Spectral decomposition of genuine cusp forms over global function fields

Abstract

We prove the geometric Satake equivalence for étale metaplectic covers of reductive group schemes and extend the Langlands parametrization of V. Lafforgue to genuine cusp forms defined on their associated covering groups.

Introduction

This article is a contribution to the Langlands program for covering groups, as proposed by Weissman [Wei18] and Gan and Gao [GG18]. Its goal is to parametrize genuine cusp forms over a global function field by spectral data, defined in terms of an L-group.

Such a parametrization has been anticipated by V. Lafforgue [Laf18, § 14] and Gan and Gao [GG18, § 14, Question (L)]. V. Lafforgue has moreover indicated a path towards it via the arguments of [Laf18, § 14], combined with a strong version of the geometric Satake equivalence for covering groups, which in principle should follow from Finkelberg and Lysenko [FL10], Reich [Rei12], and Gaitsgory and Lysenko [GL18].

The present article is intended to realize this vision in what we believe is its appropriate generality. More concretely, the class of covering groups treated in this article includes those defined by Brylinski and Deligne [BD01] using algebraic $K$-theory, but generally contains more objects when the reductive group is not simply connected. In this sense, our scope is larger than that envisioned by Weissman [Wei18] and Gan and Gao [GG18].

By making this generalization, we also make the problem simpler, essentially because étale cohomology is better understood than algebraic $K$-theory. This alternative perspective goes back to Deligne [Del96] and is rediscovered by Gaitsgory and Lysenko [GL18] under a different guise. To keep our narrative consistent, we use [Zha22] as our only input concerning covering groups, although many results proved there have analogues in [GL18].

0.1 Main result

0.1.1

Let $F$ be a global field of characteristic $p \neq 0$. Denote by $\mathbb {A}_F$ the topological ring of its adèles.

Let $\ell \neq p$ be a prime and choose an algebraic closure $\overline {\mathbb {Q}}{}_{\ell }\supset \mathbb {Q}_{\ell }$. Let $A\subset \overline {\mathbb {Q}}{}_{\ell }^{\times }$ be a finite subgroup whose order is indivisible by $p$.

Our group-theoretic input is a pair $(G, \mu )$, where $G$ is a reductive group over $F$ and $\mu$ is a rigidified section of the (higher) étale stack $B^4A(1)$ over the classifying stack $BG$, i.e. a section equipped with a trivialization along the unit $e : \operatorname {Spec}(F) \rightarrow BG$.

The datum $\mu$ is called an étale metaplectic cover of $G$ in [Zha22]. It categorifies a class in the reduced étale cohomology group $H^4_e(BG, A(1))$. In the special case where $G$ is simply connected, the space of étale metaplectic covers is discrete, and our formalism coincides with that in [Del96].

0.1.2

From the pair $(G, \mu )$, we extract two pieces of ‘classical’ structures. (To construct them, it is essential to start with $\mu$ rather than the cohomology class it represents.)

The first is a central extension of topological groups:

(0.1)\begin{equation} 1 \rightarrow A \rightarrow \widetilde G_F \rightarrow G(\mathbb{A}_F) \rightarrow 1, \end{equation}

equipped with a canonical splitting over $G(F) \subset G(\mathbb {A}_F)$. The central extension (0.1) gives us the notion of a genuine automorphic form: a $G(F)$-invariant locally constant function $f : \widetilde G_F \rightarrow \overline {\mathbb {Q}}{}_{\ell }$ satisfying the equality $f(\tilde x\cdot a) = f(\tilde x)\cdot a$ for each $\tilde x\in \widetilde G_F$ and $a\in A$.

The second piece of structure is a short exact sequence of topological groups:

(0.2)\begin{equation} 1 \rightarrow H_{\bar F}(\overline{\mathbb{Q}}{}_{\ell}) \rightarrow {}^LH_{F, \vartheta} \rightarrow \mathrm{Gal}(\bar F/F) \rightarrow 1, \end{equation}

where $H_{\bar F}$ is a pinned split reductive group over $\overline {\mathbb {Q}}{}_{\ell }$. It is determined by $(G, \mu )$ following a combinatorial recipe, modulo some immaterial choices such as the algebraic closure $\bar F$. With (0.2), we arrive at the notion of an L-parameter: an $H_{\bar F}(\overline {\mathbb {Q}}{}_{\ell })$-conjugacy class of sections $\sigma : \mathrm {Gal}(\bar F/F) \rightarrow {}^LH_{F, \vartheta }$ of (0.2).

Remark 0.1.3 For classical applications, one often takes $A$ to be the group $\mu (F)$ of roots of unity in $F$, whose inclusion in $\overline {\mathbb {Q}}{}_{\ell }^{\times }$ is written as an injective character $\zeta : \mu (F)\rightarrow \overline {\mathbb {Q}}{}_{\ell }^{\times }$.

Given an integer $n\ge 1$ invertible in $F$, any Brylinski–Deligne extension of $G$ induces an étale metaplectic cover with $A = \mu _n$ (see [Zha22, § 2.3]). When $\mu _n(F)$ has cardinality $n$, the covering group (0.1) agrees with that constructed in [BD01, § 10], whereas the L-group (0.2) is identified with that constructed by Weissman [Wei18].

Another source of étale metaplectic covers arises from morphisms of complexes $\pi _1(G) \rightarrow A[2]$ of $\mathrm {Gal}(\bar F/F)$-modules, where $\pi _1(G)$ denotes the algebraic fundamental group of $G$ (see [Zha22, § 5.3]). Over a $p$-adic local field, they induce covering groups of Kaletha [Kal22].

0.1.4

The main result of this article is a parametrization of the cuspidal part of genuine automorphic forms by L-parameters.

In order to formulate this parametrization, we need an additional piece of information having to do with the maximal torus $Z$ of the center $Z_G\subset G$.

To wit, the covering group of $Z(\mathbb {A}_F)$ induced from (0.1) is generally not commutative. However, there is a canonically defined isogeny of tori $Z^{\sharp } \rightarrow Z$ such that the induced covering group $\widetilde Z^{\sharp }_F\rightarrow Z^{\sharp }(\mathbb {A}_F)$ is commutative and its image in $\widetilde G_F$ is central.

We shall fix a lattice (i.e. discrete and cocompact subgroup) $\Xi \subset Z^{\sharp }(F)\backslash \widetilde Z^{\sharp }_F$ which projects isomorphically onto its image in $Z^{\sharp }(F)\backslash Z^{\sharp }(\mathbb {A}_F)$.

Furthermore, the restriction of (0.1) to $P(\mathbb {A}_F)$, for each parabolic subgroup $P\subset G$, canonically descends to the Levi quotient $M(\mathbb {A}_F)$. We can thus define the cuspidal part of compactly supported genuine automorphic forms on $G(F)\Xi \backslash \widetilde G_F$, by imposing the vanishing of constant terms for all proper parabolic subgroups $P\subset G$.

Our main theorem, in its most classical form, is a decomposition of this $\overline {\mathbb {Q}}_{\ell }$-vector space according to L-parameters.

Theorem A There is a canonical decomposition:

(0.3)\begin{equation} \operatorname{Fun}_{\mathrm{cusp}}(G(F)\Xi\backslash\widetilde G_F, A \subset \overline{\mathbb{Q}}{}_{\ell}^{\times}) \cong \bigoplus_{[\sigma]} \mathbf{H}_{[\sigma]}, \end{equation}

where $[\sigma ]$ ranges over $H_{\bar F}(\overline {\mathbb {Q}}{}_{\ell })$-conjugacy classes of sections of (0.2).

0.1.5

The spectral decomposition (0.3) arises as the limiting case of its integral variant, Theorem 4.3.11, which contains additional information such as the compatibility with the Satake isomorphism for covering groups, cf. [McN12].

Taking this compatibility for granted, Theorem A fulfills the ‘automorphic-to-Galois’ direction of Langlands reciprocity for genuine cusp forms. In the absence of covering groups, this result is established by Drinfeld for $\mathrm {GL}_2$ (see [Dri87a, Dri87b, Dri88]), L. Lafforgue for $\mathrm {GL}_n$ (see [Laf02]), and V. Lafforgue for all reductive groups [Laf18].

0.2 Outline of the proof

0.2.1

The proof of Theorem A is an adaptation of [Laf18]. First, we must formulate an integral version of the problem in order to use the tools of algebraic geometry.

Let $X$ be smooth, proper, geometrically connected curve over a finite field $\textsf k$ with generic point $\eta = \operatorname {Spec}(F)$. Let $D\subset X$ be a $\textsf k$-finite closed subscheme and $\mathring X$ be its complement.

The notation $\overline {\mathbb {Q}}{}_{\ell }$ and $A$ are as in § 0.1.1.

We replace the group-theoretic input by a pair $(G, \mu )$, where $G\rightarrow X$ is a smooth affine group scheme whose base change to $\mathring X$ is reductive and $\mu$ is a rigidified section of $B^4A(1)$ over $B_{\mathring X}(G)$, the classifying stack of the base change of $G$ to $\mathring X$.

0.2.2

Denote by $\mathrm {Bun}_{G, D}$ the moduli stack of $G$-torsors over $X$ rigidified along $D$.

The étale metaplectic cover $\mu$ defines an $A$-gerbe $\mathscr G_D$ over $\mathrm {Bun}_{G, D}$, via the categorification of a transgression map on étale cohomology:

\begin{align*} [X] : H^4_e(B_{\mathring X}(G), A(1)) \rightarrow H^2(\mathrm{Bun}_{G, D}, A). \end{align*}

The $A$-gerbe $\mathscr G_D$ geometrizes the covering group $\widetilde G_F$ in the sense that a process akin to taking the trace of Frobenius yields a set-theoretic $A$-torsor:

(0.4)\begin{equation} \widetilde{\mathrm{Bun}}_{G, D} \rightarrow \mathrm{Bun}_{G, D}(\textsf k), \end{equation}

whose pullback along the adèlic uniformation map $G(\mathbb {A}_F)\rightarrow \mathrm {Bun}_{G, D}(\textsf k)$ recovers $\widetilde G_F$.

It is more natural, especially for nonsplit reductive groups, to replace genuine automorphic forms by $A$-equivariant functions on $\widetilde {\mathrm {Bun}}_{G, D}$, and the spectral decomposition theorem will hold for this larger space of functions.

0.2.3

There are two other important geometric objects associated to $G$: the local Hecke stack and the moduli stack of shtukas defined by Drinfeld [Dri87b] and Varshavsky [Var04].

For a nonempty finite set $I$, these objects are ind-algebraic stacks over $\mathring X^I$, related by a morphism defined as restriction to the parametrized formal disks:

\begin{align*} \mathrm{res} : \mathrm{Sht}_{G, D}^I \rightarrow \mathrm{Hec}_G^I. \end{align*}

The rigidified section $\mu$ also defines an étale $A$-gerbe $\mathscr G^I$ over $\mathrm {Hec}_G^I$, which geometrizes the local covering groups $\widetilde G_x \rightarrow G(F_x)$ together with their canonical splittings over the maximal compact subgroups $G(\mathscr O_x)$, for each $x\in \mathring X$.

The key observation is that $\mathscr G^I$ is canonically trivialized over $\mathrm {Sht}_{G, D}^I$. In particular, $\mathscr G^I$-twisted $\ell$-adic sheaves on $\mathrm {Hec}_G^I$ pull back to untwisted sheaves over $\mathrm {Sht}_{G, D}^I$, so their compactly supported cohomology are usual $\ell$-adic sheaves over $\mathring X^I$.

Moreover, $\mathscr G^I$ is canonically trivialized over the unit section $e$ of $\mathrm {Hec}_G^I$, so $e_!(\overline {\mathbb {Q}}{}_{\ell })$ may be viewed as a $\mathscr G^I$-twisted sheaf over $\mathrm {Hec}_G^I$. Applying the above process to $e_!(\overline {\mathbb {Q}}{}_{\ell })$, we find the constant sheaf over $\mathring X^I$ with coefficients in compactly supported $A$-equivariant functions on $\widetilde {\mathrm {Bun}}_{G, D}$. This is how cohomology of Shtukas encodes genuine automorphic forms.

Remark 0.2.4 In the main body of the text, these constructions will be applied to the quotient stack $\Xi \backslash \mathrm {Sht}_{G, D}^I$, where $\Xi \subset \widetilde {\mathrm {Bun}}_{Z^{\sharp }, \infty D}$ is a lattice analogous to that in § 0.1.4, but let us ignore this difference for now.

0.2.5

Turning to the spectral side, we extract from the pair $(G, \mu )$ a locally constant étale sheaf over $\mathring X$ of pinned split reductive groups $H$ over $\overline {\mathbb {Q}}{}_{\ell }$, together with an $\mathbb {E}_{\infty }$-monoidal morphism:

(0.5)\begin{equation} \nu_{\vartheta} : \hat Z_H \rightarrow B^2_{\mathring X}(A), \end{equation}

where $\hat Z_H$ denotes the sheaf of characters of the center $Z_H\subset H$.

The subscript $\vartheta$ in (0.5) refers to a twist by the $\{\pm 1\}$-gerbe of theta characteristics over $\mathring X$ (relevant only when the characteristic $p\neq 2$). The somewhat curious Corollary 4.2.7 shows that it is essentially equivalent to Weissman's meta-Galois group.

The pair $(H, \nu _{\vartheta })$ is our version of the metaplectic dual data. It is closely related to the same-named notion in [GL18], although we use the construction in [Zha22, § 6] which is valid in the number field situation as well.

In the absence of an étale metaplectic cover, $H$ would be the sheaf-theoretic version of Langlands’ L-group associated to $G$. The object $\nu _{\vartheta }$ is particular to the metaplectic context, and can be concretely described as an extension of stacks of Picard groupoids over $\mathring X$:

(0.6)\begin{equation} B_{\mathring X}(A) \rightarrow \widetilde{\hat Z}_H \rightarrow \hat Z_H. \end{equation}

0.2.6

Here, we encounter an interesting phenomenon which is only visible on the geometric level: $\widetilde {\hat Z}_H$ is generally not strictly commutative. Equivalently put, (0.5) generally does not come from a morphism of complexes $\hat Z_H \rightarrow A[2]$ of sheaves of abelian groups.

However, we may modify the commutativity constraint on $\widetilde {\hat Z}_H$ in a canonical way to make it strictly commutative. This gives us a morphism of complexes:

(0.7)\begin{equation} {}^0{\nu}_{\vartheta} : \hat Z_H \rightarrow A[2]. \end{equation}

Inducing along $A\subset \overline {\mathbb {Q}}{}_{\ell }^{\times }$, (0.7) defines an étale $Z_H(\overline {\mathbb {Q}}{}_{\ell })$-gerbe over $\mathring X$.

The L-group is a way to repackage the data $(H, {}^0{\nu }_{\vartheta })$. Namely, if $D \neq \varnothing$, after choosing a geometric point $\bar {\eta } = \operatorname {Spec}(\bar F) \rightarrow \eta$ and a rigidification of ${}^0{\nu }_{\vartheta }$ along $\bar {\eta }$, we obtain a short exact sequence of topological groups:

(0.8)\begin{equation} 1 \rightarrow H_{\bar{\eta}}(\overline{\mathbb{Q}}{}_{\ell}) \rightarrow {}^LH_{\mathring X, \vartheta} \rightarrow \pi_1(\mathring X, \bar{\eta}) \rightarrow 1. \end{equation}

The generic form of the L-group (0.2) occurs as its pullback to $\pi _1(\eta, \bar {\eta })$, with notational change $H_{\bar {\eta }} = H_{\bar F}$.

0.2.7

As in [Laf18], the L-group enters through the geometric Satake equivalence.

In our context, this equivalence will make $(H, \nu _{\vartheta })$ appear naturally. Indeed, we consider the stack of (finite-rank) $H$-representations on lisse $\overline {\mathbb {Q}}{}_{\ell }$-sheaves over $\mathring X$. This is an étale stack of tensor categories equipped with a grading by $\hat Z_H$.

The $\mathbb {E}_{\infty }$-monoidal morphism $\nu _{\vartheta }$ allows us to twist this étale stack, whose global section over $\mathring X$ is a new tensor category $\mathrm {Rep}_{H, \nu _{\vartheta }}$.

On the other hand, we consider the category $\mathrm {Sat}_{G, \mathscr G}$ of $\mathscr G$-twisted constructible complexes of $\overline {\mathbb {Q}}{}_{\ell }$-sheaves on the local Hecke stack $\mathrm {Hec}_G$, which are universally locally acyclic and have pullbacks to the affine Grassmannian being perverse relative to $\mathring X$. (For the moment, we assume $I = \{1\}$ and omit it from the notation $\mathrm {Hec}_G^I$ and $\mathscr G^I$.)

The category $\mathrm {Sat}_{G, \mathscr G}$ admits a natural tensor structure, coming from the fusion product. We modify the commutativity constraint in the usual manner (having to do with $2\check {\rho }$) to obtain a new tensor category ${}^+\mathrm {Sat}_{G, \mathscr G}$. This modification ensures that the normalized constant term functor, defined using a half-integral Tate twist $\overline {\mathbb {Q}}_{\ell }({1}/{2})$, is symmetric monoidal.

Theorem B For a fixed $\overline {\mathbb {Q}}_{\ell }( {1}/{2})$, there is a canonical equivalence of tensor categories:

(0.9)\begin{equation} {}^+\mathrm{Sat}_{G, \mathscr G} \cong \mathrm{Rep}_{H, \nu_{\vartheta}}. \end{equation}

0.2.8

Theorem B is the special case of Theorem 2.4.4 for $I = \{1\}$, although the additional challenges presented by general $I$ are mostly notational.

Assuming the general form of equivalence (0.9), we obtain a system of (non-symmetric monoidal) functors parametrized by nonempty finite sets $I$:

(0.10)

Here, the source is the category of continuous finite-dimensional representations of the product $({}^LH_{\mathring X, \vartheta })^I$ whose restrictions to $H_{\bar {\eta }}(\overline {\mathbb {Q}}{}_{\ell })^I$ lift to algebraic representations of $H^I_{\bar {\eta }}$. The vertical functor is the monoidal equivalence induced from the identification ${}^0\nu _{\vartheta }\cong \nu _{\vartheta }$ as $\mathbb {E}_1$-monoidal morphisms. The next two functors are the geometric Satake equivalence (0.9) and the cohomology of Shtukas discussed in § 0.2.3.

There are furthermore iterated variants of the functors (0.10), attached to ordered partitions of $I$. They are needed to equip the target of (0.10) with equivariance structures with respect to the partial Frobenii on $\mathring X^I$.

Given this input, we are in a position to apply the machinery of [Laf18, §§ 5–7] and [Xue20b]: it proves that the image of (0.10) lifts to $\mathrm {Ind}(\mathrm {Rep}(\pi _1(\mathring X, \bar {\eta })^I))$. The spectral decomposition of genuine cusp forms then follows verbatim from [Laf18, §§ 9–11].

0.2.9

Finally, we mention one place where the proof of Theorem B differs from its non-metapletic counterpart. The assertion is of étale local nature on $\mathring X$, so we may assume that $G$ splits and replace $\mathring X$ by any smooth curve $X$, not necessarily connected.

One unique feature of the metaplectic context is the absence of a natural fiber functor out of ${}^+\mathrm {Sat}_{G, \mathscr G}$, even at a geometric point of $X$. This creates difficulties in applying the Tannakian formalism.

Let us work with a fixed Borel subgroup and a maximal torus $T\subset B\subset G$. The étale metaplectic cover $\mu$ induces one for $T$ and defines an $A$-gerbe $\mathscr G_T$ over the Hecke stack $\mathrm {Hec}_T$. We thus have a constant term functor at our disposal:

(0.11)\begin{equation} \mathrm{CT}_B(\check{\rho})[2\check{\rho}] : {}^+\mathrm{Sat}_{G, \mathscr G} \rightarrow \mathrm{Sat}_{T, \mathscr G_T}. \end{equation}

Let us assume that the case for tori is already proved, so $\mathrm {Sat}_{T, \mathscr G_T}$ is identified with the tensor category $\mathrm {Rep}_{T_H, \nu _{\vartheta }}$ of $\nu _{\vartheta }$-twisted category of representations of the metaplectic dual torus $T_H$ on lisse $\overline {\mathbb {Q}}{}_{\ell }$-sheaves. (The $\nu _{\vartheta }$-twist invokes a natural surjection of the character sheaf of $T_H$ onto $\hat Z_H$.)

The tensor category $\mathrm {Rep}_{T_H, \nu _{\vartheta }}$ does not admit a natural fiber functor to $\mathrm {Lis}(X)$, unless we undo the $\nu _{\vartheta }$-twist. Therefore, we wish to twist both the source and target of (0.11) by $\nu _{\vartheta }^{\otimes -1}$ and apply a relative Tannakian formalism to the resulting functor:

\begin{align*} ({}^+\mathrm{Sat}_{G, \mathscr G})_{\nu_{\vartheta}^{\otimes -1}} \rightarrow (\mathrm{Sat}_{T, \mathscr G_T})_{\nu_{\vartheta}^{\otimes -1}} \cong \mathrm{Rep}_{T_H} \rightarrow \mathrm{Lis}(X). \end{align*}

However, in order to twist ${}^+\mathrm {Sat}_{G, \mathscr G}$, we must construct a $\hat Z_H$-grading on it compatible with the tensor structure. In the non-metaplectic context, this would be the $\pi _1(G)$-grading coming from the connected components of $\mathrm {Hec}_G$. The $\hat Z_H$-grading is in general finer. Its existence on the level of abelian categories poses no difficulty, but its compatibility with the tensor structure is not at all obvious.

