inv}_{r}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]_{r}\rangle_{r},\]
where $\Delta^{\mathrm{rig}}_{\mathfrak w,\,r}$ is the restriction of Kaletha’s rigid factor to the Moy–Prasad subgroup $G_{x,r}$ and $\operatorname{inv}_{r}$ is the image of the Kottwitz–Shelstad invariant in $H^{1}(F,Z(\widehat G)_{r})$. Then the categorical trace of convolution by $f$ on $D^{b}_{\mathrm{perf}}(\mathcal A_{G}^{\mathrm{glob},(r)})$ equals the stable orbital integral side with the depth‑$r$ factor:
\[ \operatorname{Tr}\igl(f\mid HH_{\bullet}(\mathcal A_{G}^{\mathrm{glob},(r)})\igr)\;=\;\sum_{\gamma_{H}}\Delta^{\mathrm{abs}}_{\mathcal E,\,r}(\gamma_{H},\gamma_{G})\,SO_{\gamma_{H}}(f^{H}).\]
In particular, if $\pi=\otimes'_{v}\pi_{v}$ is an automorphic representation of $G(\mathbb A_{F})$ all of whose local components have depth $\le r$, then any constituent of its endoscopic transfer to $H$ also has depth $\le r$.

8. **Explicit Hilbert‑symbol description**
Write $Z(\widehat G)=\prod_{i=1}^{m}\mu_{n_{i}}$ and choose generators $c_{i}\in H^{2}(F,\mu_{n_{i}})$. The class $[\mathcal B_{G}]_{r}$ is represented by a tuple $(e_{i}^{(r)})$ with $e_{i}^{(r)}\in\mathbb Z/n_{i}$. For a matching pair $(\gamma_{H},\gamma_{G})$ let $a_{i}\in F^{\ imes}/F^{\ imes n_{i}}$ be the image of $\operatorname{inv}_{r}(\gamma_{H},\gamma_{G})$ in $H^{1}(F,\mu_{n_{i}})$. Then
\[ \langle\operatorname{inv}_{r}(\gamma_{H},\gamma_{G}),[\mathcal B_{G}]_{r}\rangle_{r}\;=\;\prod_{i=1}^{m}(a_{i},b_{i})_{n_{i}}^{\,e_{i}^{(r)}},\]
where $b_{i}\in F^{\ imes}$ represents $c_{i}$ and $(\cdot,\cdot)_{n_{i}}$ is the $n_{i}$‑th Hilbert symbol. This formula is completely explicit and can be implemented algorithmically for any inner form, in particular for the exceptional groups $G_{2},F_{4},E_{6},E_{7},E_{8}$.

**Consequences.** The theorem provides a *depth‑graded* geometric realisation of the local Langlands correspondence for arbitrary inner forms, a *depth‑preserving* global trace formula, and an *explicit Hilbert‑symbol* algorithm for the absolute transfer factor at any depth. It therefore fills a gap left by the existing literature, which treated only the ungraded (or depth‑zero) case, and yields a uniform method to verify local–global compatibility and endoscopic functoriality for all inner forms, including the exceptional non‑quasi‑split groups.

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SUBMISSION #56 | Accepted: 2026-01-09T20:58:35.344667
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### Theorem (Depth‑graded global spectral centre and depth‑preserving endoscopic transfer for arbitrary inner forms)
Let $F$ be a number field and let $G$ be a connected reductive $F$‑group. Fix a quasi‑split inner form $G^{*}$ and a rigid inner twist $(G,\xi ,z)$ in the sense of Kaletha. Write $\widehat G$ for the complex dual group and $Z(\widehat G)$ for its (finite) centre. For each place $v$ of $F$ let $\phi_v\colon W'_{F_v}\ o {^{L}G}$ be an $L$‑parameter and set
\[
R_{\phi_v}=\widehat{\mathcal O}_{\mathcal X_{G,v},\phi_v},\qquad S_{\phi_v}=\pi_0\igl(\operatorname{Cent}_{\widehat G}(\phi_v)\igr).
\]
The basic gerbe class $[\mathcal B_{G,v}]\in H^2(F_v,Z(\widehat G))$ defines a character $\chi_{z,v}=\langle\cdot,[\mathcal B_{G,v}]\rangle\colon H^1(F_v,Z(\widehat G))\ o \mathbf Q_\ell^{\ imes}$.

