Endoscopy for Hecke categories, character sheaves and representations

TL;DR

This paper establishes an equivalence between certain categories related to a split reductive group over a finite field and its endoscopic group, linking character sheaves and representations with fixed parameters.

Contribution

It proves a monoidal equivalence between blocks of the mixed Hecke category of a reductive group and its endoscopic group, extending to all blocks and connecting character sheaves and representations.

Findings

Neutral block of Hecke category is equivalent to that of the endoscopic group.

Relationship between character sheaves with fixed parameters on G and H.

Correspondence between representations of G and H with fixed semisimple parameters.

Abstract

For a split reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_q)$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.

Equations

$$1\to T\to N_G(T)\to W\to 1.$$

$$1\to {\overline{\mathbb{Q}}}_{\ell }^{\times }\to E_{\mathcal{L}}\to T\to 1$$

$${\delta }_{\mathcal{L}}:{\text{H}}^2(W_{\mathcal{L}},T)\to {\text{H}}^3(W_{\mathcal{L}},{\overline{\mathbb{Q}}}_{\ell }^{\times }).$$

$${\text{H}}^2(W,T[2])\to {\text{H}}^2(W,T)\to {\text{H}}^2(W_{\mathcal{L}},T)\stackrel{{\delta }_{\mathcal{L}}}{\to }{\text{H}}^3(W_{\mathcal{L}},{\overline{\mathbb{Q}}}_{\ell }^{\times }).$$

$$\text{IC}(w^{\gamma }{)}_{{\mathcal{L}}^{\prime }}^{†}\star \text{IC}(w^{\beta }{)}_{\mathcal{L}}^{†}\stackrel{\sim }{\to }\text{IC}(w^{\gamma \beta }{)}_{\mathcal{L}}^{†}.$$

Full text

G.L. is supported partially by the NSF grant DMS-1855773.

Z.Y. is supported partially by the Simon Foundation and the Packard Foundation.

1. The error

The published version of [1] contains an error that led to the wrong conclusion on a certain $3$-cocycle that appears in the monoidal structure of the monodromic Hecke category.

Below we use notations from [1]. The statement[1, Lemma 10.10] is wrong, i.e., the cohomology class of $(\lambda,\mu^{\natural})\in\textup{H}^{3}({\Xi,\overline{\mathbb{Q}}_{\ell}^{\times}})$ is not always trivial. The mistake in the “proof” is that, although the character sheaf $\mathcal{L}$ becomes trivial when restricted to $T(\mathbb{F}_{q})$, the trivialization cannot necessarily be made $W_{\mathcal{L}}$-equivariantly. Recall $(\lambda,\mu)$ comes from a $2$-cocycle $c\in Z^{2}(W,T)$ which in turns comes from the extension

[TABLE]

Namely, choose a lifting $\widetilde{w}\in N_{G}(T)$ for each $w\in W$, and let $c(w_{1},w_{2})\in T$ be such that $\widetilde{w}_{1}\widetilde{w}_{2}=c(w_{1},w_{2})\widetilde{w_{1}w_{2}}$. On the other hand, the datum of a character sheaf $\mathcal{L}$ on $T$ gives an extension of abelian groups

[TABLE]

where $E_{\mathcal{L}}$ consists of pairs $(t,\tau)$ where $t\in T$ and $\tau$ is a nonzero element of $\mathcal{L}_{t}$. This extension carries an action of $W_{\mathcal{L}}$. Taking $W_{\mathcal{L}}$-cohomology we get a connecting homomorphism

[TABLE]

Then $(\lambda,\mu^{\natural})=\delta_{\mathcal{L}}(c)$.

Now we can always arrange so that $c$ takes values in $T[2]$ (using Tits liftings). Restricting (1.1) to $T[2]$ the short exact sequence splits, but not necessarily $W_{\mathcal{L}}$-equivariantly. Therefore the composition

[TABLE]

is still not necessarily zero.

For example, when $G=\textup{SL}(2)$ and $\mathcal{L}$ has order $2$, we have $W_{\mathcal{L}}=W\cong\mathbb{Z}/2\mathbb{Z}$, and the composition (1.2) is nonzero.

2. Correction

The 3-cocycle responsible for the convolution structure on the monodromic Hecke category is the product of two 3-cocycles: one is $\sigma$ defined in [1, §5.8] and studied in [3], which is often nontrivial; the other one is the $\mu^{\natural}$ mentioned above, which can also be nontrivial. It turns out that the cohomology classes of these two cocycles cancel each other, so their product is cohomologically trivial.

In the new version of the paper [2], we give a construction of rigidified minimal IC sheaves that in particular imply the cancellation between $\sigma$ and $\mu^{\natural}$ (although we no longer need $\sigma$ and $\mu^{\natural}$ in the new version of the paper).

The idea is to consider a geometric Whittaker model ${}_{\psi}\mathcal{M}_{\mathcal{L}}:=D^{b}_{m}((U^{-},\psi)\backslash G/(B,\mathcal{L}))$ that is on the one hand a right module for the monodromic Hecke category and on the other hand equivalent to mixed sheaves on a point by taking stalks at the identity element $e\in G$. See [2, §5.9]. This allows us to rigidify minimal IC sheaves, denoted $\textup{IC}(w^{\beta})^{\dagger}_{\mathcal{L}}$ for blocks $\beta$. In [2, Lemma 5.12] we show that there are canonical isomorphisms

[TABLE]

for two composable blocks $\beta$ and $\gamma$, and these isomorphisms are associative. More generally, in [2, Definition 6.14] we define a rigidified IC sheaf $\textup{IC}(w)^{\dagger}_{\mathcal{L}}$ for any $w$.

As a result, the main theorems in [1] involving non-neutral blocks can be simplified and no twisting by cocycles appear in the statements. In particular, the statements of Theorems 10.7, 11.10, 12.6 and Corollary 12.7 in [2] are revised and simplified. We refer to [2] for the new statements.