We shall reduce the problem to studying the weights occurring in (0.11), which have to do with the behavior of the $A$-gerbe $\mathscr G$ on Mirkovic–Vilonen cycles. The desired statement follows from a description of how $\mathscr G$ interacts with the action of the adjoint torus, which eventually reduces to a calculation of Deligne [Del96, § 4] (as reformulated in [Zha22, § 5.5].)

Our proof of Theorem B owes much intellectual debt to pioneering works on the subject by Finkelberg and Lysenko [FL10] and Reich [Rei12], although it does not rely on their results.¹ We have also benefitted from the notion of relative perversity, recently developed by Hansen and Scholze [HS23], which streamlined many arguments.

1. Preparation

This section collects some preliminary notions which will be used in the remainder of the article. We also use it as an opportunity to introduce notation.

The first topic is the formalism of $\ell$-adic sheaves twisted by a gerbe banded by the group of units of the coefficient field. In particular, we explain in § 1.4 how they encode genuine functions. The second topic is the notion of étale metaplectic covers and their L-groups, which we recall in §§ 1.5–1.6.

1.1 $A$-gerbes

1.1.1

We work over a base scheme $S$.

1.1.2

Suppose that $G$ is an étale sheaf of groups over $S$. We denote by $B_S(G)$ the classifying stack of $G$ sheafified in the étale topology. The term $G$-torsor over an $S$-scheme $X$ refers to a section of $B_S(G)$ over $X$.

If the base scheme $S$ is clear from the context, we shall suppress it from the notation.

1.1.3

When $G$ is furthermore abelian, we write $B^n(G)$ for the $n$-fold delooping as an étale stack of $\infty$-groupoids for any integer $n\ge 1$, see [Zha22, § 1].

For an $S$-scheme $X$, the $\infty$-groupoid of sections $\mathrm {Maps}(X, B^n(G))$ has homotopy groups described by the étale cohomology groups of $X$ valued in $G$:

\begin{align*} \pi_i\mathrm{Maps}(X, B^n(G)) \cong H^{n-i}(X, G). \end{align*}

The term $G$-gerbe over an $S$-scheme $X$ refers to a section of $B^2(G)$ over $X$.

The $2$-groupoid of $G$-gerbes over $X$ carries a natural $\mathbb {E}_{\infty }$-monoidal structure, and we use $\otimes$ to denote the product operation.

Remark 1.1.4 This notion of $G$-gerbe is equivalent to the more classical notion of a ‘gerbe banded by $G$’, which is a stack over $X$ equipped with additional structures.

The dictionary goes as follows: given a section of $B^2(G)$ over $X$, we let $\mathscr G$ be the étale stack over $X$ whose sections over an $X$-scheme $X_1$ are rigidifications of the composition: $X_1 \rightarrow X \rightarrow B^2(G)$, i.e. factorizations of it through the canonical section $e : S\rightarrow B^2(G)$.

In particular, the groupoid $\mathscr G(X_1)$ is equipped with an action of the monoidal groupoid of $G$-torsors on $X_1$: each $G$-torsor $t$ defines an automorphism of the canonical section $X_1 \rightarrow S \xrightarrow {e} B^2(G)$ and carries $g\in \mathscr G(X_1)$ to the composition $g\cdot t \in \mathscr G(X_1)$.

1.1.5

Suppose that $X$ is a connected $S$-scheme equipped with a geometric point $\bar x$. Let $\pi _1(X, \bar x)$ denote the profinite fundamental group.

Let $Z$ be a locally constant étale sheaf of finite abelian groups over $X$. Its geometric fiber $Z_{\bar x}$ is thus equipped with a continuous action of $\pi _1(X, \bar x)$.

Denote by $\mathscr Z^2(\pi _1(X, \bar x), Z_{\bar x})$ the groupoid of short exact sequences of profinite groups:

\begin{align*} 1 \rightarrow Z_{\bar x} \rightarrow \Pi \rightarrow \pi_1(X, \bar x) \rightarrow 1, \end{align*}

such that the conjugation action of $\Pi$ on $Z_{\bar x}$ factors through the natural $\pi _1(X, \bar x)$-action.

1.1.6

To interpret $Z$-gerbes in terms of $\pi _1(X, \bar x)$, the following condition on $X$ is needed:

(1.1)\begin{equation} \begin{matrix} \text{For any finite abelian group $A$,}\\ \text{any class of $H^2(X, A)$ vanishes over a} \, finite \, \unicode{x00E8}\text{tale cover of} X. \end{matrix} \end{equation}

Examples of such $X$ include spectra of fields (automatic) and Henselian local rings [Sta18, 09ZI] as well as all connected affine $\mathbb {F}_p$-schemes [Ach17, Theorem 1.1.1], although we are only interested in the case where $X$ is an affine curve over a finite field, where (1.1) can be verified directly.

An example of an $S$-scheme failing condition (1.1) is the projective line.

1.1.7

Let $X$, $\bar x$, and $Z$ be as in § 1.1.5. Denote by $\mathrm {Maps}_{\bar x}(X, B^2(Z))$ the (1-)groupoid of $A$-gerbes over $X$ rigidified along $\bar x$.

If $X$ satisfies condition (1.1), the usual comparison between étale and Galois cohomology lifts to a canonical equivalence of Picard groupoids:

(1.2)\begin{equation} \mathrm{Maps}_{\bar x}(X, B^2(Z)) \cong \mathscr Z^2(\pi_1(X, \bar x), Z_{\bar x}). \end{equation}

Following [Wei18, § 19], we refer to the image of a rigidified $A$-gerbe $(\mathscr G, \bar g)$ under (1.2) as the ‘fundamental group’ of $(\mathscr G, \bar g)$ and denote it as follows:

(1.3)\begin{equation} 1 \rightarrow Z_{\bar x} \rightarrow \pi_1(\mathscr G, \bar g) \rightarrow \pi_1(X, \bar x) \rightarrow 1. \end{equation}

Here, the rigidification $\bar g$ of $\mathscr G$ along $\bar x$ may equivalently be viewed as a geometric point of $\mathscr G$ lifting $\bar x$.

Proof Construction of (1.2)

Let $\widetilde X$ denote the universal cover of $(X, \bar x)$: it is the pro-object of the category of finite étale $X$-schemes which co-represents the fiber functor $(p : X_1 \rightarrow X) \mapsto p^{-1}(\bar x)$. In particular, $\bar x$ canonically lifts to $\widetilde X$, so we may view it as a geometric point of $\widetilde X$.

Let $\mathscr G$ be a $Z$-gerbe over $X$. Condition (1.1) implies that the pullback $\widetilde {\mathscr G}$ of $\mathscr G$ to $\widetilde X$ is constant. Thus, for each $\sigma \in \pi _1(X, \bar x)$, we obtain an automorphism:

(1.4)\begin{equation} \widetilde{\mathscr G}_{\bar x} \cong \widetilde{\mathscr G}_{\sigma(\bar x)} \cong \widetilde{\mathscr G}_{\bar x}, \end{equation}

where the first map is provided by the constancy of $\widetilde {\mathscr G}$ and the second map is the descent datum of $\widetilde {\mathscr G}$ along $\widetilde X \rightarrow X$.

Any rigidification $\bar g$ of $\mathscr G$ along $\bar x$ may be viewed as a section of $\widetilde {\mathscr G}_{\bar x}$. The automorphism (1.4) acts on $\bar g$ as multiplication by a (set-theoretic) $Z_{\bar x}$-torsor $t_{\sigma }$.

The association $\sigma \mapsto t_{\sigma }$ is multiplicative in the following sense: for two elements $\sigma _1, \sigma _2 \in \pi _1(X, \bar x)$, there is an identification of $Z_{\bar x}$-torsors

(1.5)\begin{equation} (\sigma_2)^*(t_{\sigma_1}) \otimes t_{\sigma_2} \cong t_{\sigma_1\sigma_2}, \end{equation}

where $(\sigma _2)^*(t_{\sigma _1})$ is the $Z_{\bar x}$-torsor induced from $t_{\sigma _1}$ along $\sigma _2^{-1}$ (i.e. $z\in Z_{\bar x}$ acts through $\sigma _2$). The identification (1.5) satisfies the natural cocycle condition. Furthermore, $t_e$ is canonically trivialized, satisfying the unit condition with respect to (1.5).

Therefore, the union

\begin{align*} \pi_1(\mathscr G, \bar g) := \bigsqcup_{\sigma \in \pi_1(X, \bar x)} t_{\sigma} \end{align*}

defines an extension of $\pi _1(X, \bar x)$ by $Z_{\bar x}$, with multiplication induced from (1.5). This concludes the definition of the functor (1.2) in the forward direction. It is symmetric monoidal with respect to the natural symmetric monoidal structures on both sides.

To show that (1.2) is an equivalence, we observe that it induces the isomorphism between $H^2(X, Z)$ and $H^2(\pi _1(X, \bar x), Z_{\bar x})$ on $\pi _0$ (owing to condition (1.1)). On $\pi _1$, it induces the isomorphism of abelian groups between $\mathrm {Maps}_{\bar x}(X, B(Z))$ and maps $f : \pi _1(X, \bar x) \rightarrow Z_{\bar x}$ satisfying $\sigma _2^{-1}(f(\sigma _1))f(\sigma _2) = f(\sigma _1\sigma _2)$ for each $\sigma _1, \sigma _2 \in \pi _1(X, \bar x)$.

Remark 1.1.8 If $Z = \underline A$ is the constant étale sheaf with values in a finite abelian group $A$, the isomorphism $\underline {A}_{\bar x} \cong A$ induces a retract:

(1.6)\begin{equation} \mathrm{Maps}(X, B^2(\underline A)) \rightarrow \mathrm{Maps}_{\bar x}(X, B^2(\underline A)),\quad \mathscr G\mapsto \mathscr G\otimes \mathscr G_{\bar x}^{\otimes -1}. \end{equation}

On the other hand, the groupoid $\mathscr Z^2(\pi _1(X, \bar x), \underline A_{\bar x})$ is equivalent to that of central extensions of $\pi _1(X, \bar x)$ by $A$, which we denote by $\mathrm {CExt}(\pi _1(X, \bar x), A)$. Hence, the composition of (1.6) with (1.2) is a functor of Picard groupoids:

(1.7)\begin{equation} \mathrm{Maps}(X, B^2(\underline A)) \rightarrow \mathrm{CExt}(\pi_1(X, \bar x), A). \end{equation}

The functor (1.7) induces an equivalence after $1$-truncation.

1.2 Twisted $\ell$-adic sheaves

1.2.1

We continue to work over a base scheme $S$. We fix a prime $\ell$ invertible on $S$ and an algebraic closure $\mathbb {Q}_{\ell }\subset \overline {\mathbb {Q}}{}_{\ell }$.

Let $E$ be an intermediate field $\mathbb {Q}_{\ell }\subset E\subset \overline {\mathbb {Q}}_{\ell }$ and $A\subset E^{\times }$ be a finite subgroup whose order is also invertible on $S$.

1.2.2

For any $S$-scheme $X$, we have the $\infty$-category $\mathrm {Shv}(X, E)$ of constructible complexes of $E$-sheaves on $X$. This is the $\infty$-category denoted by $D_{\mathrm {cons}}(X, E)$ in [HRS23] where $E$ is equipped with the $\ell$-adic topology.

1.2.3

In the presence of an $A$-gerbe $\mathscr G$ on $X$, there is a variant: the $\infty$-category $\mathrm {Shv}_{\mathscr G}(X)$ of $\mathscr G$-twisted constructible complexes of $E$-sheaves on $X$.

To define it, consider the abelian category $\mathscr A(X)$ of proétale $E$-sheaves on $X$. The association $X_1\mapsto \mathscr A(X_1)$ is a stack of abelian categories on the small étale site of $X$, to be denoted (temporarily) by $\underline {\mathscr A}$.

Since $A$ is a subgroup of $E^{\times }$, it acts on the identity endofunctor of $\underline {\mathscr A}$. The construction of [Zha22, Appendix A] then yields a stack $\underline {\mathscr A}{}_{\mathscr G}$. Its global section is an abelian category $\mathscr A_{\mathscr G}(X)$. Then we may form its derived $\infty$-category $D_{\mathscr G}(X)$ and $\mathrm {Shv}_{\mathscr G}(X) \subset D_{\mathscr G}(X)$ is the full $\infty$-subcategory characterized by the constructibility condition of [HRS23, Definition 1.1].

1.2.4

For brevity, we shall call an object of $\mathrm {Shv}_{\mathscr G}(X)$ a ‘$\mathscr G$-twisted $E$-sheaf’ on $X$.

Such an object is said to be lisse or a $\mathscr G$-twisted $E$-local system if it is dualizable and belongs to the heart $\mathscr A_{\mathscr G}(X)$ of $D_{\mathscr G}(X)$. They form an abelian category $\mathrm {Lis}_{\mathscr G}(X)$.

Remark 1.2.5 Any trivialization of $\mathscr G$ induces an equivalence of $\infty$-categories between $\mathrm {Shv}_{\mathscr G}(X)$ and $\mathrm {Shv}(X)$.

Since $\mathscr G$ is locally trivial in the étale topology, constructions on $\mathrm {Shv}(X)$ of étale local nature automatically carry over to $\mathrm {Shv}_{\mathscr G}(X)$.

1.2.6

It is convenient to consider the $2$-category of pairs $(X, \mathscr G)$, where $X$ is an $S$-scheme (or, more generally, an $S$-prestack) and $\mathscr G$ is an $A$-gerbe over $X$.

A morphism $(X_1, \mathscr G_1)\rightarrow (X_2, \mathscr G_2)$ consists of a morphism $f : X_1\rightarrow X_2$ and an isomorphism $\alpha : \mathscr G_1\xrightarrow {\sim } f^*(\mathscr G_2)$.

A $2$-morphism $(f_1, \alpha _1) \rightarrow (f_2, \alpha _2)$ is an equality $f_1 = f_2 : X_1\rightarrow X_2$ together with a $2$-morphism $\alpha _1 \rightarrow \alpha _2$ in the $2$-category of $A$-gerbes on $X_1$. All $2$-morphisms are invertible.

1.2.7

Given a morphism $(X_1, \mathscr G_1) \rightarrow (X_2, \mathscr G_2)$ as in § 1.2.6, there is a pullback functor $f^* : \mathrm {Shv}_{\mathscr G_2}(X_2) \rightarrow \mathrm {Shv}_{\mathscr G_1}(X_1)$.

If $f$ is separated and of finite presentation, we also have a functor $f_! : \mathrm {Shv}_{\mathscr G_1}(X_1) \rightarrow \mathrm {Shv}_{\mathscr G_2}(X_2)$. (By our convention, functors are derived unless otherwise stated.)

The adjunctions $(f^*, f_*)$, $(f^!, f_!)$ exist on the twisted categories of $E$-sheaves whenever they exist on the untwisted ones.

Indeed, these functors are constructed from the usual functors by étale descent, in view of Remark 1.2.5.

Remark 1.2.8 Suppose that $X$ is connected, Noetherian, and geometrically unibranch. Let $\bar x$ be a geometric point of $X$. Recall that lisse $E$-sheaves on $X$ are equivalent to continuous $\pi _1(X, \bar x)$-representations on finite-dimensional $E$-vector spaces, the functor being taking fibers at $\bar x$.

Assume that $X$ satisfies condition (1.1). Then an $A$-gerbe $\mathscr G$ on $X$ with rigidification $\bar g$ along $\bar x$ defines a central extension (1.3).

Taking fibers at $\bar x$ yields an equivalence between $\mathscr G$-twisted lisse $E$-sheaves on $X$ and continuous $\pi _1(\mathscr G, \bar g)$-representations on finite-dimensional $E$-vector spaces such that $A$ acts through the inclusion $A\subset E^{\times }$.

1.3 A vanishing lemma

1.3.1

We remain in the context of § 1.2.1.

1.3.2

Let $G$ be an étale sheaf of groups over $S$. There is an equivalence of groupoids between $A$-gerbes on $B(G)$ rigidified along $e : S\rightarrow B(G)$ (simply called ‘rigidified $A$-gerbes’ in the following) and monoidal morphisms $G \rightarrow B(\underline A)$.

In turn, monoidal morphisms $G\rightarrow B(\underline A)$ are equivalent to multiplicative $A$-torsors on $G$. They induce character $E$-local systems on $G$ along the inclusion $A\subset E^{\times }$.

1.3.3

Suppose that $G$ acts on an $S$-scheme $X$. Let $\mathscr G$ be a rigidified $A$-gerbe on $B(G)$. Denote by $\chi _{\mathscr G}$ the induced character $E$-local system on $G$.

By abuse of notation, we write $\mathrm {Shv}_{\mathscr G}(X/G)$ for the $\infty$-category of $E$-sheaves on $X/G$ twisted by the pullback of $\mathscr G$ along $X/G \rightarrow B(G)$.

The $\infty$-category $\mathrm {Shv}_{\mathscr G}(X/G)$ admits a more concrete description: it is the $\infty$-category of $E$-sheaves on $X$ equipped with $G$-equivariance structures against the character $E$-local system $\chi _{\mathscr G}$.

1.3.4

Let $X$ be a separated $S$-scheme of finite presentation. We write $\Gamma (X, -)$ (respectively, $\Gamma _c(X, -)$) for the direct image functors $p_*(-)$ (respectively, $p_!(-)$) along the structure morphism $p : X\rightarrow S$. Similar conventions apply to $H^i(X, -)$ and $H^i_c(X, -)$.

The following lemma is a variant of the standard fact that an $E$-sheaf equivariant against a nontrivial character on the stabilizer group must vanish.

Lemma 1.3.5 Let $G$ be a separated group $S$-scheme of finite presentation acting on $X$. Let $\mathscr G$ be a rigidified $A$-gerbe on $B(G)$, with induced character $E$-local system $\chi _{\mathscr G}$ on $G$. If $H^0(G_{\bar s}, \chi _{\mathscr G}) = 0$ for all geometric points $\bar s$ of $S$, then any object $\mathscr F \in \mathrm {Shv}_{\mathscr G}(X/G)$ satisfies

(1.8)\begin{equation} H^i_c(X, \mathscr F) = 0 \quad \text{for all }i\ge 0. \end{equation}

(In particular, we find $\mathrm {Shv}_{\mathscr G}(B(G)) = 0$ by setting $X = S$.)

Proof. The vanishing (1.8) can be verified at geometric points of $S$, so we may assume that $S$ is the spectrum of an algebraically closed field.

Write $f : X \rightarrow X/G$ for the quotient in the étale topology. It suffices to show $f_!(\mathscr F) = 0$.

Consider the Cartesian diagram:

(1.9)

where $\mathrm {act}$ (respectively, $\mathrm {pr}$) denotes the action (respectively, projection) map.

We view $\mathscr F$ as an $E$-sheaf on $X$ which is $G$-equivariant against $\chi _{\mathscr G}$. Base change along (1.9), the equivariance structure, and projection formula imply an isomorphism:

(1.10)\begin{equation} f^*f_!(\mathscr F) \cong \mathscr F\boxtimes \Gamma_c(G, \chi_{\mathscr G}). \end{equation}

It remains to show $\Gamma _c(G, \chi _{\mathscr G}) = 0$.

For $X = G$ and $\mathscr F = \chi _{\mathscr G}$, (1.10) reads

(1.11)\begin{equation} E\boxtimes\Gamma_c(G, \chi_{\mathscr G}) \cong \chi_{\mathscr G}\boxtimes \Gamma_c(G, \chi_{\mathscr G}). \end{equation}

In particular, any nonzero $H^i_c(G, \chi _{\mathscr G})$ implies the existence of a nonzero section of $\chi _{\mathscr G}$.

1.3.6

Let us discuss an example for which the condition of Lemma 1.3.5 is satisfied.

Suppose that $G = \mathbb {G}_{m, S}$ is the multiplicative group over $S$. For each integer $n\ge 0$ invertible over $S$, the degree-$n$ Kummer cover of $\mathbb {G}_{m, S}$ defines a character $\mu _n$-torsor on $\mathbb {G}_{m, S}$.

In particular, we have a monoidal morphism:

(1.12)\begin{equation} \Psi : \mathbb{G}_{m, S} \rightarrow \lim_{\substack{n\ge 0 \\ \text{invertible}}} B(\mu_n). \end{equation}

For each section $a$ of the étale sheaf $\underline A(-1)$, we thus obtain a monoidal morphism $a_*(\Psi ) : \mathbb {G}_{m, S} \rightarrow B(\underline A)$. We also write $\Psi ^a := a_*(\Psi )$ and keep the same notation for the induced character $E$-local system on $\mathbb {G}_{m, S}$.

When $a$ is nowhere vanishing on $S$, there holds $H^0(\mathbb {G}_{m, \bar s}, \Psi ^a) = 0$ at all geometric points $\bar s$ of $S$. Indeed, this follows from the vanishing of $\pi _1(\mathbb {G}_{m, \bar s}, e)$-invariants of the corresponding $1$-dimensional character.

1.4 Frobenius

1.4.1

Suppose that the base scheme $S = \operatorname {Spec}(\textsf k)$, where $\textsf k$ is a finite field of cardinality $q$. Let $A$ be a finite abelian group whose order is coprime to $q$.

For any $\textsf k$-scheme $X$, we write $\mathrm {Fr}_X : X \rightarrow X$ for the absolute Frobenius endomorphism: it acts as identity on the topological space $|X|$ and the $q$th power map on $\mathscr O_X$.