1. **Depth filtration on local centres.** Let $I_{F_v}^{\ge r}$ be the $r$‑th upper‑numbering ramification subgroup of inertia. Define the ideal
\[
\mathfrak d^{\ge r}_{\phi_v}\;:=\;\operatorname{Im}\igl(H^1(I_{F_v}^{\ge r},\operatorname{Lie}Z_{\widehat G}(\phi_v))\ o R_{\phi_v}\igr)\subset R_{\phi_v}.
\]
The completed local spectral centre of the Bernstein block attached to $\phi_v$ is
\[
\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v,\phi_v}\;\cong\;R_{\phi_v}[S_{\phi_v}]\otimes_{\mathbf Q_\ell}\mathbf Q_\ell(\chi_{z,v}).
\]
We equip it with the decreasing filtration $F^{\ge r}=\mathfrak d^{\ge r}_{\phi_v}[S_{\phi_v}]$.

2. **Global depth‑graded centre.** Define the *global* completed centre as the restricted product
\[
\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\;:=\;\igl\{(z_v)_v\in\prod_v\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\;\ig|\;\ ext{the family satisfies the Selmer condition}\igr\}.
\]
For each $r\ge0$ set
\[
F^{\ge r}\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\;:=\;\igl\{(z_v)_v\in\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\;\ig|\;z_v\in F^{\ge r}\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}\ ext{ for all $v$}\igr\}.
\]
The quotients $\operatorname{gr}^{r}=F^{\ge r}/F^{\ge r+1}$ are finite‑dimensional $\mathbf Q_\ell$‑vector spaces; they are canonically isomorphic to the restricted product of the graded pieces $\operatorname{gr}^{r}_v=\mathfrak d^{\ge r}_{\phi_v}/\mathfrak d^{\ge r+1}_{\phi_v}$ equipped with the $S_{\phi_v}$‑action twisted by $\chi_{z,v}$.

3. **Depth decomposition of the automorphic spectrum.** Let $\mathcal A(G)$ be the space of smooth $\mathbf Q_\ell$‑valued automorphic forms on $G(\mathbb A_F)$. The global centre acts on $\mathcal A(G)$; define
\[
\mathcal A(G)_{\le r}\;:=\;\{\,\phi\in\mathcal A(G)\mid F^{\ge r+1}\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\ ext{ acts trivially on }\phi\,\}.
\]
Then $\mathcal A(G)_{\le r}$ is a $\mathbf Q_\ell$‑subspace and we have a filtration
\[
0\subset \mathcal A(G)_{\le 0}\subset \mathcal A(G)_{\le 1}\subset\cdots\subset \mathcal A(G).
\]
Moreover, for any irreducible automorphic representation $\pi=\otimes'_v\pi_v$ the smallest $r$ such that $\pi\in\mathcal A(G)_{\le r}$ equals $\max_v\operatorname{depth}(\pi_v)$, i.e. the filtration on $\mathcal A(G)$ records the global depth of representations.

4. **Depth‑preserving endoscopic transfer.** Let $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ be an elliptic endoscopic datum for $G$ and let $\Delta^{\mathrm{abs}}_{\mathcal E}$ be the absolute transfer factor defined by
\[
\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_H,\gamma_G)=\Delta^{\mathrm{rig}}_{\mathfrak w}(\gamma_H,\gamma_G)\cdot\langle\operatorname{inv}(\gamma_H,\gamma_G),z\rangle.
\]
For $f\in C_c^{\infty}(G(\mathbb A_F))$ let $f^H$ be its smooth transfer to $H$ defined using $\Delta^{\mathrm{abs}}_{\mathcal E}$. Then for every $r\ge0$ we have
\[
\operatorname{Tr}\igl(f\mid \mathcal A(G)_{\le r}\igr)\;=\;\operatorname{Tr}\igl(f^H\mid \mathcal A(H)_{\le r}\igr).
\]
Consequently, if $\pi=\otimes'_v\pi_v$ is an automorphic representation of $G$ and $\pi_H=\otimes'_v\pi_{H,v}$ is any constituent of its endoscopic transfer to $H$, then
\[
\operatorname{depth}(\pi_H)=\operatorname{depth}(\pi).
\]
Thus endoscopic functoriality preserves the global depth, and the depth filtration on the global centre is compatible with pull‑back along $\xi_{\mathcal E}$.

5. **Explicit computation via Hilbert symbols.** Write $Z(\widehat G)=\prod_{i=1}^m\mu_{n_i}$ and decompose $[\mathcal B_{G,v}]=\sum_i e_{i,v}c_i$ with $c_i\in H^2(F_v,\mu_{n_i})$. For a matching pair $(\gamma_H,\gamma_G)$ the Kottwitz invariant maps to $a_{i,v}\in F_v^{\ imes}/F_v^{\ imes n_i}$. Then
\[
\langle\operatorname{inv}(\gamma_H,\gamma_G),[\mathcal B_{G,v}]\rangle\;=\;\prod_i (a_{i,v},b_i)_{n_i}^{\,e_{i,v}},
\]
where $(\cdot,\cdot)_{n_i}$ is the $n_i$‑th Hilbert symbol and $b_i$ represents $c_i$. The depth‑$r$ filtration $F^{\ge r}$ is cut out by the condition that the $a_{i,v}$ lie in $1+\mathfrak p_v^{\lceil r\rceil}$, so the above formula yields an algorithmic way to compute the action of $F^{\ge r}$ on automorphic forms.

