Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters

Question

In Zhu's Coherent sheaves on the stack of Langlands parameters theorem 4.7.1 relates the cohomology of the moduli stack of shtukas to global sections of a certain sheaf on the stack of global Langlands parameters:

Let $H_{I,V}^i=H^iC_c\left(\mathrm{Sht}_{\Delta(\bar\eta),K},\mathrm{Sat}(V)\right)$. By [Xu20, Xu1, Xu2], the natural Galois action and the partial Frobenii action together induce a canonical $W_{F,S}^I$-action on $H_{I,V}^i$. The following statement can be regarded as a generalization of the main construction of [LZ].

Theorem 4.7.1. Assume that $k=\mathbb Q_\ell$ and regard $\mathrm{Loc}_{^cG,F,S}$ as an algebraic stack over $\mathbb Q_\ell$. Then for each $i$, there is a quasi-coherent sheaf $\mathfrak A_K^i$ on $^{cl}\mathrm{Loc}_{^cG,F,S}$, equipped with an action of $H_K$, such that for every finite dimensional representation $V$ of $(^cG)^I$, there is a natural $(H_K\times W_{F,S}^I)$-equivariant isomorphism $$ H_{I,V}^i\cong\Gamma(^{cl}\mathrm{Loc}_{^cG,F,S},(w_{F,S}V)\otimes\mathfrak A_K^i)\tag{4.26} $$ where $w_{F,S}V$ is the vector bundle on $\mathrm{Loc}_{^cG,F,S}$ equipped with an action by $W_{F,S}^I$ as in Remark 2.2.7.

This is a generalization of a construction in section 6 of Lafforgue-Zhu's Décomposition au-dessus des paramètres de Langlands elliptiques (also recapitulated in remark 8.5 of V. Lafforgue's ICM survey). The idea is that a regular function of the Langlands parameter determines an endomorphism of $H_{\lbrace 0\rbrace,\mathrm{Reg}}$, a subspace of the cohomology of the moduli of shtukas with one leg, with relative position of the modification bounded by the regular representation of $\widehat{G}$ (see section 7 of 4).

Let $V$ be a representation of $\widehat{G}$, let $x\in V$, $\xi\in V^{*}$. Any function $f$ of $\widehat{G}$ can be expressed as a matrix coefficient $\langle\xi,g.x\rangle$, and for $\gamma\in\mathrm{Gal}(\overline{F}/F)$ the functions $F_{f,\gamma}:\sigma\mapsto f(\sigma(\gamma))$ are supposed to topologically generate all such functions on the stack of Langlands parameters, as $f$ and $\gamma$ vary.

Letting $\underline{V}$ be the underlying vector space of the representation $V$, the tensor product $H_{\lbrace 0\rbrace,\mathrm{Reg}}\otimes \underline{V}$ will have an action of $\mathrm{Gal}(\overline{F}/F)$, obtained from an isomorphism with $H_{\lbrace 0,1\rbrace,\mathrm{Reg}\boxtimes V}$ (see remark 8.5 of 4).

Since the data of $F_{f,\gamma}$ is the same as the data of $x$, $\xi$, and $\gamma$, we may now construct the endomorphism of $H_{\lbrace 0\rbrace,\mathrm{Reg}}$ as follows: $$H_{\lbrace 0\rbrace,\mathrm{Reg}}\xrightarrow{\mathrm{Id}\otimes x} H_{\lbrace 0\rbrace,\mathrm{Reg}}\otimes \underline{V}\xrightarrow{\gamma} H_{\lbrace 0\rbrace,\mathrm{Reg}}\otimes \underline{V}\xrightarrow{\mathrm{Id}\otimes\xi} H_{\lbrace 0\rbrace,\mathrm{Reg}}$$

This allows us to construct a sheaf of $\mathcal{O}$-modules on the stack of Langlands parameters, whose global sections is $H_{\lbrace 0\rbrace,\mathrm{Reg}}^{i}$.

In several other places in 1, Zhu also hints at the hope of finding an analogue of theorem 4.7.1 for the cohomology of Shimura varieties, following ongoing work in progress of Zhu and Emerton on an analogous stack of Langlands parameters for number fields.

How might such an analogue proceed? In the setting of Shimura varieties, as opposed to shtukas, it appears we do not have the notion of legs, or of bounding relative positions of modifications with representations of $\widehat{G}$ (I do not know whether there are any existing analogies). How might such an endomorphism be constructed?

Answer 1

The anticipated analogue is as follows:

There is a map $f: \mathcal X \to \prod_{v \in S} \mathcal X_v,$ where $\mathcal X$ is the stack of $p$-adic representations of $G_{E,S}$ into ${}^LG$ (the $L$-group over $E$ of some group $G$ that is part of a Shimura datum, with reflex field $E$) unramified outside $S$, and each $\mathcal X_v$ is the corresponding local version at the place $v$. (We choose $S$ to include all primes $v$ over $p$, and over $\infty$. And probably any primes where $G$ is ramified.)

Choosing a level structure at each place $v$ of $S$ will give rise to a coherent sheaf $\mathfrak A_v$ on $\mathcal X_v$ as in Xinwen's paper. (Strictly speaking, his paper only treats the case where $v \not\mid p$ or $\infty$, but there should be sheaves for the other $v$ too. The case of $v \mid \infty$ is conceptually a bit mysterious, but there are some ideas. The case of $v \mid p$ is related to the conjectural $p$-adic Langlands program; in this case $\mathcal X_v$ will be an appropriate EG-type stack.)

We can then form $\mathfrak A := \boxtimes_{v \in S} \mathfrak A_v$, the exterior tensor product, a coherent sheaf on $\prod_{v \in S} \mathcal X_v$.

We can consider $f^! \mathfrak A$ on $\mathcal X$.

The Hodge cocharacter $\mu$ from the Shimura datum gives rise to a representation $V_{\mu}$ of ${}^LG$, and composing this with the universal representation over $\mathcal X$, we obtain a locally free sheaf $\mathcal V$ of Galois representations over $\mathcal X$.

Now we have the conjectural formula:

$$R\Gamma_c(\text{ Shimura variety at given level}, \mathbb Z_p)[\text{shift by dimension of Shimura variety}] $$ $$ = R\Gamma(\mathcal X, f^!\mathfrak A \otimes \mathcal V).$$

The analogous formula in the function field case, for cohomology of stacks of shtuka, is stated in Xinwen's paper, and this is the Shimura varieties version.

The isomorphism should be compatible with Galois and Hecke actions on the two sides.

The main difference with the shtuka case is that we just have the single $V_{\mu}$ that we can consider, so there is less structure in the Shimura variety case than in that case. (We can get a bit more structure by allowing non-trivial coefficient systems in the cohomology, or even going to $p$-adically completed cohomology; this will be reflected in the precise choice of sheaf $\mathfrak A_v$ at the places $v|p$.)

This will all be discussed in (hopefully) forthcoming papers of mine and Xinwen's, and of mine, Toby Gee, and Xinwen.