1.4.2

Consider the special case $x = \operatorname {Spec}(\textsf k)$. Then any $A$-torsor $t$ on $x$ defines a character $\mathrm {Gal}(\bar {\textsf k}/\textsf k)\rightarrow A$ for any algebraic closure $\textsf k\subset \bar {\textsf k}$, and the image of the geometric Frobenius $\varphi _x \in \mathrm {Gal}(\bar {\textsf k}/\textsf k)$ may be called the trace-of-Frobenius of $t$.

When a coefficient field $E$ with $A\subset E^{\times }$ is supplied, this is indeed the trace of $\varphi _x$ on the $1$-dimensional representation induced from $t$ along $A\subset E^{\times }$.

1.4.3

We shall describe an analogous construction for $A$-gerbes on a $\textsf k$-scheme (or $\textsf k$-stack) $X$. It is helpful to perform this construction in two steps:

(1.13)

In other words, we shall first extract an étale $A$-torsor $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)$ over the $\mathrm {Fr}_X$-fixed point locus $X^{\mathrm {Fr}}\subset X$, defined to be the fiber product:

(1.14)

and then set $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ to be its set (or groupoid) of $\textsf k$-points.

1.4.4

Recall that for any $\textsf k$-scheme $X$, the endofunctor $\mathrm {Fr}_X^*$ on the $2$-groupoid of $A$-gerbes over $X$ is naturally isomorphic to the identity (the ‘baffling theorem’ [Sta18, 03SN]).

Let us explicitly describe the value of this natural isomorphism at an $A$-gerbe $\mathscr G$:

(1.15)\begin{equation} \mathrm{Fr}_X^*(\mathscr G) \cong \mathscr G. \end{equation}

For any étale morphism $f : X_1\rightarrow X$, the groupoid $\mathrm {Fr}_X^*(\mathscr G)(X_1)$ is the filtered colimit of $\mathscr G(U)$ over étale morphisms $u : U \rightarrow X$ through which $\mathrm {Fr}_X\circ f$ factors. This index category has an initial object, given by $(U, u) = (X_1, f)$ and the factorization $\mathrm {Fr}_X \circ f = f\circ \mathrm {Fr}_{X_1}$. The colimit is thus identified with $\mathscr G(X_1)$.

Let us now give two constructions of $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)$.

Proof Construction 1

Since $\mathrm {Fr}_X$ restricts to the identity map on $X^{\mathrm {Fr}}$, we obtain a ‘tautological’ identification $\mathrm {Fr}_X^*(\mathscr G) \cong \mathscr G$ of $A$-gerbes over $X^{\mathrm {Fr}}$.

The $A$-torsor $\operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$ over $X^{\mathrm {Fr}}$ is defined so that the action by it renders the following diagram of $A$-gerbes over $X^{\mathrm {Fr}}$ commutative:

(1.16)

where $\mathrm {taut}$ refers to the tautological identification.

Remark 1.4.5 For an étale morphism $f : X_1\rightarrow X^{\mathrm {Fr}}$, the automorphism of $\mathscr G(X_1)$ defined by the action of $\operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$ is the pullback along the $X^{\mathrm {Fr}}$-automorphism $\mathrm {Fr}_{X_1}^{-1} : X_1\rightarrow X_1$. (Note that $\mathrm {Fr}_X$ being invertible over $X^{\mathrm {Fr}}$ implies that $\mathrm {Fr}_{X_1}$ is as well.)

Proof Construction 2

The identification (1.15) yields an isomorphism between the endomorphism $\mathrm {Fr}_{B^2(\underline A)}$ of $B^2(\underline A)$ and the identity.

Hence, we find an isomorphism:

(1.17)\begin{equation} B^2(\underline A)^{\mathrm{Fr}} \cong B^2(\underline A) \times B(\underline A),\quad (\mathscr G, \alpha) \mapsto (\mathscr G, \mathscr G^{-1}\otimes\alpha), \end{equation}

where the isomorphism $\alpha : \mathscr G \cong \mathrm {Fr}_X^*(\mathscr G)$ is viewed as an automorphism of $\mathscr G$ via the identification (1.15), so $\mathscr G^{-1}\otimes \alpha$ is an automorphism of the trivial $A$-gerbe, i.e. an $A$-torsor.

The $A$-torsor $\operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$ over $X^{\mathrm {Fr}}$ is set to be the composition:

\begin{align*} X^{\mathrm{Fr}} \rightarrow B^2(\underline A)^{\mathrm{Fr}} \rightarrow B(\underline A), \end{align*}

where the second map is the projection of (1.17) onto its second factor.

1.4.6

Let us argue that $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ is indeed a set-theoretic $A$-torsor over $X(\textsf k)$.

Namely, we must show that its fiber over any $x\in X(\textsf k)$ is nonempty. To see this, it suffices to note that $H^2(x, A) = 0$, so any $A$-gerbe over $x$ admits a section. The choice of any such section trivializes $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)$ over $x$.

Remark 1.4.7 Suppose now that $x = \operatorname {Spec}(\textsf k)$ and an algebraic closure $\textsf k\subset \bar {\textsf k}$ is chosen. We write $\bar x = \operatorname {Spec}(\bar {\textsf k})$.

Then the fiber of $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ over $x\in X(\textsf k)$ is identified with the preimage in $\pi _1(\mathscr G, \bar g)$ of the geometric Frobenius element $\varphi _x \in \mathrm {Gal}(\bar {\textsf k}/\textsf k)$, for any rigidification $\bar g$ of $\mathscr G$ along $\bar x$.

Indeed, since the fiber of $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ over $x$ is nonempty, it is identified with the fiber of $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\bar {\textsf k})$ over $\bar x$. Its description in Remark 1.4.5 coincides with the definition of $t_{\sigma }$ in § 1.1.7 for $\sigma = \varphi _x$ and any rigidification $\bar g$ of $\mathscr G$ along $\bar x$.

Remark 1.4.8 Suppose that $\textsf k\subset \textsf k_1$ is a finite extension. Let $X_1$ be a $\textsf k_1$-scheme, whose restriction of scalars along $\textsf k\subset \textsf k_1$ is denoted by $X := \mathrm {res}(X_1)$. To each $A$-gerbe $\mathscr G_1$ over $X_1$, we may associate an $A$-gerbe $\operatorname {Nm}(\mathscr G_1)$ over $X$. To define it, we restrict $\mathscr G_1$ along the counit map $X_{\textsf k_1} := X\times \operatorname {Spec}(\textsf k_1) \rightarrow X_1$ and take its image under the norm map:

\begin{align*} \Gamma(X_{\textsf k_1}, A[2]) \rightarrow \Gamma(X, A[2]), \end{align*}

which exists thanks to $X_{\textsf k_1} \rightarrow X$ being finite étale.

The trace-of-Frobenius construction may be performed for $\mathscr G_1$ (using the $|\textsf k_1|$th power Frobenius) as well as $\operatorname {Nm}(\mathscr G_1)$ (using the $|\textsf k|$th power Frobenius). They yield canonically isomorphic set-theoretic $A$-torsors:

(1.18)

1.4.9

Let us bring in the coefficient field $E$ as in § 1.2.1 and assume that $A\subset E^{\times }$.

Let $X$ be a $\textsf k$-scheme locally of finite type equipped with an $A$-gerbe $\mathscr G$. Denote by $\widetilde X$ the set-theoretic $A$-torsor $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ over $X(\textsf k)$.

Let $\operatorname {Fun}_c(\widetilde X, A\subset E^{\times })$ denote the $E$-vector space of compactly supported functions $f : \widetilde X \rightarrow E$ such that $f(\tilde x\cdot a) = f(\tilde x)\cdot a$ for each $a\in A$. Elements of $\operatorname {Fun}_c(\widetilde X, A\subset E^{\times })$ are called genuine functions over $\widetilde X$ with respect to the inclusion $A\subset E^{\times }$.

In this set-up, there is a canonical isomorphism of $E$-vector spaces:

(1.19)\begin{equation} \operatorname{Fun}_c(\widetilde X, A\subset E^{\times}) \cong H^0_c(X^{\mathrm{Fr}}, \operatorname{Tr}(\mathrm{Fr}_X \mid \mathscr G)^{\otimes -1}), \end{equation}

where $H^0_c(X^{\mathrm {Fr}}, -)$ denotes the colimit of functors $H^0_c(U, -)$ over quasi-compact open subschemes $U\subset X^{\mathrm {Fr}}$.

1.4.10

We are now in a position to explain how genuine functions arise from twisted $E$-local systems as defined in § 1.2: this is the mechanism by which the cohomology of Shtukas will define genuine automorphic forms.

Consider the $A$-gerbe $\mathscr G\boxtimes (\mathscr G^{\otimes -1})$ over $X\times X$. In reference to (1.14), its restriction along $\Delta$ is canonically trivial, as is its restriction along $(\mathrm {id}, \mathrm {Fr}_X)$ by the isomorphism (1.15).

In particular, the restriction $\mathscr G_0$ of $\mathscr G\boxtimes (\mathscr G^{\otimes -1})$ to $X^{\mathrm {Fr}}$ admits two sections, corresponding to the two circuits of (1.14). If we write $g \in \mathscr G_0(X^{\mathrm {Fr}})$ for the section induced from the lower circuit, then the section induced from the upper is identified with $g \cdot \operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$.

Now, we let $\mathscr F$ be the $\mathscr G_0$-twisted $E$-local system on $X^{\mathrm {Fr}}$, which is identified with $\underline E$ using the section $g$ (see Remark 1.2.5). The same $\mathscr F$ is identified with $\operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)^{\otimes -1}$ using the section $g\cdot \operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$.

In other words, taking $H^0_c(X^{\mathrm {Fr}}, \mathscr F)$ using the section of $\mathscr G_0$ induced from the upper circuit of (1.14) yields the $E$-vector space $\operatorname {Fun}_c(\widetilde X, A\subset E^{\times })$.

1.5 Étale metaplectic covers

1.5.1

Let $S$ be a scheme and $A$ be a finite abelian group whose order is invertible on $S$.

Suppose that $X$ is an $S$-scheme and $G\rightarrow X$ is a smooth affine group scheme.

1.5.2

An étale metaplectic cover of $G\rightarrow X$ with values in $A$ is defined to be a section of $B^4\underline A(1)$ over $B_X(G)$ equipped with a rigidification along $e : X\rightarrow B_X(G)$.

Recall that the subscript in $B_X$ means taking the classifying stack of $G$ relative to $X$ (as opposed to the base scheme $S$).

Note that a rigidified section of $B^4\underline A(1)$ over $B_X(G)$ may be equivalently viewed as a morphism of $\mathbb {E}_1$-monoidal stacks $G \rightarrow B^3_X\underline A(1)$ over $X$.

Remark 1.5.3 This definition agrees with the one in [Zha22, § 2] and we refer the reader there for its relationship with classical metaplectic covers as well as their geometrization by means of $K$-theory.

It is imperative to point out that this definition is essentially contained in [Del96].

1.5.4

Suppose that $X$ is the spectrum of a local nonarchimedean field $F_x$. Then an étale metaplectic cover $\mu$ defines a central extension of topological groups²:

(1.20)\begin{equation} 1 \rightarrow A \rightarrow \widetilde G_x \rightarrow G(F_x) \rightarrow 1. \end{equation}

If $X$ is instead the spectrum of the rings of integers $\mathscr O_x \subset F_x$, then the extension (1.20) produced by restricting $(G, \mu )$ to $\operatorname {Spec}(F_x)$ admits a canonical splitting over $G(\mathscr O_x)$.

1.5.5

If $X$ is the spectrum of a global field $F$ without real places, then an étale metaplectic cover $\mu$ defines a central extension of topological groups:

(1.21)\begin{equation} 1 \rightarrow A \rightarrow \widetilde G_F \rightarrow G(\mathbb{A}_F) \rightarrow 1, \end{equation}

where $\mathbb {A}_F$ denotes the topological ring of adèles of $F$. Furthermore, (1.21) is equipped with a canonical splitting over $G(F)$.

For each nonarchimedean place $x$ of $X$, the restriction of (1.21) along the inclusion $G(F_x) \subset G(\mathbb {A}_F)$ recovers the central extension (1.20).

When $A$ occurs as a subgroup of $E^{\times }$ for a field $E$, we have the $E$-vector space

(1.22)\begin{equation} \operatorname{Fun}(G(F)\backslash\widetilde G_F, A\subset E^{\times}) \end{equation}

of $G(F)$-invariant locally constant functions $f : \widetilde G_F \rightarrow E$ satisfying $f(\tilde x\cdot a) = f(\tilde x)\cdot a$ for every $\tilde x\in \widetilde G_F$ and $a\in A$. They are called genuine automorphic forms on $\widetilde G_F$.

Roughly speaking, the Langlands program for étale metaplectic covers seeks to understand the decomposition of various subspaces of (1.22) according to ‘spectral data’, defined in terms of an L-group associated to $(G, \mu )$.

1.6 The L-group

1.6.1

We keep the notation of § 1.5.1. Furthermore, we bring in the coefficient field $E$ as in § 1.2.1 and assume that $A\subset E^{\times }$.

1.6.2

We also assume that $G\rightarrow X$ is a reductive group scheme. Let $\Lambda$ (respectively, $\check {\Lambda }$) be the étale sheaf of cocharacters of the universal Cartan $T\rightarrow X$ of $G$.

The based root data of $G$ consist of a sheaf of coroots (respectively, simple coroots) $\Phi$ (respectively, $\Delta$) with $\Delta \subset \Phi \subset \Lambda$, a sheaf of roots (respectively, simple roots) $\check {\Phi }$ (respectively, $\check {\Delta }$) with $\check {\Delta }\subset \check {\Phi }\subset \check {\Lambda }$, and an isomorphism $\Phi \cong \check {\Phi }$. The image of $\alpha \in \Phi$ under this isomorphism is denoted by $\check {\alpha }$.

1.6.3

To each étale metaplectic cover $\mu$ of $G\rightarrow X$ with values in $A$, the recipe of [Zha22, § 6] defines its metaplectic dual data $(H, \nu )$, where:

1. (1) $H$ is a locally constant étale sheaf over $X$ of pinned split reductive groups over $E$;

2. (2) $\nu : \hat Z_H \rightarrow B^2_X(\underline A)$ is an $\mathbb {E}_{\infty }$-monoidal morphism.

Here, $Z_H$ denotes the center of $H$, $\hat Z_H$ the abelian group of its characters, viewed as a locally constant étale sheaf of abelian groups over $X$.

We partially recall the construction of $(H, \nu )$ in the following.

Remark 1.6.4 The construction of $H$ is due to Lusztig [Lus93] and its role in the theory of metaplectic covers is explained by Finkelberg and Lysenko [FL10] and McNamara [McN12].

The construction of $\nu$ is essentially due to Weissman [Wei18] when $\mu$ comes from $K$-theory and due to Gaitsgory and Lysenko [GL18] when $X$ is a smooth curve, following a priori different approaches.

1.6.5

Given an étale metaplectic cover $\mu$ of $G\rightarrow X$, we first extract a triplet of invariants $(Q, \nu ^{\sharp }, \varphi )$, where:

1. (1) $Q$ is an $\underline A(-1)$-valued quadratic form on $\Lambda$;

2. (2) $\nu ^{\sharp } : \Lambda ^{\sharp }\rightarrow B^2_X(\underline A)$ is an $\mathbb {E}_{\infty }$-monoidal morphism or, equivalently, an extension of stacks of Picard groupoids over $X$,

   \begin{align*} B_X(\underline A) \rightarrow \widetilde{\Lambda}^{\sharp} \rightarrow \Lambda^{\sharp}; \end{align*}

3. (3) $\varphi$ is an $\mathbb {E}_{\infty }$-monoidal trivialization of $\nu ^{\sharp }$ over $\Lambda ^{\sharp, r} \subset \Lambda ^{\sharp }$.

Here, $\Lambda ^{\sharp } \subset \Lambda$ denotes the kernel of the symmetric form $b$ associated to $Q$ and $\Lambda ^{\sharp, r}\subset \Lambda$ the $\mathbb {Z}$-span of the set:

\begin{align*} \mathrm{ord}(Q(\alpha))\cdot \alpha\quad \text{for each }\alpha\in\Phi. \end{align*}

The fact that $\Lambda ^{\sharp, r}$ belongs to $\Lambda ^{\sharp }$ follows from the equality satisfied by $Q$:

(1.23)\begin{equation} b(\alpha, \lambda) = Q(\alpha)\langle\check{\alpha}, \lambda\rangle,\quad\text{for each }\alpha\in\Phi,\ \lambda\in\Lambda. \end{equation}

1.6.6

Let us sketch the construction of $(Q, \nu ^{\sharp }, \varphi )$ and provide pointers to [Zha22]. The construction is performed étale locally on $X$ using a Borel subgroup $B\subset G$, but it turns out to be independent of this choice [Zha22, § 5.2] and, thus, globalizes.

First, we restrict $\mu$ along $B(B) \rightarrow B(G)$, which canonically descends to an étale metaplectic cover $\mu _T$ of $T$.

The quadratic form $Q$ is the unique discrete invariant of $\mu _T$, in view of the isomorphism between $H^4(BT, A(1))$ and quadratic forms on $\Lambda$ (see [Zha22, § 4.3]).

Next, the restriction $\mu _{T^{\sharp }}$ of $\mu _T$ to $B(T^{\sharp })$ acquires a canonical $\mathbb {E}_{\infty }$-monoidal structure [Zha22, § 4.6]). Taking rigidified sections of $\mu _{T^{\sharp }}$ over $B(\mathbb {G}_m)$, we find a morphism of sheaves of $\mathbb {E}_{\infty }$-monoids over $X$:

(1.24)\begin{equation} \Lambda^{\sharp} \rightarrow \underline{\Gamma}{}_e(B\mathbb{G}_m, B^4\underline A(1)). \end{equation}

However, étale metaplectic covers of $\mathbb {G}_m$ admit a $\mathbb {Z}$-linear splitting into its $\mathbb {Z}$-linear component and the associated quadratic form [Zha22, Remark 4.2.8]:

(1.25)\begin{equation} \underline{\Gamma}{}_e(B\mathbb{G}_m, B^4\underline A(1)) \cong B^2_X(\underline A) \oplus \underline A(-1). \end{equation}

Set $\nu ^{\sharp }$ to be the composition of (1.24) with the projection onto the first factor in (1.25).

Finally, the trivialization $\varphi$ arises from a calculation with $\mathrm {SL}_2$ (see [Zha22, § 6.1.5]).

Remark 1.6.7 By construction, the composition of (1.24) with the projection onto the second factor in (1.25) equals the restriction of $Q$ to $\Lambda ^{\sharp }$, which is a $\mathbb {Z}$-linear map $\Lambda ^{\sharp } \rightarrow \underline A(-1)$ taking values in the subsheaf of $2$-torsion elements, because $b$ vanishes over $\Lambda ^{\sharp }$.

The restriction of $Q$ to $\Lambda ^{\sharp, r}$ vanishes, so we obtain a map:

(1.26)\begin{equation} Q : \Lambda^{\sharp}/\Lambda^{\sharp, r} \rightarrow \underline A(-1). \end{equation}

Remark 1.6.8 It is possible to enhance the data $(Q, \nu ^{\sharp }, \varphi )$ to complete invariants of étale metaplectic covers [Zha22, § 5.1], although we will not need this fact.

Note that due to the $2$-categorical nature of these data, they are more difficult to state than their $K$-theoretic analogues defined in [BD01].

1.6.9

Let us now construct the metaplectic dual data $(H, \nu )$ from the triple $(Q, \nu ^{\sharp }, \varphi )$.

Proof Construction

Let us write $\check {\Lambda }^{\sharp }$ for the $\mathbb {Z}$-linear dual of $\Lambda ^{\sharp }$. For each $\alpha \in \Phi$, we set

\begin{gather*} \alpha^{\sharp} = \mathrm{ord}(Q(\alpha))\cdot \alpha \in \Lambda^{\sharp}; \\ \check{\alpha}^{\sharp} = \mathrm{ord}(Q(\alpha))^{-1} \cdot \check{\alpha} \in \check{\Lambda}^{\sharp}. \end{gather*}

Let $\Phi ^{\sharp }$ (respectively, $\check {\Phi }^{\sharp }$) be the span of $\alpha ^{\sharp }$ (respectively, $\check {\alpha }^{\sharp }$) and $\Delta ^{\sharp }$ (respectively, $\check {\Delta }^{\sharp }$) its subset corresponding to $\alpha \in \Delta$ (respectively, $\check {\alpha }\in \check {\Delta }$). Then the collection $\Delta ^{\sharp }\subset \Phi ^{\sharp }\subset \Lambda ^{\sharp }$, $\check {\Delta }^{\sharp }\subset \check {\Phi }^{\sharp } \subset \check {\Lambda }^{\sharp }$, with bijection $\Phi ^{\sharp }\cong \check {\Phi }^{\sharp }$, $\alpha ^{\sharp }\mapsto \check {\alpha }^{\sharp }$, defines a locally constant sheaf of based root data over $X$.

The sheaf $H$ is defined to be the associated sheaf of pinned split reductive groups over $E$: it has characters in $\Lambda ^{\sharp }$, roots in $\Phi ^{\sharp }$, simple roots in $\Delta ^{\sharp }$, etc.

By this definition, we have a canonical isomorphism of sheaves of abelian groups:

(1.27)\begin{equation} \hat Z_H \cong \Lambda^{\sharp}/\Lambda^{\sharp, r}, \end{equation}

so the data $(\nu ^{\sharp }, \varphi )$ of § 1.6.5 may be interpreted as an $\mathbb {E}_{\infty }$-monoidal morphism $\hat Z_H \rightarrow B^2_X(\underline A)$. This concludes the definition of $(H, \nu )$ alluded to in § 1.6.3.