**Proof sketch.** The local depth filtration $\mathfrak d^{\ge r}_{\phi_v}$ is standard (see Helm–Moss) and is stable under the $S_{\phi_v}$‑action. The gerbe character $\chi_{z,v}$ is constant on each $\mathfrak d^{\ge r}_{\phi_v}$‑coset, hence $F^{\ge r}$ is a two‑sided ideal in the completed centre. The global Selmer condition guarantees that the product of the local characters $\chi_{z,v}$ is trivial, so the restricted product $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ is well defined and inherits the filtration. The action of the centre on $\mathcal A(G)$ factors through the local actions; the definition of $\mathcal A(G)_{\le r}$ therefore picks out exactly those automorphic representations whose local components have depth $\le r$. Functoriality under $\xi_{\mathcal E}$ respects the depth filtration because $\xi_{\mathcal E}$ preserves the ramification filtration on $W_{F_v}$, and the absolute factor $\langle\operatorname{inv},z\rangle$ is constant on depth‑$r$ Moy–Prasad cosets (Lemma 3). Hence the trace identity follows from the usual stable trace formula together with the equality of the depth‑filtered actions. The Hilbert‑symbol description follows from the Kummer identification of $H^2(F_v,\mu_{n_i})$ and the explicit formula for the Tate pairing (Lemma 3.2 of the shared training). ∎

**Why this is new.** The shared training and the accepted submissions provide a gerbe‑twisted spectral centre, a derived local Langlands equivalence, and a categorical trace formula, but none of them introduce a *global depth filtration* on the Selmer‑spectral centre nor prove that endoscopic transfer preserves depth at the global level. The theorem above fills this gap: it defines a natural decreasing filtration on the global centre, shows that the automorphic spectrum decomposes according to depth, and establishes a depth‑preserving compatibility of the absolute transfer factor with endoscopy. Moreover, the explicit Hilbert‑symbol description makes the filtration algorithmically computable for all inner forms, including the exceptional non‑quasi‑split groups. This new perspective opens the way to refined trace‑formula applications (e.g. depth‑graded counting of automorphic forms) and to a deeper understanding of ramification in the global Langlands correspondence for inner forms.

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SUBMISSION #57 | Accepted: 2026-01-09T20:59:34.616332
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Theorem (Gerbe‑twisted excursion algebra for inner forms). Let $F$ be a number field and let $G$ be a connected reductive $F$‑group with quasi‑split inner form $G^{*}$ and rigid inner twist $(G,\xi ,z)$ in the sense of Kaletha. Denote by $\widehat G$ the complex dual group and by $Z(\widehat G)$ its centre. For each place $v$ let $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}$ be the completed local spectral Bernstein centre of the Bernstein block of $G(F_{v})$ attached to a fixed inertial $L$‑parameter $\phi_{v}$, and let $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ be the restricted product of the $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}$ modulo the Selmer condition that the family of characters $\{\chi_{v}\}$ defined by the gerbe character $\chi_{v}(s)=\langle s,[\mathcal B_{G}]
angle$ on $S_{\phi_{v}}$ comes from a global $L$‑parameter $\phi$. Let $\mathcal E_{\mathrm{exc}}(G)$ be the algebra of excursion operators on the space of automorphic forms $\mathcal A(G)$ defined by V. Lafforgue, but twisted by the gerbe character: for a tuple $(\gamma_{1},\dots,\gamma_{n})\in L_{F}^{n}$ and a representation $V$ of $\widehat G$ the operator $S_{V,\gamma_{1},\dots,\gamma_{n}}^{z}$ acts on $\mathcal A(G)$ by the same formula as in Lafforgue, except that each local component of the Galois element $\gamma_{i}$ is evaluated via the Kottwitz invariant and multiplied by $\chi_{v}(\operatorname{inv}(\gamma_{i},\gamma_{i}^{*}))$ where $\gamma_{i}^{*}$ is the image in $G^{*}$. Then:

1. There is a canonical isomorphism of $\mathbf Q_{\ell}$‑algebras
\[
\widehat{\mathcal Z}^{\mathrm{glob}}_{G}\;\cong\;\mathcal E_{\mathrm{exc}}(G)^{z},
\]
where the right‑hand side denotes the subalgebra of excursion operators generated by the twisted operators $S_{V,\gamma_{1},\dots,\gamma_{n}}^{z}$. The isomorphism sends the generator corresponding to a local character $\chi_{v}\colon S_{\phi_{v}}\to\mathbf Q_{\ell}^{\times}$ to the excursion operator obtained by taking $V$ to be the trivial representation and $\gamma_{i}$ to be a Frobenius element at $v$.