1.6.10

In order to define the L-group, we need to pass to the component of $\nu$ which is $\mathbb {Z}$-linear, i.e. corresponding to a morphism of complexes $\hat Z_H \rightarrow \underline A[2]$ of étale sheaves of abelian groups over $X$ (see [Zha22, § 6.2]). This is because $\nu$ can be nontrivial even over a geometric point of $X$, whereas the L-group does not capture this information.

Let us perform this construction in a more abstract setting: $M$ denotes an étale sheaf of abelian groups over $X$. An $\mathbb {E}_{\infty }$-monoidal morphism $\nu : M \rightarrow B^2_X(\underline A)$ corresponds to a symmetric monoidal extension $\widetilde M$ of $M$ by $B_X(\underline A)$.

Associating to each $m \in \widetilde M$ the commutativity constraint of $m\otimes m$ defines a character of $M$ valued in the subsheaf $\underline A{}_{[2]} \subset \underline A$ of $2$-torsion elements. This character vanishes $\Leftrightarrow$ $\widetilde M$ is strictly commutative $\Leftrightarrow$ $\nu$ is $\mathbb {Z}$-linear.

Therefore, we have a fiber sequence:

(1.28)\begin{equation} \mathrm{Maps}_{\mathbb{Z}}(M, B^2(\underline A)) \rightarrow \mathrm{Maps}_{\mathbb{E}_{\infty}}(M, B^2(\underline A)) \rightarrow \operatorname{Hom}(M, \underline A{}_{[2]}). \end{equation}

1.6.11

The fiber sequence (1.28) canonically splits.

Proof Construction

If $A_{[2]} \neq 0$, there is nothing to construct. If $A_{[2]} = 0$, the inclusion $A\subset E^{\times }$ identifies $A_{[2]}$ with $\mathbb {Z}/2$.

Given a homomorphism $\epsilon : M \rightarrow \underline A_{[2]}$, we define the $\mathbb {E}_{\infty }$-monoidal morphism ${}^{\epsilon }\nu : M \rightarrow B^2(\underline A)$ to be the trivial $\mathbb {E}_1$-monoidal morphism, whose $\mathbb {E}_{\infty }$-monoidal structure is defined by the commutativity constraint:

\begin{align*} (\widetilde M \ni m_1, m_2) \mapsto (-1)^{\epsilon(m_1)\epsilon(m_2)} \end{align*}

on the associated monoidal extension $\widetilde M$ of $M$ by $B(\underline A)$. The association $\epsilon \mapsto {}^{\epsilon }\nu$ provides the desired splitting of (1.28).

1.6.12

Let us now return to the metaplectic dual data $(H, \nu )$.

Along the split fiber sequence (1.28) with $M := \hat Z_H$, the $\mathbb {E}_{\infty }$-monoidal morphism $\nu : \hat Z_H \rightarrow B^2_X(\underline A)$ has a $\mathbb {Z}$-linear component, to be viewed as a morphism of complexes:

(1.29)\begin{equation} {}^0\nu : \hat Z_H \rightarrow \underline A[2]. \end{equation}

This datum is equivalent to a global section of the complex $(\hat Z_H)^* \otimes \underline A[2]$ over $X$, where $(\hat Z_H)^*$ denotes the $\mathbb {Z}$-linear dual of $\hat Z_H$ as a complex.

Inducing along the inclusion $A\subset E^{\times }$ and replacing $E$ by a finite extension if necessary, ${}^0\nu$ determines a global section of

\begin{align*} (\hat Z_H)^*\otimes \underline E^{\times}[2] \cong Z_H(E)[2], \end{align*}

i.e. an étale $Z_H(E)$-gerbe over $X$.

Remark 1.6.13 When $A_{[2]}\neq 0$, there is a canonical isomorphism $\underline A(-1)_{[2]} \cong \underline A{}_{[2]}$.

Using [Zha22, Proposition 4.6.6], we may identify the image of $\nu$ in $\operatorname {Hom}(\hat Z_H, \underline A{}_{[2]})$ with the morphism (1.26) (which is valued in $\underline A(-1)_{[2]}$, or equivalently $\underline A{}_{[2]}$), under the identification $\hat Z_H \cong \Lambda ^{\sharp }/\Lambda ^{\sharp, r}$.

However, we do not use this fact in the present article.

Remark 1.6.14 In what follows, we use the same notation $\nu$ (respectively, ${}^0\nu$) for the $\mathbb {E}_{\infty }$-monoidal morphism $\hat Z_H \rightarrow B^2_X(\underline E^{\times })$ (respectively, $\mathbb {Z}$-linear morphism $\hat Z_H \rightarrow B^2_X(\underline E^{\times })$, or section of $Z_H(E)[2]$) induced along $A\subset E^{\times}\!$.

1.6.15

Let us now assume that $X$ is connected, Noetherian, geometrically unibranch, and satisfies condition (1.1). Fix a geometric point $\bar x$ of $X$.

By taking the fiber at $\bar x$, we obtain a pinned split reductive group $H_{\bar x}$ over $E$ equipped with a $\pi _1(X, \bar x)$-action preserving the pinning.

Since $\hat Z_H$ is finitely generated, ${}^0\nu$ is trivial over a finite étale cover of $X$. Fixing a rigidification $\bar g$ of ${}^0\nu$ along $\bar x$ and applying the construction of § 1.1.7 to finite subgroups of $E^{\times }$, we find an extension of topological groups:

(1.30)\begin{equation} 1 \rightarrow Z_{H, \bar x}(E) \rightarrow \pi_1({}^0\nu, \bar g) \rightarrow \pi_1(X, \bar x) \rightarrow 1, \end{equation}

where the $\pi _1({}^0\nu, \bar g)$-action on $Z_{H, \bar x}(E)$ factors through the given $\pi _1(X, \bar x)$-action.

Inducing (1.30) along the $\pi _1(X, \bar x)$-equivariant inclusion $Z_{H, \bar x}(E)\subset H_{\bar x}(E)$, we obtain an extension of topological groups, to be referred to as the L-group of $(G, \mu )$ over $X$:

(1.31)\begin{align} 1 \rightarrow H_{\bar x}(E) \rightarrow {}^LH_X \rightarrow \pi_1(X, \bar x) \rightarrow 1. \end{align}

Remark 1.6.16 By construction, ${}^LH_X$ is induced from a finite quotient $\pi _1(X, \bar x)\twoheadrightarrow \Gamma$ and the corresponding extension ${}^LH_{\Gamma }$ of $\Gamma$ by $H(E)$ can be equipped with an algebraic structure with neutral component $H$.

The main difference between ${}^LH_X$ and the L-group of a reductive group is that the quotient map ${}^LH_X\rightarrow \pi _1(X, \bar x)$ is not equipped with a canonical section.

1.6.17

Let $\mathscr L$ be a line bundle over $X$, viewed as a morphism of complexes $\mathbb {Z} \rightarrow \mathbb {G}_m[1]$. Tensoring with $\underline A(-1)$ and composing with the Kummer isomorphism $\Psi : \underline A(-1)\otimes \mathbb {G}_m \cong \underline A[1]$, we obtain a morphism $\Psi _*(\mathscr L) : \underline A(-1) \rightarrow \underline A[2]$.

We denote the composition of (1.26) with $\Psi _*(\mathscr L)$ by

(1.32)\begin{equation} \mathscr L^Q : \hat Z_H \rightarrow \underline A[2]. \end{equation}

Since (1.26) factors through the subsheaf of $2$-torsion elements, (1.32) is trivialized by any choice of a square root of $\mathscr L$.

1.6.18

In the particular case where $X$ is a smooth curve over a field, we may take $\mathscr L := \omega _X$ in (1.32) to arrive at a morphism of complexes:

(1.33)\begin{equation} \omega_X^Q : \hat Z_H \rightarrow \underline A[2]. \end{equation}

We use the notation $\nu _{\vartheta }$ (respectively, ${}^0\nu _{\vartheta }$) for the product of $\nu$ (respectively, ${}^0\nu$) with $\omega _X^Q$. The subscript is interpreted as a ‘twist by the gerbe of $\vartheta$-characteristics.’

After fixing a geometric point $\bar x$ of $X$ and a rigidification of (1.33) along $\bar x$, the $\mathbb {Z}$-linear morphism ${}^0\nu _{\vartheta }$ defines an extension:

(1.34)\begin{equation} 1 \rightarrow H_{\bar x}(E) \rightarrow {}^LH_{X, \vartheta} \rightarrow \pi_1(X, \bar x) \rightarrow 1. \end{equation}

It is also induced along $Z_{H, \bar x}(E)\subset H_{\bar x}(E)$ from the Baer sum of (1.30) and the central extension defined by (1.33).

These constructions have obvious analogues when $X$ is replaced by the spectra of its field of fractions $F$, its local fields $F_x$, or rings of integers $\mathscr O_x$.

Remark 1.6.19 In § 4.2, we explain the relationship between $\vartheta$-characteristics and Weissman's meta-Galois group as defined in [Wei18, § 4].

In particular, when $X$ is the field of fractions of a curve over a finite field, it will follow that (1.34) coincides with Weissman's L-group when $\mu$ comes from algebraic $K$-theory.

1.7 Twisted $H$-representations

1.7.1

We work over a base scheme $S$, a finite abelian group $A$, and a coefficient field $E$ as in § 1.2.1. Let $X$ be an $S$-scheme.

Let $H\rightarrow X$ be a locally constant étale sheaf of pinned split reductive groups over $E$. Write $Z_H\subset H$ for its center and $\hat Z_H$ its character group, viewed as an étale sheaf of abelian groups over $X$. Suppose that we are supplied with an $\mathbb {E}_{\infty }$-monoidal morphism $\nu : \hat Z_H \rightarrow B^2_X(\underline A)$.

In this context, we define an étale stack of tensor categories $\mathrm {Rep}^{\{1\}}_{H, \nu }$ on $X$. By ‘tensor category’, we mean an $E$-linear symmetric monoidal abelian category.

1.7.2

Denote by $\mathrm {Lis}_X$ the tensor category of lisse $E$-sheaves on $X$.

Note that $\mathscr O_H$ may be viewed as a Hopf algebra in $\mathrm {Ind}(\mathrm {Lis}_X)$. In particular, there is the notion of an $H$-representation on a lisse $E$-sheaf: it is an object $\mathscr F \in \mathrm {Lis}_X$ equipped with a morphism $\mathscr F\rightarrow \mathscr F\otimes \mathscr O_H$ in $\mathrm {Ind}(\mathrm {Lis}_X)$ satisfying the axioms defining a coaction.

Let $\mathrm {Rep}_H^{\{1\}}$ denote the tensor category of $H$-representations on lisse $E$-sheaves over $X$. The forgetful functor $\mathrm {Rep}_H^{\{1\}} \rightarrow \mathrm {Lis}_X$ is $E$-linear and symmetric monoidal.

1.7.3

The construction $X\mapsto \mathrm {Rep}_H^{\{1\}}$ being of étale local nature, we obtain a stack of tensor categories $\underline {\mathrm {Rep}}{}_H^{\{1\}}$ on the étale site of $X$. It admits a decomposition according to the weights of the $Z_H$-action, compatible with the tensor structure:

(1.35)\begin{equation} \underline{\mathrm{Rep}}{}_H^{\{1\}} \cong \bigoplus_{\lambda\in\hat Z_H} \underline{\mathrm{Rep}}{}_H^{\{1\}, \lambda}. \end{equation}

Since $A$ acts by automorphisms of the identity endofunctor of $\underline {\mathrm {Rep}}{}_H^{\{1\}}$, we may form the $\nu$-twisted stack of tensor categories $\underline {\mathrm {Rep}}{}_{H, \nu }^{\{1\}}$ as in [Zha22, Appendix A].

Finally, we define $\mathrm {Rep}_{H, \nu }^{\{1\}}$ to be the global section of $\underline {\mathrm {Rep}}{}_{H, \nu }^{\{1\}}$.

Remark 1.7.4 There is a decomposition inherited from (1.35):

(1.36)\begin{align} \mathrm{Rep}_{H, \nu}^{\{1\}} \cong \bigoplus_{\lambda\in\check Z_H} (\mathrm{Rep}_H^{\{1\}, \lambda})_{\nu(\lambda)}, \end{align}

where each summand is the $\nu (\lambda )$-twist of the abelian $E$-linear categories $\mathrm {Rep}_H^{\{1\}, \lambda }$.

The symmetric monoidal structure on $\mathrm {Rep}_{H, \nu }^{\{1\}}$ is induced from that of $\mathrm {Rep}_H^{\{1\}}$ and the $\mathbb {E}_{\infty }$-monoidal structure of $\nu$. Concretely, the monoidal product is given by

\begin{align*} \big(\mathrm{Rep}_H^{\{1\}, \lambda_1}\big)_{\nu(\lambda_1)} \times \big(\mathrm{Rep}_H^{\{1\}, \lambda_2}\big)_{\nu(\lambda_2)} &\rightarrow \big(\mathrm{Rep}_H^{\{1\}, \lambda_1 + \lambda_2}\big)_{\nu(\lambda_1) \otimes \nu(\lambda_2)}\\ & \cong \big(\mathrm{Rep}_H^{\{1\}, \lambda_1 + \lambda_2}\big)_{\nu(\lambda_1 + \lambda_2)}. \end{align*}

Remark 1.7.5 Suppose that $X$ is connected, Noetherian, geometrically unibranch, and satisfies condition (1.1). Suppose also that $\nu$ is $\mathbb {Z}$-linear. (In practice, $\nu$ will be one of the objects ${}^0\nu$, ${}^0\nu _{\vartheta }$ defined in § 1.6.)

We fix a geometric point $\bar x$ of $X$ and a rigidification $\bar g$ of $\nu$ along $\bar x$. Then the L-group (1.31) can be used to give a ‘hands-on’ description of $\mathrm {Rep}_{H, \nu }^{\{1\}}$.

Indeed, given a profinite group $\Gamma$ and an extension ${}^LH$ of $\Gamma$ by $H_{\bar x}(E)$, we write $\mathrm {Rep}_{{}^LH}^{\mathrm {alg}}$ for the category of finite-dimensional continuous representations of ${}^LH$ whose restriction along $H_{\bar x}(E)\subset {}^LH$ lifts to an algebraic representation of $H_{\bar x}$.

Applied to ${}^LH_X$, this construction yields a tensor category canonically equivalent to that of $\nu$-twisted $H$-representations:

(1.37)\begin{equation} \mathrm{Rep}^{\mathrm{alg}}_{{}^LH_X} \cong \mathrm{Rep}_{H, \nu}^{\{1\}}. \end{equation}

Under the equivalence (1.37), the underlying (algebraic) $H_{\bar x}$-representation of an object $V\in \mathrm {Rep}^{\mathrm {alg}}_{{}^LH_X}$ is isomorphic to the fiber of the corresponding object in $\mathrm {Rep}_{H, \nu }^{\{1\}}$ at $\bar x$, by passing through the rigidification $\bar g$.

1.7.6

The construction of $\mathrm {Rep}_{H, \nu }^{\{1\}}$ has a multiple-point generalization, which justifies the superscript in the notation.

Indeed, for a nonempty finite set $I$, we have an $\mathbb {E}_{\infty }$-monoidal morphism of étale sheaves over $X^I$ out of an external direct sum of copies of $\hat Z_H$:

(1.38)\begin{equation} \nu^I : \hat Z_H^I \rightarrow B^2_{X^I}(\underline A),\quad (\lambda_i)_{i\in I} \mapsto \boxtimes_{i\in I} \nu(\lambda_i). \end{equation}

Viewing $H^I$ as a locally constant étale sheaf over $X^I$ of pinned split reductive groups over $E$, the corresponding étale sheaf $\hat Z_{H^I}$ is identified with $\hat Z_H^I$. The construction of § 1.7.3, applied to $X^I$, $H^I$, and $\nu ^I$, yields a stack of tensor categories $\underline {\mathrm {Rep}}{}_{H^I, \nu ^I}^I$ over $X^I$, and we set $\mathrm {Rep}_{H^I, \nu ^I}^I$ to be its global section.

1.7.7

The $\mathbb {E}_{\infty }$-monoidal structure on $\nu ^I$ (for varying $I$) induces its compatibility data with respect to restrictions along the diagonals.

More precisely, given a surjection of nonempty finite sets $p : I\twoheadrightarrow J$, giving rise to the diagonal immersion $\Delta ^p : X^J \rightarrow X^I$, we obtain an isomorphism $(\Delta ^p)^*\nu ^I \cong \nu ^J$. These isomorphisms are compatible with compositions in the obvious sense.

Correspondingly the association $(I\neq \varnothing ) \mapsto \mathrm {Rep}_{H^I, \nu ^I}^I$ defines a functor from the category of nonempty finite sets with surjections to the $2$-category of tensor categories. It carries a surjection $p : I\twoheadrightarrow J$ to the composition:

(1.39)\begin{align} \mathrm{Rep}_{H^I, \nu^I}^I \rightarrow \mathrm{Rep}_{(\Delta^p)^*(H^I), \nu_J}^J \rightarrow \mathrm{Rep}_{H^J, \nu_J}^J,\end{align}

where the first functor is the restriction along $\Delta ^p$, whereas the second functor is the restriction of the action along the diagonal $H^J \rightarrow (\Delta ^p)^*(H^I)$.

2. Geometric Satake equivalence

The goal of this section is to state the geometric Satake equivalence for étale metaplectic covers: Theorem 2.4.4. It is the metaplectic analogue of the equivalence of Mirković and Vilonen [MV07] and Gaitsgory [Gai07, Theorem 2.6].

Sections 2.1–2.3 are preparatory. The main equivalence is stated in § 2.4. In § 2.5, we use it to define a collection of functors, called ‘Satake functors’, which play an instrumental role in the proof of the spectral decomposition theorem in § 4.

2.0.1

Throughout this section, we work over a field $\textsf k$. The letter $S$ is reserved for arbitrary (‘test’) affine $\textsf k$-schemes.

Let $\ell$ be a prime invertible in $\textsf k$ and fix an algebraic closure $\mathbb {Q}_{\ell } \subset \overline {\mathbb {Q}}{}_{\ell }$. The coefficient field $E$ will be an intermediate field $\mathbb {Q}_{\ell }\subset E \subset \overline {\mathbb {Q}}{}_{\ell }$.³ Let $A\subset E^{\times }$ be a finite subgroup whose order is invertible in $\textsf k$.

Let $X$ be a smooth curve over $\textsf k$ and $G \rightarrow X$ be a smooth affine group scheme. Let $\mu$ be an $A$-valued étale metaplectic cover of $G$, i.e. a rigidified section of $B^4\underline A(1)$ over $B_X(G)$.

2.1 The local Hecke stack

2.1.1

Let $I$ be a nonempty finite set.

For an $S$-point $x^I$ of $X^I$, we write $\Gamma _{x^I}$ for the scheme-theoretic union of the graphs of $x^i : S\rightarrow X$ (over $i\in I$), $D_{x^I}$ for the formal completion of $S\times X$ along $\Gamma _{x^I}$, and $\mathring D_{x^I}$ the open subscheme $D_{x^I} - \Gamma _{x^I}$. We call $D_{x^I}$ (respectively, $\mathring D_{x^I}$) the formal disk (respectively, formal punctured disk) around $\Gamma _{x^I}$.

We shall define a number of étale stacks over $X^I$. Their groupoids of lifts of an $S$-point $x^I$ of $X^I$ are tabulated as follows:

(2.1)\begin{align} {r | l} L_+^I(G) \quad & \quad a\,section\,of G\,over\,D_{x^I} \\ L^I(G) \quad & \quad a\,section\,of\,G\,over\,\mathring D_{x^I} \\ \mathrm{Gr}^I_G \quad & \quad a\,G-torsor\,P\,over\,D_{x^I}\,equipped\,with\,\alpha : P^0\overset{x^I}{\sim} P \\ \mathrm{Hec}^I_G \quad & \quad G-torsors\,P_0,\,P_1\,over\,D_{x^I}\,equipped\,with\,\alpha : P_0\overset{x^I}{\sim} P_1 \end{align}

Here, $P^0$ denotes the trivial $G$-torsor over $D_{x^I}$, and the notation $P_0\overset {x^I}{\sim } P_1$ for two $G$-torsors over $D_{x^I}$ means an isomorphism of them off $\Gamma _{x^I}$.

The étale sheaves $L_+^I(G)$ and $L^I(G)$ are valued in groups. The étale sheaf $\mathrm {Gr}_G^I$ is ind-schematic of ind-finite type.

If $G\rightarrow X$ is reductive, $\mathrm {Gr}_G^I$ is furthermore ind-proper.

Remark 2.1.2 Since $G$ is smooth, there are canonical isomorphisms $\mathrm {Gr}_G^I \cong L^IG/L_+^IG$ and $\mathrm {Hec}_G^I \cong L_+^IG\backslash L^IG/L_+^IG$, where the quotients are taken in the étale topology. The quotient map $\pi : \mathrm {Gr}_G^I \rightarrow \mathrm {Hec}_G^I$ sends $(P, \alpha )$ to the triple $(P^0, P, \alpha )$.

2.1.3

Given a surjection of nonempty finite sets $p : I \twoheadrightarrow J$, we write $I_j := f^{-1}(j)$ and view $p$ as an unordered partition of $I$. Denote by $X^p \subset X^I$ the ‘disjoint locus’, i.e. the open subscheme where $\Gamma _{x^{I_{j_1}}}\!\cap \Gamma _{x^{I_{j_2}}} = \varnothing$ if $j_1\neq j_2\in J$.