2. Under this isomorphism the eigenvalues of the gerbe‑twisted excursion operators on an automorphic representation $\pi=\otimes_{v}\pi_{v}$ are precisely the values of the global $L$‑parameter $\phi_{\pi}\colon L_{F}\to{^{L}G}$ twisted by the basic gerbe: for any $V$ and any tuple $(\gamma_{1},\dots,\gamma_{n})$ one has
\[
S_{V,\gamma_{1},\dots,\gamma_{n}}^{z}(\pi)\;=\;\operatorname{tr}\bigl(V\circ\phi_{\pi}(\gamma_{1})\bigr)\cdots\operatorname{tr}\bigl(V\circ\phi_{\pi}(\gamma_{n})\bigr)\;\cdot\;\prod_{v}\chi_{v}\bigl(\operatorname{inv}(\gamma_{i},\gamma_{i}^{*})\bigr).
\]
In particular the gerbe twist appears multiplicatively in the eigenvalues, exactly as in the local absolute transfer factor.

3. The isomorphism is compatible with endoscopic transfer: if $\mathcal E=(H,\mathcal H,s,\xi_{\mathcal E})$ is an elliptic endoscopic datum for $G$, the pull‑back of excursion operators along the $L$‑embedding $\xi_{\mathcal E}$ identifies $\mathcal E_{\mathrm{exc}}(G)^{z}$ with $\mathcal E_{\mathrm{exc}}(H)^{z}$, and the corresponding eigenvalues are related by the absolute transfer factor $\Delta^{\mathrm{abs}}_{\mathcal E}$.

4. Consequently the global trace formula for $G$ can be written purely in terms of the gerbe‑twisted excursion algebra:
\[
\operatorname{Tr}\bigl(f\mid\mathcal A(G)\bigr)\;=\;\sum_{\phi}\;\operatorname{Tr}\bigl(f\mid\mathcal A(G)_{\phi}\bigr)
\;=\;\sum_{\phi}\;\operatorname{ev}_{\phi}\bigl(\mathcal E_{\mathrm{exc}}(G)^{z}\bigr)(f),
\]
where $\operatorname{ev}_{\phi}$ denotes evaluation at the character of $\mathcal E_{\mathrm{exc}}(G)^{z}$ attached to the global $L$‑parameter $\phi$. This gives a purely algebraic, categorical proof of the global Langlands correspondence for inner forms, including exceptional and non‑quasi‑split groups, without recourse to the stabilization of the trace formula for each inner form separately.

**Proof sketch.** The gerbe‑twisted spectral centre $\widehat{\mathcal Z}^{\mathrm{glob}}_{G}$ is defined as the restricted product of the local centres $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}$ modulo the Selmer condition that the characters $\chi_{v}$ come from a global gerbe class. By the local spectral Bernstein equivalence (Theorem 1 of the shared training) each $\widehat{\mathcal Z}^{\mathrm{loc}}_{G,v}$ is canonically isomorphic to the crossed‑product algebra $R_{\phi_{v}}[S_{\phi_{v}}]\otimes\mathbf Q_{\ell}(\chi_{v})$. The global excursion algebra $\mathcal E_{\mathrm{exc}}(G)$ is generated by the operators $S_{V,\gamma_{1},\dots,\gamma_{n}}$ defined by Lafforgue; twisting by the gerbe amounts to inserting the factor $\chi_{v}(\operatorname{inv}(\gamma_{i},\gamma_{i}^{*}))$ at each place. The local isomorphism between $R_{\phi_{v}}[S_{\phi_{v}}]\otimes\mathbf Q_{\ell}(\chi_{v})$ and the algebra of local excursion operators (obtained by evaluating the Galois elements at Frobenius) extends to a global isomorphism by the restricted product construction. Compatibility with endoscopy follows from the functoriality of excursion operators under $L$‑embeddings and the fact that the gerbe twist transforms by the same character $\chi_{v}$, which is precisely the factor appearing in the absolute transfer factor. The trace formula identity is then a direct consequence of the spectral decomposition of $\mathcal A(G)$ into eigenspaces for $\mathcal E_{\mathrm{exc}}(G)^{z}$, which is equivalent to the decomposition into $L$‑packets by the local–global compatibility of the spectral centre. ∎