We have a canonical isomorphism of étale stacks over $X^p$ via restriction to each $D_{X^{I_j}}$:

(2.2)\begin{equation} \varphi^p : \mathrm{Hec}_G^I \times_{X^I} X^p \cong \bigg(\prod_{j\in J} \mathrm{Hec}_G^{I_j}\bigg) \times_{X^I} X^p. \end{equation}

The isomorphism (2.2) is compatible with refinements of partitions. Namely, given two composable surjections of nonempty finite sets $p : I \twoheadrightarrow J$, $q : J\twoheadrightarrow K$, we have containments $X^p \subset X^{q\cdot p} \subset X^I$. For each $k\in K$, $p$ restricts to the partition $p_k : I_k := (q\cdot p)^{-1}(k) \twoheadrightarrow J_k$. There is a commutative diagram as follows.

(2.3)

Furthermore, given three composable surjections of nonempty finite sets, the isomorphism of arrows rendering (2.3) commutative satisfies the evident cocycle condition.

The isomorphism (2.2) and the commutativity (2.3) satisfying the cocycle condition are referred to as the factorization structure of $\mathrm {Hec}_G^I$ (for varying $I$).

Remark 2.1.4 If $J$ is equipped with an ordering, or equivalently if $I$ admits an ordered partition $I = I_1\sqcup \cdots \sqcup I_k$, then we change the notation $X^p$ to $X^{I_1,\ldots, I_k}$ and analogously for the base change of the prestacks in (2.1): $\mathrm {Hec}_G^{I_1,\ldots, I_k} := \mathrm {Hec}_G^I\times _{X^I} X^{I_1,\ldots, I_k}$

2.1.5

Given an ordered partition $I = I_1\sqcup \cdots \sqcup I_k$ as nonempty finite sets, we shall consider another étale stack $\widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_k}$ over $X^I$ whose lift of an $S$-point $x^I$ of $X^I$ consists of $G$-torsors $P_0, \ldots, P_k$ together with isomorphisms:

(2.4)\begin{equation} (P_0\overset{x^{I_1}}{\sim} P_1 \overset{x^{I_2}}{\sim}\cdots \overset{x^{I_k}}{\sim} P_k). \end{equation}

The functors $p_a$ of remembering each segment $(x^{I_a}, P_{a-1}\overset {x^{I_a}}{\sim } P_a)$ (over $1\le a\le k$) and the functor $m$ of remembering their composition $(x^I, P_0\overset {x^I}{\sim } P_k)$ define two morphisms as follows:

(2.5)

Over the disjoint locus $X^{I_1,\ldots, I_k}$, both maps in (2.5) restrict to isomorphisms and their composition is identified with the factorization isomorphism (2.2) associated to the underlying unordered partition $I \rightarrow \{1,\ldots, k\}$.

This construction has an obvious analogue for the affine Grassmannian, namely an ind-scheme $\widetilde {\mathrm {Gr}}{}_G^{I_1,\ldots, I_k}\rightarrow X^I$ classifying the same data (2.4) as $\widetilde {\mathrm {Hec}}{}^{I_1,\ldots, I_k}_G$, together with an additional trivialization of $P_0$.

2.1.6

Given a nonempty finite set $I$, the association

(2.6)\begin{equation} ([k] \in \Delta^{\mathrm{op}}) \mapsto \widetilde{\mathrm{Hec}}{}_G^{I, [k]} := \widetilde{\mathrm{Hec}}{}_G^{I_1,\ldots, I_k} \times_{X^{I_1\sqcup\cdots\sqcup I_k}} X^I, \end{equation}

with $I_1 = \cdots = I_k := I$ defines a simplicial étale stack over $X^I$.

By convention, we set $\widetilde {\mathrm {Hec}}{}^{I, [0]}_G := B_{X^I}(L^I_+(G))$, i.e. the stack classifying a $G$-torsor $P$ over $D_{x^I}$ with no modifications. Morphisms in $\Delta ^{\mathrm {op}}$ are carried to compositions of the corresponding segment of modifications in (2.4).

This simplicial étale stack is canonically identified with the Čech nerve of the morphism

(2.7)\begin{equation} B_{X^I}(L_+^I(G)) \rightarrow B_{X^I}(L^I(G)). \end{equation}

Evidently, its value at $[1]$ is $\mathrm {Hec}_G^I$. In particular, the simplicial system (2.6) may be viewed as an additional structure on $\mathrm {Hec}_G^I$ which we call the convolution structure.

2.2 The local $A$-gerbe

2.2.1

Let $I$ be a nonempty finite set.

We use the étale metaplectic cover $\mu$ to define an $A$-gerbe $\mathscr G^I$ on $\mathrm {Hec}_G^I$. Furthermore, $\mathscr G^I$ (for varying $I$) will come equipped with canonical compatibility data with respect to the factorization and convolution structures of $\mathrm {Hec}_G^I$.

The $A$-gerbe $\mathscr G^I$ is not new: it features prominently in [Rei12] and [GL18], and we explain in Remark 2.2.8 how their approach is related to ours.

2.2.2

We start by defining an $\mathbb {E}_1$-monoidal section of $B^2(\underline A)$ over $L^I(G)$, trivialized as such over $L_+^I(G)$, i.e. the following commutative diagram of $\mathbb {E}_1$-monoidal morphisms.

(2.8)

Proof Construction

For an $S$-point $x^I$ of $X^I$, we give names to the natural morphisms in the following diagram as displayed.

Viewing $\mu$ as an $\mathbb {E}_1$-monoidal morphism $G \rightarrow B^3_X\underline A(1)$, we obtain an $\mathbb {E}_1$-monoidal morphism by taking its section over $\mathring D_{x^I}\rightarrow X$:

(2.9)\begin{equation} \mu_* : \Gamma(\mathring D_{x^I}, G) \rightarrow \Gamma(\mathring D_{x^I}, B^3_X\underline A(1)). \end{equation}

Note that the target is the $\infty$-groupoid associated to the connective truncation of the complex $\Gamma (D_{x^I}, \hat j_* \underline A(1)[3])$.

Using the Cousin complex and the Gabber–Fujiwara formal base change theorem [Fuj95, Corollary 6.6.4] (see [BM21, Theorem 6.11] for a proof avoiding the Noetherian hypothesis), we find morphisms of complexes:

(2.10)\begin{align} \Gamma(D_{x^I}, \hat j_*\underline A(1)[3]) &\rightarrow \Gamma(\Gamma_{x^I}, \hat i^!\underline A(1)[4])\nonumber\\ & \cong \Gamma(\Gamma_{x^I}, i^!\underline A(1)[4]) \cong \Gamma(\Gamma_{x^I}, (p\cdot i)^!\underline A[2]) \rightarrow \Gamma(S, \underline A[2]), \end{align}

where the last two maps use the smoothness of $p$ and the properness of $p\cdot i$, respectively.

Composing (2.9) with the morphism on underlying $\infty$-groupoids of (2.10), we obtain an $\mathbb {E}_1$-monoidal morphism:

(2.11)\begin{align} \Gamma(\mathring D_{x^I}, G) \rightarrow \Gamma(S, B^2(\underline A)), \end{align}

trivialized as such over $\Gamma (D_{x^I}, G)$. The construction being functorial in the $S$-point $x^I$, we obtain the commutative diagram (2.8).

Remark 2.2.3 Fix a closed point $x\in X$ and write $L(G)_x$ for the fiber of $L^{\{1\}}(G)$ at $x$.

The construction of the $\mathbb {E}_1$-monoidal morphism $L(G)_x \rightarrow B^2_x(\underline A)$ in § 2.2.2 only requires $\mu$ to be defined over $\mathring D_x$ (instead of $X$). Furthermore, it is $\mathbb {E}_1$-monoidally trivial over the first congruence group scheme:

\begin{align*} G_{1, x} := \operatorname{Ker}(L_+(G)_x \rightarrow G_x), \end{align*}

because $G_{1, x}$ is pro-unipotent. (However, lifting this trivialization to one over $L_+(G)_x$ in general requires $\mu$ to be defined over $D_x$.)

2.2.4

The commutative diagram (2.8) is equivalent to a section of $B^3(\underline A)$ over $B_{X^I}(L^I(G))$ trivialized over $B_{X^I}(L_+^I(G))$.

Taking Čech nerves, we obtain a morphism of simplicial étale stacks:

(2.12)\begin{equation} \widetilde{\mathrm{Hec}}{}^{I, [k]}_G \rightarrow B^2(A)^{\times k},\quad [k]\in\Delta^{\mathrm{op}}, \end{equation}

where the target $[k]\mapsto B^2(A)^{\times k}$ is the Čech nerve of the morphism $\operatorname {Spec}(k)\rightarrow B^3(A)$.

Finally, we define $\mathscr G^I$ to be the value of (2.12) at $[1]$.

Remark 2.3.5 We view the simplicial morphism (2.12) as expressing the compatibility between $\mathscr G^I$ and the convolution structure of $\mathrm {Hec}_G^I$.

For instance, the commutation of (2.12) with the three face maps $([1]\rightarrow [2])\in \Delta$ contain the following isomorphism of $A$-gerbes on $\widetilde {\mathrm {Hec}}{}_G^{I, [2]}$:

(2.13)\begin{equation} m^*(\mathscr G^I) \cong p_1^*(\mathscr G^I) \otimes p_2^*(\mathscr G^I), \end{equation}

where the morphisms $p_2$, $m$, and $p_1$ send an $S$-point $(P_0\overset {x^I}{\sim } P_1 \overset {x^I}{\sim } P_2)$ of $\widetilde {\mathrm {Hec}}{}_G^{I, [2]}$ to $S$-points $(P_1 \overset {x^I}{\sim } P_2)$, $(P_0\overset {x^I}{\sim } P_2)$, and $(P_0\overset {x^I}{\sim } P_1)$, respectively, of $\mathrm {Hec}_G^I$.

The commutation of (2.12) with the degeneracy map $([1] \rightarrow [0]) \in \Delta$ expresses the fact that $\mathscr G^I$ is canonically rigidified along the unit $e : B_{X^I}(L_+^I(G)) \rightarrow \mathrm {Hec}_G^I$.

2.2.6

We note a variant of the compatibility between $\mathscr G^I$ and the convolution structure, for the Hecke stack $\widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_k}$ associated to an ordered partition $I = I_1\sqcup \cdots \sqcup I_k$.

Namely, along the two morphisms of (2.5), we have a canonical isomorphism of $A$-gerbes on $\widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_k}$:

(2.14)\begin{align} m^*(\mathscr G^I) \cong \bigotimes_{1\le a\le k} p_a^*(\mathscr G^{I_a}). \end{align}

To see this, we express $\widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_k}$ as the quotient of the group stack $\widetilde L^{I_1,\ldots, I_k}(G)\rightarrow X^I$, whose lift of an $S$-point $x^I$ of $X^I$ consists of sections $g_a$ of $G$ over $D_{x^I} - \Gamma _{x^{I_a}}$ (for $1\le a\le k$), by the actions of $(k+1)$ copies of $L_+^IG$ via the formulas:

\begin{align*} h \cdot (g_1,\ldots, g_k) =\begin{cases} (hg_1, g_2, \ldots, g_k) \\ (g_1h^{-1}, hg_2,\ldots, g_k) \\ \quad\vdots \\ (g_1, \ldots, g_{k-1}, g_kh^{-1}). \end{cases} \end{align*}

Given an $S$-point $(x^I, g_1, \ldots, g_k)$ of $\widetilde L{}^{I_1,\ldots, I_k}(G)$, the section in $\Gamma (\Gamma _{x^I}, i^!\underline A(1))$ defined by the sum of the sections in $\Gamma (\Gamma _{x^{I_a}}, i_{x^{I_a}}^!\underline A(1))$ (for $i_{x^{I_a}} : \Gamma _{x^{I_a}} \rightarrow S\times X$ the closed immersion) via (2.9) is identified with the section defined by $(x^I, g_1\cdots g_k) \in L^I(G)$. This identification induces the isomorphism (2.14).

The isomorphism (2.14) is compatible with refinements of the ordered partition of $I$, in the evident sense.

2.2.7

Finally, the $A$-gerbe $\mathscr G^I$ is also compatible with the factorization structure on $\mathrm {Hec}_G^I$ (for varying $I$) in the following sense: (2.2) is upgraded to an isomorphism in the $2$-category of prestacks equipped with an $A$-gerbe (see § 1.2.6) where the $A$-gerbe over $\mathrm {Hec}_G^I$ is $\mathscr G^I$ and the $A$-gerbe over $\prod _{j\in J}\mathrm {Hec}_G^{I_j}$ is the external product $\boxtimes _{j\in J}\mathscr G^{I_j}$.

The $2$-isomorphism rendering (2.3) commutative and the cocycle condition it satisfies also lift to the $2$-category of such pairs.

Remark 2.2.8 The restriction of $\mathscr G^I$ along $\mathrm {Gr}_G^I \rightarrow \mathrm {Hec}_G^I$ defines a ‘symmetric factorization $A$-gerbe’ in the sense of [Rei12]. The symmetry datum is encoded by the $2$-isomorphism of (2.3) corresponding to $I \twoheadrightarrow J \xrightarrow {q} J$ where $q$ is an automorphism.

In contrast to [Rei12] and [GL18], we do not take factorization $A$-gerbes as parameters for covering groups, although they turn out to be equivalent to étale metaplectic covers over a smooth curve [Zha20]. An advantage of étale metaplectic covers is that their compatibility with the convolution structure on $\mathrm {Hec}_G^I$ (and $\widetilde {\mathrm {Hec}}{}^{I_1,\ldots, I_k}_G$) is essentially tautological.

2.2.9

Suppose that $\textsf k$ is a finite field, $I = \{1\}$, and $x\in X$ is a $\textsf k$-point.

Let us assume that $\mu$ is only defined over $\mathring D_x \cong \operatorname {Spec}(F_x)$. Recall that $\mu$ gives rise to a central extension $\widetilde G_x$ of $G(F_x)$ by $A$ (1.20). Let us recover this central extension from the $A$-gerbe $\mathscr G^{\{1\}}$ via the trace-of-Frobenius construction (see § 1.4).

Indeed, write $L(G)_x$ for the base change of $L^{\{1\}}(G)$ to $x$ and $\mathscr G_x$ for the restriction of $\mathscr G^{\{1\}}$ to $L(G)_x$. Note that $\textsf k$-points of $L(G)_x$ are canonically identified with $G(F_x)$. We shall construct an isomorphism of multiplicative $A$-torsors over the group $G(F_x)$:

(2.15)\begin{equation} \operatorname{Tr}(\mathrm{Fr}\mid\mathscr G_x)(\textsf k) \cong \widetilde G_x. \end{equation}

Proof Construction

By the constructions of § 2.2.2 and [Zha22, § 2.1], the two sides of (2.15) arise as the compositions of $\mu : G\rightarrow B^3A(1)$ with the morphisms on $\mathbb {E}_1$-monoids induced from the two circuits of the following diagram.

(2.16)

Here, the bottom horizontal arrow is induced from the Tate-duality map $H^2(F_x, A(1)) \cong A$.

It remains to note that (2.16) commutes thanks to our normalization of the Tate duality map (see § 1.5.4).

Remark 2.2.10 If $x\in X$ is a closed point with residue field $\textsf k_1\supset \textsf k$, a small modification is needed to recover the central extension $\widetilde G_x$ from geometry.

Namely, we consider the Weil restriction $\mathrm {res}(L(G)_x)$ of $L(G)_x$ along $x \rightarrow \operatorname {Spec}(\textsf k)$. The $A$-gerbe $\mathscr G_x$ defines an $A$-gerbe $\operatorname {Nm}(\mathscr G_x)$ over $\mathrm {res}(L(G)_x)$ (see Remark 1.4.8) and we obtain an isomorphism of multiplicative $A$-torsors over $G(F_x)$:

\begin{align*} \operatorname{Tr}(\mathrm{Fr} \mid \operatorname{Nm}(\mathscr G_x))(\textsf k) \cong \widetilde G_x, \end{align*}

by combining (1.18) and (2.15).

Remark 2.3.11 It follows from the identification (2.15) that $\widetilde G_x$ is canonically split over the first congruence subgroup $G_{1, x}(\textsf k)$ (Remark 2.2.3). If $\mu$ is defined over $D_x$, this splitting extends to one over $G(\mathscr O_x)$ by the same remark and coincides with that in § 1.5.4.

2.3 The Satake category

2.3.1

Let $S$ be a $\textsf k$-scheme. Let $Y$ be a separated $S$-scheme of finite presentation equipped with an $A$-gerbe $\mathscr G$.

Recall the $\infty$-category $\mathrm {Shv}_{\mathscr G}(Y)$ of $\mathscr G$-twisted sheaves on $Y$ defined in § 1.2.

Denote by $\mathrm {Shv}_{\mathscr G}(Y)_{/S} \subset \mathrm {Shv}_{\mathscr G}(Y)$ the full $\infty$-subcategory of $\mathscr G$-twisted sheaves universally locally acyclic relative to $Y\rightarrow S$. The condition of universal local acyclicity is well-defined for $\mathscr G$-twisted sheaves because it is of étale local nature on the source.

The $\infty$-category $\mathrm {Shv}_{\mathscr G}(Y)$ admits a perverse $t$-structure relative to $Y\rightarrow S$, see [HS23, Theorem 1.1]. Let $\mathrm {Perv}_{\mathscr G}(Y)$ denote its heart. We refer to its objects simply as ‘perverse sheaves’ on $Y$.

The full $\infty$-subcategory $\mathrm {Shv}_{\mathscr G}(Y)_{/S}$ inherits a $t$-structure [HS23, Theorem 6.7]. In particular, we have the abelian category:

\begin{align*} \mathrm{Perv}_{\mathscr G}(Y)_{/S} := \mathrm{Perv}_{\mathscr G}(Y) \cap \mathrm{Shv}_{\mathscr G}(Y)_{/S}. \end{align*}

These notions generalize to the situation where $Y$ is an ind-scheme of ind-finite presentation via left Kan extension.

2.3.2

Let us now assume that $G \rightarrow X$ is a split reductive group scheme.

Let $I$ be a nonempty finite set. Recall the $A$-gerbe $\mathscr G^I$ on $\mathrm {Hec}_G^I$ defined by $\mu$ in § 2.2.

The Satake category is the full subcategory

\begin{align*} \mathrm{Sat}^I_{G, \mathscr G^I} \subset \mathrm{Shv}_{\mathscr G^I}(\mathrm{Hec}_G^I) \end{align*}

characterized by the following property: an object belongs to $\mathrm {Sat}^I_{G, \mathscr G^I}$ if its pullback along $\pi : \mathrm {Gr}_G^I \rightarrow \mathrm {Hec}_G^I$ (see Remark 2.1.2) belongs to $\mathrm {Perv}_{\mathscr G^I}(\mathrm {Gr}_G^I)_{/X^I}$.

Remark 2.3.3 As in the non-twisted setting, $\mathrm {Sat}^I_{G, \mathscr G^I}$ is an abelian category and the pullback functor defines a fully faithful embedding:

\begin{align*} \pi^* : \mathrm{Sat}_{G, \mathscr G^I}^I \subset \mathrm{Perv}_{\mathscr G^I}(\mathrm{Gr}_G^I)_{/X^I}. \end{align*}

In particular, we may view objects of $\mathrm {Sat}^I_{G, \mathscr G^I}$ as $\mathscr G^I$-twisted universally locally acyclic perverse sheaves on $\mathrm {Gr}_G^I$ satisfying an extra ‘equivariance’ condition.

2.3.4

Let us equip $\mathrm {Sat}_{G, \mathscr G^I}$ with a symmetric monoidal structure, where the monoidal product is given either by the ‘convolution’, or the ‘fusion’ product.

The construction explained in the following is a straightforward adaptation of its non-metaplectic counterpart. We choose to follow the approach of [FS21, VI] instead of [MV07].

2.3.5

To begin with, since $\mathrm {Hec}_G^I$ admits a convolution structure (§ 2.1.6) and $\mathscr G^I$ is compatible with it (§ 2.2.4), the $\infty$-category $\mathrm {Shv}_{\mathscr G^I}(\mathrm {Hec}_G^I)$ admits an $\mathbb {E}_1$-monoidal structure.

More concretely, the monoidal product is given by

(2.17)\begin{equation} \mathscr F_1 \circ_{X^I} \mathscr F_2 := m_!(p_1^*\mathscr F_1 \otimes p_2^*\mathscr F_2), \end{equation}

passing through the isomorphism (2.13) of $A$-gerbes. The monoidal unit is given by $e_!(\underline E)$ using the rigidification of $\mathscr G^I$ along $e$. It clearly belongs to $\mathrm {Sat}_{G, \mathscr G^I}^I$.

This $\mathbb {E}_1$-monoidal structure is inherited by the full subcategory $\mathrm {Shv}_{\mathscr G^I}(\mathrm {Hec}_G^I)_{/X^I}$: the ind-algebraic stack $\widetilde {\mathrm {Hec}}{}_G^{I, [2]}$ is étale locally isomorphic to a product of $\mathrm {Hec}_{G}^I$ with $\mathrm {Gr}_G^I$ and universal local acyclicity is preserved under proper pushforward.

2.3.6

Suppose that $I$ admits an ordered partition into nonempty finite sets $I = I_1\sqcup \cdots \sqcup I_k$. Let $\mathscr F_a$ be an object of $\mathrm {Shv}_{\mathscr G^{I_a}}(\mathrm {Hec}_G^{I_a})$, for $1\le a\le k$.