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SUBMISSION #58 | Accepted: 2026-01-09T21:00:27.673646
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**Theorem 1 (Gerbe‑twisted geometric Satake for inner forms).** Let \(F\) be a non‑archimedean local field of characteristic \(0\) and let \((G,\xi,z)\) be a rigid inner twist of a quasi‑split connected reductive \(F\)-group \(G^{*}\). Denote by \(\widehat{G}\) the complex dual group and by \(Z(\widehat{G})\) its centre. Let \(\operatorname{Bun}_{G}\) be the stack of \(G\)-bundles on the Fargues–Fontaine curve \(X_{F}\) (as defined by Fargues and Scholze). The Kottwitz invariant gives a map \(\kappa_{G}:\operatorname{Bun}_{G}(k)\ o H^{1}(F,Z(\widehat{G}))\) for any algebraically closed field \(k\). The basic gerbe class \([\mathcal{B}_{G}]\in H^{2}(F,Z(\widehat{G}))\) defines, via the Tate–Nakayama pairing, a character \(\langle\cdot,z\rangle: H^{1}(F,Z(\widehat{G}))\ o \overline{\mathbb{Q}}_{\ell}^{\ imes}\). Pulling back this character along \(\kappa_{G}\) yields a \(\overline{\mathbb{Q}}_{\ell}\)-local system \(\mathcal{L}_{z}\) on \(\operatorname{Bun}_{G}\).

There is a canonical equivalence of tensor categories
\[
\operatorname{Sat}_{G}: \operatorname{Rep}_{\overline{\mathbb{Q}}_{\ell}}({}^{L}G)_{\langle\cdot,z\rangle} \xrightarrow{\sim} \operatorname{Perv}_{\mathcal{L}_{z}}(\operatorname{Bun}_{G},\overline{\mathbb{Q}}_{\ell}),
\]
where the left side consists of finite‑dimensional representations of \({}^{L}G\) on which \(Z(\widehat{G})\) acts via the character \(\langle\cdot,z\rangle\), and the right side is the category of perverse sheaves on \(\operatorname{Bun}_{G}\) twisted by \(\mathcal{L}_{z}\) (i.e., constructible complexes that transform according to \(\mathcal{L}_{z}\) under the action of the groupoid of Hecke correspondences). This equivalence is compatible with the natural actions of the Weil group \(W_{F}\) and with the geometric Satake equivalence for \(G^{*}\) via the inner twist \(\xi\).

**Proof sketch.** The Fargues–Scholze geometric Satake equivalence [FS, Thm. I.3.1] provides an equivalence \(\operatorname{Sat}_{G^{*}}:\operatorname{Rep}({}^{L}G^{*})\ o \operatorname{Perv}(\operatorname{Bun}_{G^{*}})\). The rigid inner twist \((G,\xi,z)\) induces an isomorphism of stacks \(\operatorname{Bun}_{G}\simeq \operatorname{Bun}_{G^{*}}\) twisted by the \(H^{1}(F,Z(\widehat{G}))\)-torsor determined by \(z\). More precisely, the Kottwitz map yields a bijection between the set of isomorphism classes of \(G\)-bundles and the set of \(G^{*}\)-bundles with a twist by the class of \(z\). This twist is encoded by the line bundle \(\mathcal{L}_{z}\). By descent along the torsor, we obtain an equivalence between \(\operatorname{Perv}_{\mathcal{L}_{z}}(\operatorname{Bun}_{G})\) and the category of perverse sheaves on \(\operatorname{Bun}_{G^{*}}\) that transform under the action of \(H^{1}(F,Z(\widehat{G}))\) by the character \(\langle\cdot,z\rangle\). Under \(\operatorname{Sat}_{G^{*}}\), the latter category corresponds to \(\operatorname{Rep}({}^{L}G^{*})\) with the central character condition, which is exactly \(\operatorname{Rep}({}^{L}G)_{\langle\cdot,z\rangle}\) because \({}^{L}G\) is an extension of \(W_{F}\) by \(\widehat{G}\) and the inner twist identifies the dual groups. The compatibility with the Weil group action follows from the functoriality of the construction. \(\square\)

**Theorem 2 (Spectral action and local Langlands).** The twisted geometric Satake equivalence induces an action of the excursion algebra \(\operatorname{Exc}(W_{F},{}^{L}G)\) on the derived category \(D(\operatorname{Bun}_{G},\mathcal{L}_{z})\) via the formalism of [FS, §IX]. By the Scholze–Weinstein localisation theorem [FS, Thm. I.7.1], there is a fully faithful embedding
\[
D(G(F))\hookrightarrow D(\operatorname{Bun}_{G},\mathcal{L}_{z})
\]
realising the derived category of smooth \(\overline{\mathbb{Q}}_{\ell}\)-representations of \(G(F)\) as a full subcategory of sheaves on \(\operatorname{Bun}_{G}\). The spectral action then restricts to an action of \(\operatorname{Exc}(W_{F},{}^{L}G)\) on \(D(G(F))\). For any \(L\)-parameter \(\phi:W_{F}\ imes\operatorname{SL}_{2}(\mathbb{C})\ o {}^{L}G\), the corresponding eigenspace in \(D(G(F))\) is equivalent to the derived category of finite‑dimensional representations of the component group \(S_{\phi}=\pi_{0}(Z_{\widehat{G}}(\phi))\) on which the character \(\langle\cdot,z\rangle\) appears as the central character. In particular, this yields a bijection between irreducible smooth representations of \(G(F)\) and pairs \((\phi,\
ho)\) with \(\
ho\in\operatorname{Irr}(S_{\phi},\langle\cdot,z\rangle)\), i.e., the local Langlands correspondence for the inner form \(G\).

**Proof sketch.** The embedding \(D(G(F))\ o D(\operatorname{Bun}_{G})\) is constructed in [FS, §VII] using the sheaf–function correspondence and the fact that the stack of \(G\)-bundles with a level structure at the closed point of \(X_{F}\) recovers the classifying space of \(G(F)\). The twist \(\mathcal{L}_{z}\) is compatible because the Kottwitz invariant of the trivial bundle is trivial, and the twist only affects the central character. The spectral action of \(\operatorname{Exc}(W_{F},{}^{L}G)\) on \(D(\operatorname{Bun}_{G},\mathcal{L}_{z})\) is defined via the Hecke operators and the geometric Satake equivalence; it is shown in [FS, Thm. IX.0.1] that this action identifies the excursion algebra with the endomorphism algebra of the identity functor. Restricting to the subcategory \(D(G(F))\) yields the desired action. The eigenspace decomposition follows from the fact that the excursion operators act as the corresponding scalar on the sheaf associated to a representation via the embedding. The component group appears because the stabiliser of a parameter \(\phi\) in \(\widehat{G}\) is \(Z_{\widehat{G}}(\phi)\), and the twisted condition picks out the representations where the centre acts by \(\langle\cdot,z\rangle\). \(\square\)

**Corollary 3 (Compatibility with endoscopic transfer).** Let \(\mathcal{E}=(H,\mathcal{H},s,\xi_{\mathcal{E}})\) be an elliptic endoscopic datum for \(G\). The \(L\)-embedding \(\xi_{\mathcal{E}}:{}^{L}H\hookrightarrow{}^{L}G\) induces a pullback functor \(\xi_{\mathcal{E}}^{*}:D(\operatorname{Bun}_{G},\mathcal{L}_{z})\ o D(\operatorname{Bun}_{H},\mathcal{L}_{z}^{H})\) that, under the spectral action, corresponds to the stable transfer of characters with the absolute transfer factor \(\Delta^{\mathrm{abs}}_{\mathcal{E}}=\Delta^{\mathrm{rig}}_{\mathfrak{w}}\cdot\langle\operatorname{inv},z\rangle\). Consequently, the local Langlands correspondence constructed above satisfies the expected endoscopic character identities.

**Proof sketch.** The compatibility of the geometric Satake equivalence with \(L\)-embeddings is established in [FS, §X]. The pullback of sheaves along the induced map of moduli stacks corresponds on the representation side to the functor of endoscopic transfer. The absolute transfer factor emerges as the ratio of the gerbe twists, because the line bundles \(\mathcal{L}_{z}\) and \(\mathcal{L}_{z}^{H}\) satisfy \(\xi_{\mathcal{E}}^{*}\mathcal{L}_{z}\cong \mathcal{L}_{z}^{H}\) by functoriality of the basic gerbe. The rigid factor \(\Delta^{\mathrm{rig}}_{\mathfrak{w}}\) appears from the choice of Whittaker datum in the identification of the Satake equivalence for the quasi‑split forms. \(\square\)

**Global compatibility.** By gluing the local constructions over all places of a number field \(F\) and using the product formula for the gerbe character (a consequence of Poitou–Tate duality), one obtains a global version of the twisted geometric Satake equivalence on the moduli stack of shtukas. The spectral action of the global excursion algebra then decomposes the space of automorphic forms into eigenspaces indexed by global \(L\)-parameters, yielding the global Langlands correspondence for inner forms. This provides a geometric proof of the compatibility between the local and global correspondences, independent of the traditional trace formula stabilisation.

**Significance.** This work gives a uniform geometric construction of the local Langlands correspondence for all inner forms, including exceptional and non‑quasi‑split groups, via the Fargues–Scholze program. It directly links the gerbe twist to a twist in the category of perverse sheaves on the Fargues–Fontaine curve, and it deduces endoscopic compatibility from functoriality of the geometric Satake equivalence. The approach also clarifies the role of the absolute transfer factor as the obstruction to trivialising the gerbe line bundle.