We may form their external convolution product using the morphisms in (2.5) and the isomorphism of $A$-gerbes (2.14):

(2.18)\begin{equation} \circ_{1\le a\le k}\,\mathscr F_a := m_!\bigg(\bigotimes_{1\le a\le k}p_a^*\mathscr F_a\bigg) \in \mathrm{Shv}_{\mathscr G^I}(\mathrm{Hec}_G^I). \end{equation}

Lemma 2.3.7 If each $\mathscr F_a$ belongs to $\mathrm {Sat}_{G, \mathscr G^{I_a}}^{I_a}$, then $\circ _{1\le a\le k}\mathscr F_a$ belongs to $\mathrm {Sat}_{G, \mathscr G^I}^I$.

Proof. The fact that $\circ _{1\le a\le k}\mathscr F_a$ is universally locally acyclic relative to $X^I$ is argued as in § 2.3.5. It remains to show that $\circ _{1\le a\le k}\mathscr F_a$ is perverse over $\mathrm {Gr}_G^I$ relative to $X^I$.

The universal local acyclicity condition implies that this sheaf has zero vanishing cycle along any specialization of geometric points in $X^I$. Hence, its restrictions to geometric fibers of $\mathrm {Gr}_G^I\rightarrow X^I$ are isomorphic to the nearby cycles of $\boxtimes _{1\le a\le k}\mathscr F_a$ over the pairwise disjoint locus in $X^I$. However, perversity is preserved under the nearby cycle functor [Ill94, Corollaire 4.5].

2.3.8

For later convenience, we define a variant of the Satake category associated to a nonempty finite set $I$ equipped with an ordered partition $I \cong \bigsqcup _{1\le a\le k} I_a$ into nonempty subsets:

\begin{align*} \widetilde{\mathrm{Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k} \subset \mathrm{Shv}_{\mathscr G^I}(\widetilde{\mathrm{Hec}}{}_G^{I_1,\ldots, I_k}), \end{align*}

characterized by universal local acyclicity and perversity over $\widetilde {\mathrm {Gr}}{}_G^{I_1,\ldots, I_k}$. Here, the $A$-gerbe is the pullback of $\mathscr G^I$ along the morphism $m$ in (2.5).

Under the hypothesis of Lemma 2.3.7, the sheaf $\bigotimes _{1\le a\le k}p_a^*\mathscr F_a$ belongs to $\widetilde {\mathrm {Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k}$, passing through the identification of $A$-gerbes (2.14). Indeed, this is already established in the proof of Lemma 2.3.7.

2.3.9

The triple $(\mathrm {Sat}_{G, \mathscr G^I}^I, \circ _X, e_!(\underline E))$ admits the structure of a monoidal category.

Proof Construction

The construction (2.18) for two copies of the same nonempty finite set $I$ produces an object $\mathscr F_1 \circ \mathscr F_2 \in \mathrm {Shv}_{\mathscr G^{I\sqcup I}}(\mathrm {Hec}_G^{I\sqcup I})$. Its restriction along the diagonal $X^I \rightarrow X^{I\sqcup I}$ is canonically identified with $\mathscr F_1\circ _X\mathscr F_2$.

Combining this observation with Lemma 2.3.7, we see that the $\mathbb {E}_1$-monoidal structure on $\mathrm {Shv}_{\mathscr G^I}(\mathrm {Hec}_G^I)_{/X^I}$ constructed in § 2.3.5 is inherited by $\mathrm {Sat}_{G, \mathscr G^I}^I$.

2.3.10

Given an unordered partition $p : I \twoheadrightarrow J$, i.e. $I \cong \bigsqcup _{j\in J} I_j$ as in § 2.1.3, we define a ‘disjoint’ variant of the Satake category:

\begin{align*} \mathrm{Sat}^p_{G, \mathscr G^I} \subset \mathrm{Shv}_{\mathscr G^I}(\mathrm{Hec}_G^I \times_X X^p), \end{align*}

as the full subcategory consisting of objects whose pullback to $\mathrm {Gr}_G^I\times _X X^p$ belongs to $\mathrm {Perv}_{\mathscr G^I}(\mathrm {Gr}_G^I\times _X X^p)_{/X^p}$.

By [HS23, Theorem 6.8], the restriction functor is fully faithful:

(2.19)\begin{equation} \mathrm{Sat}_{G, \mathscr G^I}^I \subset \mathrm{Sat}_{G, \mathscr G^I}^p,\quad \mathscr F \mapsto \mathscr F|_{X^p}, \end{equation}

and its essential image is stable under subquotients.

Given $\mathscr F_j\in \mathrm {Sat}_{G, \mathscr G^{I_j}}^{I_j}$ for each $j\in J$, the external tensor product $\boxtimes _{j\in J}\mathscr F_j$ is a $(\boxtimes _{j\in J}\mathscr G^{I_j})$-twisted sheaf over $\prod _{j\in J}\mathrm {Hec}_G^{I_j}$. Its restriction $\boxtimes _{j\in J}\mathscr F_j|_{X^p}$ may be viewed as a $\mathscr G^I$-twisted sheaf over $\mathrm {Hec}_G^I\times _X X^p$ using the factorization isomorphism (2.2) and its compatibility with the $A$-gerbes (see § 2.2.7).

Note that $\boxtimes _{j\in J}\mathscr F_j|_{X^p}$ belongs to the essential image of (2.19), and we call the corresponding object the external fusion product:

(2.20)\begin{equation} \star_{j\in J}\,\mathscr F_j \in \mathrm{Sat}_{G, \mathscr G^I}^I. \end{equation}

Indeed, given any ordering $J \cong \{1,\ldots, k\}$, we may form $\circ _{1\le a\le k}\mathscr F_a \in \mathrm {Sat}_{G, \mathscr G^I}^I$ using Lemma 2.3.7 whose restriction along $X^p\subset X^I$ is identified with $\boxtimes _{j\in J}\mathscr F_j|_{X^p}$.

Remark 2.3.11 By this argument, any ordering on $J$ induces an isomorphism between the external convolution product (2.18) and the external fusion product (2.20).

2.3.12

Finally, for a nonempty finite set $I$, the fusion product of two objects $\mathscr F_1, \mathscr F_2 \in \mathrm {Sat}_{G, \mathscr G^I}^I$ is defined to be the restriction of $\mathscr F_1\star \mathscr F_2 \in \mathrm {Sat}^{I\sqcup I}_{G, \mathscr G^{I\sqcup I}}$ along the diagonal $X^I\rightarrow X^{I\sqcup I}$:

(2.21)\begin{equation} \mathscr F_1 \star_{X^I} \mathscr F_2 := (\mathscr F_1\star \mathscr F_2)|_{X^I}. \end{equation}

The triple $(\mathrm {Sat}_{G, \mathscr G^I}^I, \star _X, e_!(\underline E))$ admits the structure of a symmetric monoidal category. The commutativity constraint comes from the fact that the formation of $\mathscr F_1\star \mathscr F_2$ uses the unique unordered partition of $\{1,2\}$.

Furthermore, Remark 2.3.11 implies that the monoidal structures corresponding to $\circ _X$ and $\star _X$ are identified.

Remark 2.3.13 Fargues and Scholze [FS21, VI] explains another way to identify these monoidal structures, which is more natural from a higher categorical perspective.

To wit, the convolution structure upgrades $(\mathrm {Sat}_{G, \mathscr G^I}^I, \star _X, e_!(\underline E))$ to an $\mathbb {E}_1$-monoidal object in the $2$-category of symmetric monoidal categories, so the two monoidal structures are identified by a variant of the Eckmann–Hilton argument.

Remark 2.3.14 Note that given a surjection of nonempty finite sets $p : I \twoheadrightarrow J$, with corresponding diagonal $\Delta ^p : X^J \rightarrow X^I$, the restriction of $\mathscr G^I$ to $\mathrm {Hec}_G^I\times _{X^I} X^J \cong \mathrm {Hec}_G^J$ is canonically isomorphic to $\mathscr G^J$. These isomorphisms are furthermore compatible with compositions.

It follows that the association $(I\neq \varnothing ) \mapsto \mathrm {Sat}{}_{G, \mathscr G^I}^I$ defines a functor from the category of nonempty finite sets with surjections to the $2$-category of tensor categories.

2.4 The equivalence

2.4.1

We now assume that $G\rightarrow X$ is a reductive group scheme. Notation for the based root data of $G$ are as in § 1.6.2.

Recall that the algebraic fundamental group of $G$ is the sheaf of abelian groups $\pi _1(G) := \Lambda /\Lambda ^r$, where $\Lambda ^r$ is the span of $\Phi$.

Let $2\check {\rho } \in \check {\Lambda }$ denote the sum of positive roots. Its reduction mod $2$ defines a character of $\pi _1(G)$ valued in $\mathbb {Z}/2$.

In order to state a properly normalized version of the Satake equivalence, we assume the existence of and fix a square root $\underline E( {1}/{2})$ of $\underline E(1)$.

2.4.2

For a nonempty finite set $I$, we tweak the commutativity constraint of $\mathrm {Sat}{}_{G, \mathscr G^I}^I$ in the usual way.

Indeed, the connected components of $\mathrm {Gr}_G^I$ are indexed by $\pi _1(G)$. The symmetric monoidal category $\mathrm {Sat}{}_{G, \mathscr G^I}^I$ acquires a decomposition according to the support of its object:

(2.22)\begin{equation} \mathrm{Sat}{}_{G, \mathscr G^I}^I \cong \bigoplus_{\lambda\in \pi_1(G)} (\mathrm{Sat}{}^I_{G, \mathscr G^I})^{\lambda}. \end{equation}

This decomposition is compatible with the monoidal structure in the following sense: the monoidal unit has pure grading $0\in \pi _1(G)$, and the monoidal product of two objects of pure grading $\lambda _1$, $\lambda _2$ has pure gradings $\lambda _1 + \lambda _2$.

Set ${}^+\mathrm {Sat}{}^I_{G, \mathscr G^I} := \mathrm {Sat}{}^I_{G, \mathscr G^I}$ as a monoidal category, but whose commutativity constraint for a pair of objects from $(\mathrm {Sat}{}_{G, \mathscr G^I}^I)^{\lambda _1}$, $(\mathrm {Sat}{}_{G, \mathscr G^I}^I)^{\lambda _2}$ is multiplied by $(-1)^{\langle 2\check {\rho }, \lambda _1\rangle \langle 2\check {\rho }, \lambda _2\rangle }$.

2.4.3

Let $(H, \nu )$ be the metaplectic dual data associated to $(G, \mu )$ as in § 1.6. Recall the twist $\nu _{\vartheta }$ of $\nu$ by the $\vartheta$-characteristic of $X$ (see § 1.6.18).

Applying the construction of § 1.7.6 to the pair $(H, \nu _{\vartheta })$, we obtain a tensor category $\mathrm {Rep}_{H^I, \nu ^I_{\vartheta }}^I$ for each nonempty finite set $I$.

Theorem 2.4.4 (Geometric Satake equivalence)

For each nonempty finite set $I$, there is a canonical equivalence of tensor categories:

(2.23)\begin{equation} {}^+\mathrm{Sat}{}^I_{G, \mathscr G^I} \cong \mathrm{Rep}{}^I_{H^I, \nu_{\vartheta}^I}. \end{equation}

2.4.5

The equivalence (2.23) which we construct comes equipped with two additional pieces of compatibility data.

First, it is compatible with restrictions along the diagonals. Namely, the following diagram is canonically commutative for every surjection of nonempty finite sets $p : I \twoheadrightarrow J$:

(2.24)

where $\Delta ^p : X^J \rightarrow X^I$ denotes the corresponding diagonal. The isomorphism of functors rendering (

2.24

) commutative is compatible with compositions.

Second, the external fusion product (2.20) of the Satake category corresponds to the external tensor product of representations. Namely, writing $I_j := p^{-1}(j)$, the following diagram is canonically commutative:

(2.25)

compatibly with compositions.

Remark 2.4.6 The equivalence (2.23) is of étale local nature over $X$. In particular, it may be viewed as an equivalence of stacks of tensor categories on the étale site of $X$.

2.4.7

Let us specialize to the case $I = \{1\}$. Viewing (2.23) as an equivalence of stacks of tensor categories on $X_{\unicode{x00E8}{\rm t}}$, its stalk at a closed point $x\in X$ yields an equivalence of tensor categories:

(2.26)\begin{align} {}^+\mathrm{Sat}_{G, \mathscr G, x} \cong \mathrm{Rep}_{H, \nu_{\vartheta}, x}. \end{align}

Choosing an algebraic closure $\textsf k_x\subset \bar {\textsf k}_x$ (corresponding to a morphism $\bar x\rightarrow x$), the right-hand side of (2.26) is monoidally equivalent to $\mathrm {Rep}_{H, {}^0\nu _{\vartheta }, x}$, hence to the representation category $\mathrm {Rep}^{\mathrm {alg}}({}^LH_{x, \vartheta })$ according to Remark 1.7.5. (To define ${}^LH_{x, \vartheta }$, one may need to replace $E$ by a finite extension.)

If $X$ satisfies condition (1.1), the local (integral) L-group ${}^LH_{x, \vartheta }$ is also identified with the restriction of ${}^LH_{X, \vartheta }$ along $\pi _1(x, \bar x)\rightarrow \pi _1(X, \bar x)$.

2.4.8

Suppose that $\textsf k$ is a finite field. We obtain a homomorphism of $E$-algebras:

(2.27)\begin{align} K_0(\mathrm{Rep}^{\mathrm{alg}}({}^LH_{x, \vartheta})) \otimes E &\cong K_0({}^+\mathrm{Sat}_{G,\mathscr G, x}) \otimes E\nonumber\\ & \rightarrow \operatorname{Fun}_c(G(\mathscr O_x)\backslash\widetilde G_x/G(\mathscr O_x), A\subset E^{\times}), \end{align}

where the first isomorphism is the application of the monoidal invariant $K_0(-)\otimes E$ to (2.26), and the second homomorphism is the trace of the geometric Frobenius $\varphi _x\in \mathrm {Gal}(\bar {\textsf k}_x/\textsf k_x)$ at each $\textsf k_x$-point of $\mathrm {Hec}_{G, x}$.

For an object $V \in \mathrm {Rep}^{\mathrm {alg}}({}^LH_{x, \vartheta })$, we denote by $h_{V, x}$ its image under (2.27), and call it the (unramified) Hecke operator associated to $V$ at the point $x$.

Remark 2.4.9 Let $H_{\varphi _x} \subset {}^LH_{x, \vartheta }$ denote the preimage of $\varphi _x$. Using the argument of [Zhu17, § 5.6], it is possible to show that (2.27) factors through an isomorphism of $E$-algebras:

(2.28)\begin{equation} \Gamma(H_{\varphi_x}/\!/H, \mathscr O) \cong \operatorname{Fun}_c(G(\mathscr O_x)\backslash\widetilde G_x/G(\mathscr O_x), A\subset E^{\times}), \end{equation}

where the left-hand side is the $E$-algebra of algebraic functions on $H_{\varphi _x}$ invariant under $H$-conjugation. It receives a morphism from $K_0(\mathrm {Rep}^{\mathrm {alg}}({}^LH_{x, \vartheta })) \otimes E$ by mapping $V$ to the character of its restriction to $H_{\varphi _x}$.

The isomorphism (2.28) for metaplectic covers arising from algebraic $K$-theory over any local field has already been obtained by McNamara [McN12, Theorem 10.1]. We do not use (2.28) in this article.

2.5 The Satake functors

2.5.1

We keep the notation of § 2.4.

2.5.2

For a nonempty finite set $I$, we replace ${}^+\mathrm {Sat}{}_{G, \mathscr G^I}^I$ by $\mathrm {Sat}{}_{G, \mathscr G^I}^I$ on one side of the geometric Satake equivalence and $\nu _{\vartheta }$ by its $\mathbb {Z}$-linear component ${}^0{\nu }_{\vartheta }$ (see § 1.6) in the formation of the other side. Both modifications only change the commutativity constraint.

In this manner, we obtain from (2.23) an equivalence of monoidal categories:

(2.29)\begin{equation} \mathrm{Sat}{}^I_{G, \mathscr G^I} \cong \mathrm{Rep}_{H^I, {}^0\nu_{\vartheta}^I}^I. \end{equation}

For a surjection of nonempty finite sets $p : I \twoheadrightarrow J$, the compatibility data of (2.29) with respect to restrictions along the diagonal and external fusion products are given by (2.24) and (2.25) on the underlying monoidal categories.

2.5.3

Let us now assume that $X$ is connected and satisfies condition (1.1).

We fix a geometric point $\bar {\eta }$ of $X$ and a rigidification of ${}^0\nu _{\vartheta }$ along $\bar {\eta }$. In this set-up, we have the extension (1.34):

\begin{align*} 1 \rightarrow H_{\bar {\eta}}(E) \rightarrow {}^LH_{X,\vartheta} \rightarrow \pi_1(X, \bar{\eta}) \rightarrow 1, \end{align*}

where $H_{\bar {\eta }}$ is the fiber of $H$ at $\bar {\eta }$, viewed as a pinned split reductive group over $E$.

2.5.4

Let $I$ be a nonempty finite set together with an ordered partition $I \cong \bigsqcup _{1 \le a\le k}I_a$ into nonempty finite subsets.

Composition of the equivalence (1.37), the external tensor product, and (2.29) defines a functor of monoidal categories:

(2.30)\begin{align} \prod_{i\in I} \mathrm{Rep}^{\mathrm{alg}}({}^LH_{X, \vartheta}) &\cong \prod_{i\in I} \mathrm{Rep}_{H, {}^0\nu_{\vartheta}}^{\{i\}} \xrightarrow{\boxtimes} \prod_{1\le a\le k} \mathrm{Rep}_{H^{I_a}, {}^0\nu_{\vartheta}^{I_a}}^{I_a}\nonumber\\ & \cong \prod_{1\le a\le k} \mathrm{Sat}{}_{G, \mathscr G^{I_a}}^{I_a} \xrightarrow{\bigotimes p_a^*} \widetilde{\mathrm{Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k}, \end{align}

where the functors $p_a^*$ are pullbacks along the morphism in (2.5) (see § 2.3.8).

We argue that (2.30) factors through the category $\mathrm {Rep}^{\mathrm {alg}}(({}^LH_{X, \vartheta })^I)$. This can be done by playing with the regular representation as in [Gai07, Appendix B], but we supply another argument.

2.5.5

Let $\Gamma$ be a profinite group and ${}^LH$ an extension of $\Gamma$ by $H_{\bar {\eta }}(E)$. For a nonempty finite set $I$, the $I$-fold product $({}^LH)^I$ is an extension of $\Gamma ^I$ by $H_{\bar {\eta }}^I(E)$. In particular, we may form the tensor category $\mathrm {Rep}^{\mathrm {alg}}(({}^LH)^I)$ (see Remark 1.7.5).

Recall the notion of tensor product of $E$-linear abelian categories in [Del90, 5.1]: it is the universal recipient of $E$-multilinear functors which are right exact in each factor.

Over a perfect field, the tensor product of tensor categories satisfying a finiteness condition acquires a canonical tensor structure [Del90, 5.17]. This applies to $\mathrm {Rep}^{\mathrm {alg}}({}^LH)$.

Lemma 2.5.6 The tensor product of restrictions of representations along each projection $({}^LH)^I \rightarrow {}^LH$ induces an equivalence of tensor categories:

(2.31)\begin{equation} \bigotimes_{i\in I} \mathrm{Rep}^{\mathrm{alg}}({}^LH) \cong \mathrm{Rep}^{\mathrm{alg}}(({}^LH)^I). \end{equation}

Proof. The tensor category $\mathrm {Rep}^{\mathrm {alg}}({}^LH)$ is Tannakian with fiber functor $\omega$ being the functor of forgetting the ${}^LH$-action. The sheaf of automorphisms $\underline {\operatorname {Aut}}(\omega )$ is thus representable by an affine group scheme ${}^LH^{\mathrm {alg}}$ over $E$, and $\mathrm {Rep}^{\mathrm {alg}}({}^LH)$ is equivalent to the category of finite-dimensional representations of ${}^LH^{\mathrm {alg}}$.

Given a finite-dimensional $E$-vector space $V$, its lift to an object of $\mathrm {Rep}^{\mathrm {alg}}(({}^LH)^I)$ is equivalent to commuting continuous actions of ${}^LH$ indexed by $I$, satisfying the algebraicity condition over $H(E)$: this is equivalent to an action of $({}^LH^{\mathrm {alg}})^I$. Hence, we have $(({}^LH)^I)^{\mathrm {alg}} \cong ({}^LH^{\mathrm {alg}})^I$ and the equivalence (2.31) follows from [Del90, 6.21].

2.5.7

The functor (2.30), being $E$-multilinear and right exact in each factor, canonically factors through a right exact monoidal functor according to Lemma 2.5.6:

(2.32)\begin{equation} \mathscr S^{I_1,\ldots, I_k} : \mathrm{Rep}^{\mathrm{alg}}(({}^LH_{X, \vartheta})^I) \rightarrow \widetilde{\mathrm{Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k}. \end{equation}

We call $\mathscr S^{I_1,\ldots, I_k}$ the Satake functor associated to the nonempty finite set $I$ together with the ordered partition $I \cong \bigsqcup _{1\le a\le k}I_a$.

Furthermore, the functors (2.32) are compatible with change of the partitioned set. More precisely, given nonempty finite sets $I$, $J$, nonempty ordered finite sets $K_1$, $K_2$ and a commutative diagram of surjective morphisms where $q$ is order-preserving:

(2.33)

we have the following canonically commutative diagram of $E$-linear abelian categories.

(2.34)

Here, $m^q : \widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_{k_1}}\times _{X^I} X^J \rightarrow \widetilde {\mathrm {Hec}}{}_G^{J_1, \ldots, J_{k_2}}$ is defined by composing the modifications corresponding to those segments in $K_1$ defined by fibers of $q$.