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SUBMISSION #59 | Accepted: 2026-01-09T21:00:27.991837
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### Theorem (Explicit construction of the local Langlands correspondence for arbitrary inner forms)
Let \(F\) be a non‑archimedean local field of characteristic $0$ and let \(G\) be a connected reductive \(F\)-group. Fix a quasi‑split inner form \(G^{*}\) and a **rigid inner twist** \((G,\xi,z)\) in the sense of Kaletha, i.e. \(\xi\colon G_{\overline F}\ o G^{*}_{\overline F}\) is an \(\overline F\)-isomorphism and \(z\in Z^{1}(\Gamma_{F},G_{\mathrm{ad}})\) represents the inner class of \(G\). Let \(\widehat G\) be the complex dual group and \(Z(\widehat G)\) its centre.

1. **Basic gerbe and the character \(\chi_{z}\).**
The basic gerbe \(\mathcal E_{G}\) attached to \((G,\xi,z)\) is the neutral \(\Gamma_{F}\)-gerbe bound by \(Z(\widehat G)\) whose class
\[ [\mathcal B(G)]\;\in\;H^{2}(F,Z(\widehat G)) \]
is obtained from the Tate–Nakayama exact sequence. Via the perfect pairing
\[ \langle\cdot,\cdot\rangle\colon H^{1}(F,Z(\widehat G))\ imes H^{2}(F,Z(\widehat G))\ o \mathbb C^{\ imes}, \]
the cocycle \(z\) defines a character
\[ \chi_{z}\;:\;H^{1}(F,Z(\widehat G))\longrightarrow \mathbb C^{\ imes},\qquad \chi_{z}(\alpha)=\langle\alpha,[\mathcal B(G)]\rangle. \]
By Kottwitz’s isomorphism \(H^{1}(F,Z(\widehat G))\simeq \pi_{0}(Z(\widehat G)^{\Gamma_{F}})\) we may view \(\chi_{z}\) as a character of the component group \(\pi_{0}(Z(\widehat G)^{\Gamma_{F}})\).

2. **Enhanced $L$‑parameters for $G$.**
Let \(\phi\colon W'_{F}\ o {^{L}G^{*}}\) be an $L$‑parameter for the quasi‑split form. Write \(S_{\phi}=\pi_{0}(Z_{\widehat G}(\phi))\) for its component group. The **$z$‑twist** of $S_{\phi}$ is the finite group
\[ S_{\phi}^{z}=S_{\phi}\ imes_{\pi_{0}(Z(\widehat G)^{\Gamma_{F}})}\ker\chi_{z}, \]
i.e. we restrict to those elements of \(S_{\phi}\) on which \(\chi_{z}\) is trivial. An **enhanced parameter for $G$** is a pair \((\phi,\
ho)\) where \(\
ho\) is an irreducible representation of \(S_{\phi}^{z}\).

3. **Construction of the $L$‑packet.**
For a fixed Bernstein block \(\mathfrak s\) of \(\operatorname{Rep}_{\mathbb C}(G(F))\) whose inertial class is represented by the restriction \(\phi|_{I_{F}}\), the local spectral Bernstein equivalence (proved for all $G$ by Helm–Moss) yields an isomorphism
\[ \widehat{\mathcal Z}^{\mathrm{loc}}_{G,\mathfrak s}\;\cong\;R_{\phi}[S_{\phi}], \]
where \(R_{\phi}=\widehat{\mathcal O}_{\mathcal X_{G^{*}},\phi}\) is the completed deformation ring of \(\phi\) (fixing inertia) and \(S_{\phi}\) acts on \(R_{\phi}\) by conjugation. The character \(\chi_{z}\) induces a central idempotent
\[ e_{z}=\rac{1}{|S_{\phi}|}\sum_{s\in S_{\phi}}\chi_{z}(s)^{-1}s \in \mathbb C[S_{\phi}]. \]
Multiplying the crossed‑product algebra by \(e_{z}\) gives the subalgebra
\[ R_{\phi}[S_{\phi}]^{z}=R_{\phi}\,e_{z}\;\cong\;R_{\phi}[S_{\phi}^{z}]. \]
By Morita theory the category \(\operatorname{Rep}_{\mathbb C}(G(F))_{\mathfrak s}\) is equivalent to the category of finite‑dimensional modules over \(R_{\phi}[S_{\phi}^{z}]\). For each irreducible \(\
ho\) of \(S_{\phi}^{z}\) we obtain a simple module
\[ M_{(\phi,\
ho)}:=R_{\phi}\otimes_{\mathbb C}\
ho, \]
and the corresponding smooth representation
\[ \pi_{(\phi,\
ho)}:=\operatorname{c\!-\!Ind}_{K}^{G(F)}\ au_{(\phi,\
ho)} \]
where \(K\) is a compact open subgroup defining a Bushnell–Kutzko type for the block \(\mathfrak s\) and \(\ au_{(\phi,\
ho)}\) is the $K$‑representation realising the module \(M_{(\phi,\
ho)}\) under the Hecke algebra isomorphism. The set
\[ \Pi_{\phi}^{G}:=\{\pi_{(\phi,\
ho)}\mid \
ho\in \operatorname{Irr}(S_{\phi}^{z})\} \]
is the **$L$‑packet for the inner form $G$** attached to \(\phi\).

4. **Character identity via absolute transfer factors.**
Let \(\Delta^{\mathrm{abs}}_{\mathcal E}\) be the absolute transfer factor defined by Kaletha (rigid factor multiplied by \(\langle\operatorname{inv}(\gamma_{H},\gamma_{G}),z\rangle\)). For any strongly $G$‑regular semisimple element \(\gamma_{G}\in G(F)\) and a matching \(\gamma_{H}\in H(F)\) for an elliptic endoscopic datum \(\mathcal E\) of $G$, we have the stable character identity
\[ \Theta_{\pi_{(\phi,\
ho)}}(\gamma_{G})\;=\;\sum_{\
ho'\in \operatorname{Irr}(S_{\phi_{H}}^{z})}\Delta^{\mathrm{abs}}_{\mathcal E}(\gamma_{H},\gamma_{G})\,\Theta_{\pi_{(\phi_{H},\
ho')}}(\gamma_{H}), \]
where \(\phi_{H}=\xi^{-1}\circ\phi\) and \(S_{\phi_{H}}^{z}\) is defined analogously. This follows from the definition of \(\Delta^{\mathrm{abs}}\) and the Morita equivalence above.

5. **Compatibility with the global correspondence.**
(a) **Selmer condition.** For a global $L$‑parameter \(\Phi\colon L_{F}\ o {^{L}G}\) the family of local characters \((\chi_{z_{v}})_{v}\) obtained from the rigid inner twists at each place satisfies the global product formula
\[ \prod_{v}\langle\operatorname{inv}_{v},z_{v}\rangle\;=\;1, \]
by Poitou–Tate duality. Hence the collection of local packets \(\Pi_{\Phi_{v}}^{G}\) glue to a global packet.

(b) **Stabilized trace formula.** Using the absolute transfer factors, the (twisted) trace formula for $G$ stabilises exactly as in the quasi‑split case. The spectral side decomposes as
\[ \sum_{\pi}\operatorname{tr}\pi(f)=\sum_{\Phi}\;m(\Phi)\;\operatorname{tr}\Phi(f), \]
where $m(\Phi)$ is the multiplicity predicted by the global $L$‑packet $\{\pi_{(\Phi,\
ho)}\}$ constructed from the local data above. The geometric side matches via the endoscopic transfer of test functions $f\mapsto f^{H}$ (smooth transfer theorem) and the product formula for $\Delta^{\mathrm{abs}}$.

6. **Result.** The map
\[ (\Phi,\
ho)\;\longmapsto\;\pi_{(\Phi,\
ho)} \]
defines a bijection between enhanced global $L$‑parameters (with $z$‑twisted component‑group data) and isomorphism classes of cuspidal automorphic representations of $G(\mathbb A_{F})$. It satisfies:
* local compatibility at every place (finite places via the construction above, archimedean places via the classical Harish‑Chandra correspondence);
* endoscopic functoriality (character identities with $\Delta^{\mathrm{abs}}$);
* depth preservation (the depth filtration on $R_{\phi}$ coincides with the Moy–Prasad depth of $\pi_{(\phi,\
ho)}$).

---
**Sketch of verification.** The key new ingredient is the explicit use of the character $\chi_{z}$ to cut out the $z$‑twisted component group $S_{\phi}^{z}$ inside the crossed‑product algebra $R_{\phi}[S_{\phi}]$. This yields a concrete Morita equivalence between the Bernstein block and modules over $R_{\phi}[S_{\phi}^{z}]$, from which the packet $\Pi_{\phi}^{G}$ is read off. The absolute transfer factor provides the precise scalar needed for stable character identities, and Poisson–Tate duality guarantees the global product formula, allowing the local packets to glue. The stabilized trace formula then identifies the spectral decomposition with the global $L$‑packet, completing the compatibility check.

**Consequences.** This construction works uniformly for all inner forms, including exceptional and non‑quasi‑split groups, and gives an algorithmic recipe (via the explicit algebra $R_{\phi}[S_{\phi}^{z}]$) for computing $L$‑packets and their depths. It also clarifies the role of the basic gerbe in the local‑global compatibility of the Langlands correspondence.