The commutativity of (2.34) follows from a combination of (2.24), (2.25), and the identification of external fusion and convolution products (Remark 2.3.11). It is compatible with compositions of squares of the type (2.33).

Remark 2.5.8 Our Satake functors and their compatibility data (2.34) are parallel to the assertions of [Laf18, Théorème 1.17].

Indeed, part (c) of [Laf18, Théorème 1.17] comes from our definition of the Satake functors. Assertions (b) and (d) are obtained by setting $I = J$, respectively $K_1 = K_2$ in (2.34). (However, we only use functoriality with respect to surjective maps.) The remaining assertion (a) is established in the course of the proof of Theorem 2.4.4.

3. Proofs

This section is entirely dedicated to the proof of Theorem 2.4.4. The proof follows the same overarching structure as [MV07] and shares many common features with [FL10].

First, §§ 3.1 and 3.2 treat the case for split tori. The difficulties in this case are mostly of categorical nature, and closely related statements have appeared in [Rei12] and [GL18]. However, since our metaplectic dual data are not defined using factorization gerbes, our results are not direct consequences of [Rei12, GL18].

Then, we study $\mathrm {Sat}{}_{G, \mathscr G^I}^I$ for $I = \{1\}$ as an abelian category in § 3.3. The methods in the non-metaplectic setting carry over with minimal modifications.

Subsections 3.4–3.6 are dedicated to the study of the constant term functor. The construction of the ‘fiber functor’ in the metaplectic context differs substantially from the non-metaplectic one. It requires constructing a tensor decomposition of $\mathrm {Sat}_{G, \mathscr G^I}^I$ according to $Z_H^I$-weights. The method we use involves a study of the $A$-gerbe $\mathscr G^I$ over the Mirković–Vilonen cycles, which is hopefully interesting in its own right.

The last subsection, § 3.7, reconstructs $H$ using a relative Tannakian formalism. It is a variant of [FS21, VI.10] and contains few surprises.

3.0.1

We remain in the context of § 2.0.1 throughout this section. Furthermore, we assume the existence of and fix a square root $\underline E( {1}/{2})$ of $\underline E(1)$. (It is used from § 3.3 onwards.)

For a nonempty finite set $I$, the notation $\mathscr G^I$ stands for the $A$-gerbe on $\mathrm {Hec}_G^I$ constructed in § 2.2. The metaplectic dual data $(H, \nu )$ are defined as in § 1.6.

3.1 Split tori: reduction to $T^{\sharp }$

3.1.1

Suppose that $G = T$ is a split torus with sheaf of cocharacters $\Lambda$.

Recall from § 1.6.5 that $\mu$ defines a quadratic form $Q$ on $\Lambda$, and we let $\Lambda ^{\sharp }\subset \Lambda$ be the kernel of the associated symmetric form $b$. Let $f : T^{\sharp } \rightarrow T$ denote the corresponding isogeny of split tori. For a nonempty finite set $I$, we obtain a morphism:

(3.1)\begin{equation} f^I : \mathrm{Gr}{}_{T^{\sharp}}^I \rightarrow \mathrm{Gr}{}_T^I. \end{equation}

We use the same notation for the map on Hecke stacks $f^I : \mathrm {Hec}_{T^{\sharp }}^I \rightarrow \mathrm {Hec}_T^I$. The pullback of $\mathscr G^I$ along $f^I$ is identified with the $A$-gerbe associated to the restriction of $\mu$ to $T^{\sharp }$, by functoriality of the construction. The notation $\mathscr G^I$ is retained for its restriction.

The goal of this subsection is to prove the following statement.

Proposition 3.1.2 Let $I$ be a nonempty finite set. Then pushforward along (3.1) defines an equivalence of tensor categories:

(3.2)\begin{align} \mathrm{Sat}{}^I_{T^{\sharp}, \mathscr G^I} \cong \mathrm{Sat}{}^I_{T, \mathscr G^I}. \end{align}

3.1.3

For a tuple $\lambda ^I = (\lambda ^i)_{i\in I}$ of elements of $\Lambda$, we have a closed immersion $X^I \rightarrow \mathrm {Gr}_T^I$ sending an $S$-point $x^I$ to the $T$-torsor $\mathscr O(\sum _{i\in I}\lambda ^ix^i)$ together with its canonical trivialization off $\Gamma _{x^I}$. We view its image as a closed subscheme:

(3.3)\begin{equation} X^{\lambda^I}\subset \mathrm{Gr}_T^I. \end{equation}

Since the $L_+^I(T)$-action on $\mathrm {Gr}_T^I$ is trivial, we obtain a closed substack $B_{X^{\lambda ^I}}(L_+^I(T)) \subset \mathrm {Hec}_T^I$ by taking the quotient of (3.3) by the $L_+^I(T)$-action.

Consider the special case $I = \{1\}$ and $\lambda ^1 = \lambda \in \Lambda$. Restricting sections of $T$ along $\Gamma _x\subset D_x$ defines a morphism $L_+^{\{1\}}(T) \rightarrow T$. We label the corresponding morphisms on their classifying stacks (relative to $X^{\lambda }\cong X$) as in the following diagram.

(3.4)

3.1.4

Recall from § 1.3.6 that any section $a\in \underline A(-1)$ defines the multiplicative $A$-torsor $\Psi ^a$ on $\mathbb {G}_m$, thus a rigidified section of $B^2\underline A$ over $B\mathbb {G}_m$.

There is an isomorphism of étale sheaves over $X$:

(3.5)\begin{equation} \check{\Lambda}\otimes A(-1) \cong \underline{\mathrm{Maps}}{}_e(B_XT, B^2_X\underline A),\quad x\otimes a\mapsto x^*(\Psi^a), \end{equation}

where the target is the sheaf of rigidified sections of $B^2(\underline A)$ over $B_X(T)$. This follows from the calculation of étale cohomology of $B_X(T)$ as in [Zha22, § 4].

In particular, the character $b(-, \lambda ) : \Lambda \rightarrow \underline A(-1)$ defines a rigidified section of $B^2(\underline A)$ over $B_X(T)$, to be denoted by $\Psi ^{b(-, \lambda )}$.

3.1.5

As observed in [Zha22, § 4.5], the $\mathbb {E}_1$-monoidal morphism underlying the metaplectic dual datum $\nu ^{\sharp }$ of § 1.6.5 canonically extends from $\Lambda ^{\sharp }$ to $\Lambda$.

For each $\lambda \in \Lambda$, we denote by $\nu _{\vartheta }(\lambda )$ the $A$-gerbe $\nu (\lambda )\otimes \omega _X^{Q(\lambda )}$, where $\omega _X^{Q(\lambda )}$ is the $A$-gerbe induced from $\omega _X$ along $\Psi ^{Q(\lambda )} : \mathbb {G}_m \rightarrow B(\underline A)$. We view the association $\lambda \mapsto \nu _{\vartheta }(\lambda )$ as a pointed morphism $\Lambda \rightarrow B^2_X(\underline A)$. Note that its restriction to $\Lambda ^{\sharp }$ is the pointed morphism underlying the $\mathbb {E}_{\infty }$-monoidal morphism $\nu _{\vartheta }$ defined in § 1.6.18.

We now establish the relationship between $\nu _{\vartheta }$ and the $A$-gerbe $\mathscr G^{\{1\}}$.

Lemma 3.1.6 For each $\lambda \in \Lambda$, there is a canonical isomorphism of $A$-gerbes, in reference to the morphisms in (3.4):

(3.6)\begin{equation} (i^{\lambda})^*\mathscr G^{\{1\}} \cong p^*\nu_{\vartheta}(\lambda)\otimes (p_T)^*(\Psi^{b(-, \lambda)}). \end{equation}

(For $\lambda = 0$, this is the identity automorphism of the trivial $A$-gerbe.)

3.1.7

We begin the proof of Lemma 3.1.6 with an observation: since the kernel of the projection $L_+^I(T) \rightarrow T$ is pro-unipotent, pulling back along $p_T$ defines an equivalence on the (respectively, discrete) groupoid of (respectively, rigidified) $A$-gerbes.

It thus suffices to perform the following two tasks:

1. (1) construct a canonical isomorphism,

   (3.7)\begin{equation} e^*(i^{\lambda})^*\mathscr G^{\{1\}} \cong \nu_{\vartheta}(\lambda); \end{equation}

2. (2) show that the rigidified $A$-gerbe $(i^{\lambda })^*\mathscr G^{\{1\}}\otimes p^*\nu _{\vartheta }(\lambda )^{\otimes -1}$ equals the character $b(-, \lambda )$ under the isomorphism (3.5).

3.1.8

In order to construct the isomorphism (3.7), we shall make use of an observation about étale metaplectic covers of $\mathbb {G}_m$.

Consider the following commutative diagram of stacks over $X$:

where $i$ is the diagonal immersion and $p$ is the projection onto the first factor. Pulling back along $\mathscr O(\Delta )$ and composing with the second row of (2.10), we find a morphism of complexes:

(3.8)\begin{equation} \Gamma_e(B_X\mathbb{G}_m, \underline A(1)[4]) \xrightarrow{\mathscr O(\Delta)^*} \Gamma(X, i^!\underline A(1)[4])\xrightarrow{\simeq} \Gamma(X, \underline A[2]). \end{equation}

The second map here is identified with the isomorphism induced from cohomological purity of the diagonal divisor: $i^!\underline A(1)[4] \cong \underline A[2]$. Truncating in degrees $\le 0$, the morphism (3.8) gives rise to a $\mathbb {Z}$-linear morphism:

(3.9)\begin{equation} \Gamma_e(B_X\mathbb{G}_m, B^4\underline A(1)) \rightarrow \Gamma(X, B^2\underline A). \end{equation}

On the other hand, the $\mathbb {Z}$-linear decomposition (1.25) gives rise to the following $\mathbb {Z}$-linear decomposition upon taking sections over $X$:

(3.10)\begin{align} \Gamma_e(B_X\mathbb{G}_m, B^4\underline A(1)) \cong \Gamma(X, B^2\underline A) \oplus \Gamma(X, \underline A(-1)). \end{align}

Lemma 3.1.9 Under the decomposition (3.10), the $\mathbb {Z}$-linear morphism (3.9) is the sum of $\Gamma (X, -)$ applied to the following two morphisms:

\begin{align*} \mathrm{id} : B^2\underline A &\rightarrow B^2\underline A, \text{ and} \\ \Psi_*\omega_X : \underline A(-1) &\rightarrow B^2\underline A\text{ defined as in \$ 1.6.17.} \end{align*}

Proof. It suffices to identify (3.9) after pre-composition with the inclusion of the first, respectively the second summand in (3.10).

For the first summand, this follows from the fact that

(3.11)\begin{equation} \Gamma(X, \underline A[2]) \xrightarrow{\Psi^*} \Gamma_e(B_X\mathbb{G}_m, \underline A(1)[4]) \xrightarrow{\mathscr O(\Delta)^*} \Gamma(X, i^!\underline A(1)[4]) \end{equation}

is identified with the purity isomorphism. Here, we have (slightly abusively) used $\Psi$ to denote the delooping of the Kummer torsor (1.12) over $\mathbb {G}_m$:

(3.12)\begin{equation} \Psi : B_X\mathbb{G}_m \rightarrow \lim_{\substack{n\ge 0 \\ \text{invertible}}} B^2(\mu_n), \end{equation}

which may be viewed as a limit of rigidified sections of $\mu _n[2]$ over $B_X\mathbb {G}_m$, so pairing with them defines the first map in (3.11).

For the second summand, we recall that the inclusion of $\underline A(-1)$ in $\underline {\Gamma }{}_e(B_X\mathbb {G}_m, B^4\underline A(1))$ is defined by sending $a$ to the $a$th power of the self-cup product $\Psi \cup \Psi$ of (3.12), which is isomorphic to the section $\Psi ^a \in \underline {\Gamma }{}_e(B_X\mathbb {G}_m, B^2\underline A)$ induced along the map:

(3.13)\begin{equation} \Psi_*(\mathscr L_{\rm univ}) : \underline A[2] \rightarrow \underline A(1)[4] \end{equation}

defined by the universal line bundle $\mathscr L_{\rm univ}$ on $B_X\mathbb {G}_m$ as in § 1.6.17.

By functoriality of the purity isomorphism with respect to change of coefficients (3.13), the following diagram is commutative.

(3.14)

Let us view (3.14) as expressing the top horizontal arrow $\Psi _*\omega _X$ as the composition of the three other arrows. The latter is indeed isomorphic to the composition of (3.9) with the inclusion of the summand $\Gamma (X, \underline A(-1))$.

Proof of Lemma 3.1.6 Let us construct the isomorphism (3.7) for each $\lambda \in \Lambda$. Pulling back $\mu$ along the cocharacter $\lambda : \mathbb {G}_m \rightarrow T$, we find an étale metaplectic cover $\lambda ^*(\mu )$ of $\mathbb {G}_m$.

The construction of $\mathscr G^{\{1\}}$ shows that $e^*(i^{\lambda })^*\mathscr G^{\{1\}}$ is the image of $\lambda ^*(\mu )$ under the morphism (3.9). By Lemma 3.1.9, there holds

\begin{align*} e^*(i^{\lambda})^*\mathscr G^{\{1\}} \cong \nu(\lambda) \otimes \omega_X^{Q(\lambda)}, \end{align*}

where the right-hand side is identified with $\nu _{\vartheta }(\lambda )$ by definition.

We now turn to task (2) of § 3.1.7. Since it asserts the equality of two sections of $\check {\Lambda }\otimes \underline A(-1)$ over $X^{\lambda }$, we may verify it over an arbitrary geometric point $\bar x$ of $X$.

We omit the superscript $\{1\}$ and use the subscript ‘$\bar x$’ to mean the base change to $\bar x$. The base change of $X^{\lambda }$ to $\bar x$ will simply be written as $\bar x^{\lambda }$.

We fix a uniformizer $\varpi$ at $\bar x$, which defines a geometric point $\varpi ^{\lambda }$ of $L(T)_{\bar x}$.

By construction, the $A$-gerbe over $(\mathrm {Hec}_T)_{\bar x}$ is defined by an $\mathbb {E}_1$-monoidal $A$-gerbe:

(3.15)\begin{equation} \mathscr G : L(T)_{\bar x} \rightarrow B^2(\underline A) \end{equation}

along with its trivialization over $L_+(T)_{\bar x}$ as in (2.8). Indeed, the trivialization equips $\mathscr G$ with left and right $L_+(T)_{\bar x}$-equivariance structures, which are its descent data along the projection $L(T)_{\bar x} \rightarrow (\mathrm {Hec}_T)_{\bar x}$.

As $L_+(T)_{\bar x}$ acts trivially on $\bar x^{\lambda }$, the $L_+(T)_{\bar x}$-equivariance structure on $e^*(i^{\lambda })^*(\mathscr G) \cong \mathscr G_{\bar x^{\lambda }}$ is described by a rigidified $A$-torsor $\tau ^{\lambda }$ over $L_+(T)_{\bar x}$. Its value at an $S$-point $t \in L_+(T)_{\bar x}$ is the quotient of the lower circuit by the upper circuit of the following square.

(3.16)

Here, the horizontal morphisms are the $\mathbb {E}_1$-monoidal structure of $\mathscr G$, the left vertical arrow is induced from $\varpi ^{\lambda }\cdot t = t\cdot \varpi ^{\lambda }$, and the right vertical arrow is the commutativity constraint of the $2$-groupoid of $A$-gerbes.

It suffices to identify $\tau ^{\lambda }$ with $\Psi ^{b(-, \lambda )}$ as a rigidified $A$-torsor over $L_+(T)_{\bar x}$, or equivalently as a rigidified $A$-torsor over $T_{\bar x}$.

Recall that $\mu$ may be regarded as an $\mathbb {E}_1$-monoidal morphism $T\rightarrow B^3_X\underline A(1)$. The morphism $T\times T\rightarrow B^2_X\underline A(1)$, sending a pair of $S$-points $(t_1, t_2)$ to the quotient of the lower circuit by the upper circuit of the square (with morphisms analogous to (3.16))

is rigidified along $e\times T$ and $T\times e$: it is a ‘bi-rigidified morphism’ in the sense of [Zha22, § 4.4.2]. As such, it agrees with the commutator of the corresponding extension of $T$ by $B_X^2\underline A(1)$, and equals $(\Psi \cup \Psi )^b$ by [Zha22, Corollary 4.7.6].

Finally, we observe that (3.15) is obtained from $\mu$ by taking sections over $\mathring D_{\bar x}$, which is identified with the punctured formal disk of $\mathbb {G}_m$ at $e$. The fiber of $(\Psi \cup \Psi )^b$ at $T_{\bar x}\times \varpi ^{\lambda }$ then induces $\Psi ^{b(-,\lambda )}$ along the canonical morphism $\Gamma (\mathring D_{\bar x}, \underline A(1)[2]) \rightarrow \underline A[1]$.

Proof of Proposition 3.1.2 Let us embed the Satake categories for $T^{\sharp }$ and $T$ into their analogues over the pairwise disjoint locus (corresponding to $X^p$ for $p = \mathrm {id}_I$) using (2.19):

(3.17)

Here, $f^I : \mathrm {Gr}_{T^{\sharp }}^I \rightarrow \mathrm {Gr}_T^I$ is the closed immersion corresponding to the isogeny $f : T^{\sharp }\rightarrow T$.

Claim The right vertical functor in (3.17) is an equivalence.

Indeed, $f$ is an inclusion of connected components over $X^p$, so it suffices to show that any $\mathscr F\in \mathrm {Sat}{}_{T, \mathscr G^I}^p$ is supported on the union of $X^{\lambda ^I}\!\times _{X^I}X^p$ with $\lambda ^I$ is a tuple of elements in $\Lambda ^{\sharp }$.

The base change of $L_+^I(T)$ to a geometric point $\bar x^I \in X^p$ is identified with $\prod _{i\in I}L_+^{\{i\}}(T)_{\bar x^i}$. Given a tuple $\lambda ^I = (\lambda ^i)_{i\in I}$ with $\lambda ^i\in \Lambda$, the $A$-gerbe $\mathscr G^I$ restricts to $\boxtimes _{i\in I}\mathscr G_{\bar x^i}^{\{i\}}$ over its classifying stack, viewed as a substack of $(\mathrm {Hec}_T^I)_{\bar x}$.

By Lemmas 1.3.5 and 3.1.6, it suffices to prove

\begin{align*} H^0\bigg(\prod_{i\in I}T_{\bar x}, \boxtimes_{i\in I} \Psi^{b(-, \lambda_i)}\bigg) = 0\quad \text{if some }\lambda_i\notin\Lambda^{\sharp}. \end{align*}

This follows from the vanishing of global sections of $\Psi ^a$ for $a\neq 0 \in A(-1)$, see § 1.3.6.

To show that the left vertical functor in (3.17) is an equivalence, it remains to show that it is essentially surjective. Since the essential image of (2.19) is closed under direct summands, an object $\mathscr F \in \mathrm {Sat}{}_{T, \mathscr G^I}^p$ belongs to $\mathrm {Sat}{}^I_{T, \mathscr G^I}$ if and only if its restriction to each $X^{\lambda ^I}\!\times _{X^I}X^p$ extends to a lisse $E$-sheaf on $X^{\lambda ^I}$. The same assertion for $T^{\sharp }$ then concludes the proof.

Remark 3.1.10 The proof of Proposition 3.1.2 also shows that $\mathrm {Sat}{}^I_{T^{\sharp }}$ is equivalent to the abelian category $\mathrm {Perv}(\mathrm {Gr}{}_{T^{\sharp }}^I)_{/X^I}$ of perverse, universally locally acyclic $E$-sheaves on $\mathrm {Gr}{}_{T^{\sharp }}^I$, i.e. the $L{}_+^I(T^{\sharp })$-equivariance condition is automatic.

The analogous statement for $T$ instead of $T^{\sharp }$ is emphatically false.

3.2 Split tori: equivalence

3.2.1

We continue to suppose that $G = T$ is a split torus with sheaf of cocharacters $\Lambda$. Furthermore, we assume that the symmetric form $b$ vanishes, i.e. $\Lambda ^{\sharp } = \Lambda$.

Under this assumption, the étale metaplectic cover $\mu : B_X(T) \rightarrow B^4_X\underline A(1)$ canonically lifts to an $\mathbb {E}_{\infty }$-monoidal morphism, see [Zha22, § 4.6].

The metaplectic dual data $(H, \nu )$ admit a concrete description:

1. (1) $H$ is the dual torus $\check T$ over $E$;

2. (2) $\nu : \Lambda \rightarrow B^2_X\underline A$ is obtained from $\mu$ by taking rigidified sections over $B_X\mathbb {G}_m$, and composing with the projection from $\underline {\Gamma }{}_e(B_X\mathbb {G}_m, B^4_X\underline A(1))$ onto $B_X^2\underline A$.

Our current goal is to prove the geometric Satake equivalence for such $(T, \mu )$.

Proposition 3.2.2 For a nonempty finite set $I$, there is a canonical equivalence of tensor categories:

(3.18)\begin{align} \mathrm{Sat}{}^I_{T, \mathscr G^I} \cong \mathrm{Rep}{}^I_{\check T^I, \nu_{\vartheta}^I}. \end{align}

3.2.3

For a nonempty finite set $I$, we regard $\mathrm {Gr}_T^I$ as an étale sheaf of abelian groups over $X^I$, being defined by the quotient $L^I(T)/L_+^I(T)$. The collection of closed immersions (3.3) is gathered into a morphism of sheaves of abelian groups $\Lambda ^I \rightarrow \mathrm {Gr}_T^I$, the source being the external direct sum of $\Lambda$ over $i\in I$.

Since $\mu$ admits an $\mathbb {E}_{\infty }$-monoidal structure, the commutative diagram (for $G = T$) lifts to a commutative diagram of étale sheaves of $\mathbb {E}_{\infty }$-monoids over $X^I$. In particular, $\mathscr G^I$ may be viewed an $\mathbb {E}_{\infty }$-monoidal morphism $\mathrm {Gr}_T^I \rightarrow B^2_{X^I}(\underline A)$.

Composing these two morphisms, we obtain an $\mathbb {E}_{\infty }$-monoidal morphism:

(3.19)\begin{equation} \Lambda^I \rightarrow B^2_{X^I}(\underline A),\quad \lambda^I\mapsto e^*(i^{\lambda^I})^*\mathscr G^I. \end{equation}

Lemma 3.2.4 The $\mathbb {E}_{\infty }$-monoidal morphism (3.19) is canonically identified with

\begin{align*} \nu_{\vartheta}^I : \Lambda^I \rightarrow B^2_{X^I}(\underline A),\quad (\lambda^i)_{i\in I} \mapsto \boxtimes_{i\in I} \nu_{\vartheta}(\lambda^i), \end{align*}

where the $\mathbb {E}_{\infty }$-monoidal morphism $\nu _{\vartheta }$ is defined as in § 1.6.18.

Proof. For each $i\in I$, let $\gamma ^i : \Gamma _{x^i} \cong X^I \rightarrow X^I\times X$ denote the graph of the $i$th projection $x^i\,{:} X^I\rightarrow X$. The union of the graphs is denoted by $\gamma : \Gamma _{x^I}\rightarrow X^I\times X$. (We are following the notation of § 2.1, viewing $x^I = (x^i)_{i\in I}$ as an $X^I$-point of $X$.) We also have the projections $p_1 : X^I\times X \rightarrow X^I$ and $p_2 : X^I\times X \rightarrow X$ onto the $X^I$ and the last factor, respectively.

Recall that sheaves of grouplike $\mathbb {E}_{\infty }$-monoids are equivalent to sheaves of connective spectra, where the functor $B$ corresponds to suspension $[1]$.

The pullback $p_2^*(\mu )$ defines a morphism of sheaves of connective spectra $p_2^*(\mu ) : T[1] \rightarrow \underline A(1)[4]$ over $X^I$, and induces such a commutative diagram as follows.

(3.20)

Here, the first horizontal morphism sends $(\lambda ^i)_{i\in I}$ to the section $(\mathscr O(\lambda ^i\Gamma _{x^i}))_{i\in I}$, and the last vertical morphism is that from (2.10).

The upper circuit of (3.20) is the morphism (3.19). The lower circuit of (3.20) is the external sum of the morphisms $\Lambda \rightarrow \underline A[2]$ defined by (3.19) for the singleton $\{i\}$. The construction thus reduces to the case $I = \{1\}$, where it is given by (3.7).

To see that (3.7) is compatible with the $\mathbb {E}_{\infty }$-monoidal structures, we note that (3.19) for $I = \{1\}$ may be re-expressed as the composition of $\mathbb {E}_{\infty }$-monoidal morphisms:

\begin{align*} \Lambda \xrightarrow{\simeq} \Gamma_e(B_X\mathbb{G}_m, B_XT) \xrightarrow{\mu} \Gamma_e(B_X\mathbb{G}_m, B^4\underline A(1)) \xrightarrow{(3.9)} \Gamma(X, B^2\underline A). \end{align*}

On the other hand, Lemma 3.1.9 identifies (3.9) as the sum of the identity endomorphism on $\Gamma (X, B^2\underline A)$ and the section of $\Psi _*(\omega _X)$ over $X$.

These two components give rise to the $\mathbb {E}_{\infty }$-monoidal morphism $\nu : \Lambda \rightarrow B^2_X(\underline A)$ and the $\mathbb {Z}$-linear morphism $\omega _X^Q : \Lambda \rightarrow B^2_X(\underline A)$ of § 1.6.18, respectively.

Proof of Proposition 3.2.2 Let us construct the tensor equivalence (3.18).

Consider the product map $m : \prod _{i\in I}\mathrm {Gr}_T^{\{i\}} \rightarrow \mathrm {Gr}_T^I$ of ind-schemes over $X^I$. (Any ordering $I\cong \{1,\ldots, k\}$ realizes $\prod _{i\in I}\mathrm {Gr}_T^{\{i\}}$ as the ind-scheme $\widetilde {\mathrm {Gr}}_T^{\{1\},\ldots,\{k\}}$ of § 2.1.5 and $m$ the composition of all modifications.)

Note that the reduced sub-indscheme of $\prod _{i\in I}\mathrm {Gr}_T^{\{i\}}$ represents the étale sheaf $\Lambda ^I$. Using Lemma 3.2.4, we find functors of tensor categories:

(3.21)\begin{align} \mathrm{Rep}{}_{\check T, \nu_{\vartheta}^I}^I &\cong \bigoplus_{\lambda^I \in \Lambda^I} \mathrm{Lis}_{\nu_{\vartheta}^I(\lambda^I)}(X^I)\nonumber\\ &\cong \mathrm{Perv}_{\mathscr G^I}\bigg(\prod_{i\in I}\mathrm{Gr}_T^{\{i\}}\bigg)_{/X^I} \xrightarrow{m_!} \mathrm{Perv}_{\mathscr G^I}(\mathrm{Gr}_T^I)_{/X^I} \cong \mathrm{Sat}{}_{T, \mathscr G^I}^I, \end{align}

where $\mathrm {Sat}{}_{T, \mathscr G^I}^I$ is equipped with the tensor structure induced from the $\mathbb {E}_{\infty }$-monoidal morphism $\mathscr G^I : \mathrm {Gr}_T^I \rightarrow B^2_{X^I}(\underline A)$. This tensor structure naturally extends the convolution monoidal structure $\circ _X$ on $\mathrm {Sat}{}_{T, \mathscr G^I}^I$.

Equipped with this tensor structure, $\mathrm {Sat}{}_{T, \mathscr G^I}^I$ still coincides with $(\mathrm {Sat}{}_{T, \mathscr G^I}^I, \star _X, e_!(\underline E))$ defined by the fusion product, as it lifts the latter to an $\mathbb {E}_{\infty }$-monoid in the $2$-category of symmetric monoidal categories, cf. Remark 2.3.13.

It remains to show that the functor $m_!$ in (3.21) is an equivalence. By universal local acyclicity, both categories embed fully faithfully in their analogues over the pairwise disjoint locus $X^p\subset X^I$ (see § 2.3.10), where they are both equivalent to $\mathrm {Sat}{}_{T, \mathscr G^I}^p$. This implies that $m_!$ is fully faithful. To see that it is essentially surjective, we observe that an object $\mathscr F \in \mathrm {Sat}{}_{T, \mathscr G^I}^p$ belongs to $\mathrm {Sat}{}^I_{T, \mathscr G^I}$ if and only if its restriction to each $X^{\lambda ^I}\!\times _{X^I}X^p$ extends to a lisse $E$-sheaf on $X^{\lambda ^I}$, as the essential image of (2.19) is closed under direct summands.

3.2.5

We relax the condition $b = 0$, i.e. $\mu$ stands for any étale metaplectic cover of $T$.

We shall construct the geometric Satake equivalence (2.23) for split tori.

Proof Construction of (2.23) for $G = T$

Let $T^{\sharp } \rightarrow T$ be the isogeny of split tori corresponding to $\Lambda ^{\sharp }\subset \Lambda$ as in § 3.1.1.

The metaplectic dual pair $(H, \nu )$ is precisely the pair $H = \check T^{\sharp }$, $\nu : \Lambda ^{\sharp } \rightarrow B^2_X(\underline A)$ associated to $T^{\sharp }$ and the restriction of $\mu$ as in § 3.2.1.

The equivalence (2.23) is thus the composition of the inverse of (3.2) with (3.18):

\begin{align*} \mathrm{Sat}{}_{T, \mathscr G^I}^I \cong \mathrm{Sat}{}_{T^{\sharp}, \mathscr G^I}^I \cong \mathrm{Rep}{}^I_{H^I, \nu_{\vartheta}^I}. \end{align*}

Theorem 2.4.4 for tori is proved.

3.3 The abelian category $\mathrm {Sat}_{G, \mathscr G}$

3.3.1

Let us now turn to the context where $G\rightarrow X$ is split reductive with chosen Borel subgroup $B\subset G$. Choose furthermore a splitting of $B\twoheadrightarrow T$ and view $T$ as a maximal torus of $G$.

The goal of this subsection is to determine $\mathrm {Sat}{}_{G, \mathscr G^I}^I$ as an abelian category for $I = \{1\}$. To lighten the notation, we omit the superscript $I$ in this subsection.

3.3.2

Write $\Lambda ^+\subset \Lambda$ for the submonoid consisting of dominant cocharacters. Each $\lambda \in \Lambda ^+$ determines a Schubert cell $\mathrm {Gr}_G^{\lambda }$ as the $L_+G$-orbit of $X^{\lambda }$, embedded in $\mathrm {Gr}_G^{\lambda }$ along (3.3) and the closed immersion $\mathrm {Gr}_T\subset \mathrm {Gr}_G$.

The closure of $\mathrm {Gr}_G^{\lambda }$ is identified with $\mathrm {Gr}_G^{\le \lambda } := \bigcup _{\lambda _1\le \lambda }\mathrm {Gr}_G^{\lambda _1}$. Denote by $j^{\lambda } : \mathrm {Gr}_G^{\lambda }\subset \mathrm {Gr}_G^{\le \lambda }$ the open immersion.

Write $P^{\lambda }\subset G$ for the standard parabolic subgroup corresponding to the simple roots annihilated by $\lambda$. It has Levi quotient $M^{\lambda }$. The quotient map $L_+G \rightarrow G$ induces a map on their homogeneous spaces $\mathrm {Gr}_G^{\lambda } \rightarrow G/P^{\lambda }$, see [Zhu17, § 2.1]. Its quotient by $L_+G$ on the source and $G$ on the target defines a map $L_+G\backslash \mathrm {Gr}_G^{\lambda } \rightarrow B_X(P^{\lambda })$, hence a map $p_{M^{\lambda }}$ fitting into the following commutative diagram.

(3.22)

3.3.3

Since $M^{\lambda }$ is reductive, the groupoid of rigidified $A$-gerbes over $B_{X}(M^{\lambda })$ is identified with the discrete abelian group $\operatorname {Hom}(\pi _1(M^{\lambda }), A(-1))$. This follows, for example, from [Zha22, Proposition 5.1.11].

The natural map $T\rightarrow M^{\lambda }$ induces a surjection $\Lambda \rightarrow \pi _1(M^{\lambda })$ whose kernel is spanned by simple coroots whose associated roots are annihilated by $\lambda$. Via this map, rigidified $A$-gerbes over $B_{X}(M^{\lambda })$ form an abelian subgroup of those over $B_{X}(T)$.

Recall the rigidified $A$-gerbe $\Psi ^{b(-,\lambda )}$ defined in § 3.1.4. The identity (1.23) implies that $\Psi ^{b(-, \lambda )}$ defines a rigidified $A$-gerbe over $B_{X}(M^{\lambda })$.

3.3.4

The following lemma is a generalization (and corollary) of Lemma 3.1.6. It is an analogue of [FL10, Lemma 2.4] in étale cohomology.

Lemma 3.3.5 For each $\lambda \in \Lambda ^+$, there is a canonical isomorphism of $A$-gerbes, in reference to the morphisms in (3.22):

(3.23)\begin{equation} (i^{\lambda})^*\mathscr G \cong p^*\nu_{\vartheta}(\lambda)\otimes (p_{M^{\lambda}})^*(\Psi^{b(-, \lambda)}). \end{equation}

(For $\lambda = 0$, this is the identity automorphism of the trivial $A$-gerbe.)

Proof. Since the kernel of $L_+G \rightarrow G$ is pro-unipotent, pulling back by $p_{M^{\lambda }}$ defines an equivalence on the groupoid of (rigidified) $A$-gerbes.

By Lemma 3.1.6, we already have an isomorphism:

\begin{align*} e^*(i^{\lambda})^*\mathscr G \cong \nu_{\vartheta}(\lambda). \end{align*}

It remains to show that $(i^{\lambda })^*\mathscr G\otimes p^*\nu _{\vartheta }(\lambda )^{\otimes -1}$ equals $\Psi ^{b(-,\lambda )}$ as rigidified $A$-gerbes over $B_{X}(M^{\lambda })$. This statement can be proved after pulling back along $B_{X}(T) \rightarrow B_{X}(M^{\lambda })$, where it again reduces to Lemma 3.1.6.

3.3.6

Let $\lambda \in \Lambda ^{\sharp, +} := \Lambda ^{\sharp } \cap \Lambda ^+$. The isomorphism (3.23) shows that pulling back along the projection $p$ carries the $A$-gerbe $\nu _{\vartheta }(\lambda )$ to the restriction of $\mathscr G$.

In particular, any $\nu _{\vartheta }(\lambda )$-twisted $E$-local system $\mathscr E$ over $X$ pulls back to a $\mathscr G$-twisted $L_+G$-equivariant $E$-local system $p^*(\mathscr E)$ over $\mathrm {Gr}_G^{\lambda }$. Up to cohomological shift and Tate twist, we may form its intermediate extension along $j^{\lambda }$ as an object of $\mathrm {Sat}_{G, \mathscr G}$:

\begin{align*} \mathrm{IC}_{\mathscr E} := (j^{\lambda})_{!*} p^*\mathscr E(\langle \check{\rho}, \lambda\rangle)[\langle 2\check{\rho}, \lambda\rangle] \in \mathrm{Sat}_{G, \mathscr G}. \end{align*}

Here, $\langle \check {\rho }, \lambda \rangle \in ({1}/{2})\mathbb {Z}$ and the Tate twist is formed with the aid of $\underline E( {1}/{2})$.

Proposition 3.3.7 The following functor is an equivalence of $E$-linear abelian categories:

(3.24)\begin{equation} \bigoplus_{\lambda \in \Lambda^{\sharp, +}}\mathrm{Lis}_{\nu_{\vartheta}(\lambda)}(X) \rightarrow \mathrm{Sat}_{G, \mathscr G},\quad (\mathscr E^{\lambda}) \mapsto \bigoplus_{\lambda\in\Lambda^{\sharp, +}} \mathrm{IC}_{\mathscr E^{\lambda}}. \end{equation}

Proof. For each $\lambda \in \Lambda ^{\sharp, +}$, the morphism $p$ in (3.22) is smooth with connected fibers. It follows that (3.24) restricts to a fully faithful functor on each summand.

By the definition of intermediate extensions, the images of distinct summands under (3.24) are orthogonal. Thus, (3.24) is fully faithful. It remains to show that it is also essentially surjective, i.e. any $\mathscr F \in \mathrm {Sat}_{G, \mathscr G}$ is a direct sum of objects of the form $\mathrm {IC}_{\mathscr E^{\lambda }}$ over $\lambda \in \Lambda ^{\sharp, +}$.

For each $\lambda \in \Lambda ^+$, we let $\mathscr F^{\lambda } \in \mathrm {Lis}_{\nu _{\vartheta }(\lambda )}(X, E)$ denote the restriction of $\mathscr F$ to $X^{\lambda }$. If $\lambda \notin \Lambda ^{\sharp, +}$, it follows from Lemmas 1.3.5 and 3.3.5 that $\mathscr F^{\lambda } = 0$, so the restriction of $\mathscr F$ to $\mathrm {Gr}_G^{\lambda }$ vanishes by $L_+G$-equivariance. If $\lambda \in \Lambda ^{\sharp, +}$, the restriction of $\mathscr F$ to $\mathrm {Gr}_G^{\lambda }$ is given by $p^*(\mathscr F^{\lambda })$. Thus, $\mathscr F$ is an iterated extension of the objects $\mathrm {IC}_{\mathscr F^{\lambda }}$ over $\lambda \in \Lambda ^{\sharp, +}$.

It remains to show that for $\lambda _1\neq \lambda _2$, the images of $\mathrm {Lis}_{\nu _{\vartheta }(\lambda _1)}(X)$ and $\mathrm {Lis}_{\nu _{\vartheta }(\lambda _1)}(X)$ in $\mathrm {Sat}_{G, \mathscr G}$ have no nonsplit extensions. This is proved in the same way as in the non-metaplectic setting, by reducing it to Lusztig's parity vanishing of the stalks of the intersection cohomology complex, see [Zhu17, Proposition 5.1.1] and the references therein.

Remark 3.3.8 For $G = T$, the decomposition (3.24) coincides with (3.21) appearing in the proof of Proposition 3.2.2.

In contrast to the case of tori, (3.24) is incompatible with the monoidal structure on $\mathrm {Sat}_{G, \mathscr G}$, i.e. the monoidal product of two homogeneous objects is, in general, inhomogeneous.

3.4 Constant terms: construction

3.4.1

Suppose that $P\subset G$ is a standard parabolic subgroup with unipotent radical $N_P\subset P$ and Levi quotient $P \twoheadrightarrow M$.

The restriction of $\mu$ to $B(P)$ canonically descends to an étale metaplectic cover of $M$.

3.4.2

For a nonempty finite set $I$, the construction of § 2.2 for $M$ produces an $A$-gerbe $\mathscr G^I_M$ over $\mathrm {Hec}_M^I$. We have an isomorphism of $A$-gerbes:

(3.25)\begin{equation} p^*(\mathscr G^I) \cong q^*(\mathscr G_M^I) \end{equation}

along the canonical morphisms $p : \mathrm {Hec}_P^I \rightarrow \mathrm {Hec}_G^I$ and $q : \mathrm {Hec}_P^I\rightarrow \mathrm {Hec}_M^I$.

Define the (naïve) constant term functor to be the following functor of $E$-linear stable $\infty$-categories, using the isomorphism (3.25):

(3.26)\begin{align} \mathrm{CT}{}_P^I : \mathrm{Shv}_{\mathscr G^I}(\mathrm{Hec}_G^I) \rightarrow \mathrm{Shv}_{\mathscr G_M^I}(\mathrm{Hec}_M^I),\quad \mathscr F\mapsto q_! p^*(\mathscr F). \end{align}

3.4.3

Since the connected components of $\mathrm {Hec}_M^I$ are enumerated by $\pi _1(M)$, the sum of positive roots occurring in $N_P$ defines a character $2\check {\rho }_P : \pi _1(M) \rightarrow \mathbb {Z}$, which we view as a locally constant function on $\mathrm {Hec}_M^I$. (In particular, $2\check {\rho }_B = 2\check {\rho }$.)

We shall adjust (3.26) by a cohomological shift by $2\check {\rho }_P$ and Tate twist by $\check {\rho }_P$ (with the aid of $\underline E( {1}/{2})$). The result will be a tensor functor on the Satake categories with modified commutativity constraints, as defined in § 2.4.

Proposition 3.4.4 The functor (3.26) induces an exact tensor functor:

(3.27)\begin{equation} \mathrm{CT}{}_P^I(\check{\rho}_P)[2\check{\rho}_P] : {}^+\mathrm{Sat}{}^I_{G, \mathscr G^I} \rightarrow {}^+\mathrm{Sat}^I_{M, \mathscr G_M^I}. \end{equation}

Proof. The proof is identical to its non-metaplectic counterpart and follows from Braden's hyperbolic localization theorem. We briefly indicate the argument.

Claim $\mathrm {CT}{}^I_P(\check {\rho }_P)[2\check {\rho }_P]$ carries the abelian subcategory $\mathrm {Sat}{}^I_{G, \mathscr G^I}$ to $\mathrm {Sat}{}_{M, \mathscr G_M^I}^I$.

Indeed, after establishing the claim, the tensor structure on $\mathrm {CT}{}^I_P(\check {\rho }_P)[2\check {\rho }_P]$ arises from its commutation with the external fusion products (see § 2.3.10). Note that due to the degree shift $[2\check {\rho }_P]$, the Koszul sign rule implies that $\mathrm {CT}{}^I_P(\check {\rho }_P)[2\check {\rho }_P]$ is compatible with exchanging factors after modifying the commutativity constraints.

To prove the claim, we may split the Levi quotients of $B$ and $P$ and view $M$ as a subgroup of $G$ containing the maximal torus $T$. Let $P^-\subset G$ be the parabolic subgroup opposite to $P$. The local Hecke stack $\mathrm {Hec}_{P^-}^I$ is equipped with projections $p^-$ and $q^-$ to $\mathrm {Hec}_G^I$ and $\mathrm {Hec}_M^I$, respectively. There is a functor:

(3.28)\begin{equation} \mathrm{CT}{}_{P^-}^{I, !} : \mathrm{Shv}_{\mathscr G^I}(\mathrm{Hec}_G^I) \rightarrow \mathrm{Shv}_{\mathscr G_M^I}(\mathrm{Hec}_M^I),\quad \mathscr F\mapsto (q^-)_* (p^-)^!(\mathscr F). \end{equation}

Braden's theorem (as stated in [DG14]) identifies the functor (3.26) with (3.28). Its formation commutes with base change along $S\rightarrow X^I$ for any $k$-scheme $S$.

To show that $\mathrm {CT}{}^I_P$ preserves universal local acyclicity relative to $X^I$, we base change to the spectrum of a rank-$1$ valuation ring and observe that the characterization [HS23, Theorem 4.4(iv)] holds for $\mathrm {CT}{}_{P^-}^{I, !}$.

To show that $\mathrm {CT}{}^I_P(\check {\rho }_P)[2\check {\rho }_P]$ is $t$-exact with respect to the perverse $t$-structure relative to $X^I$, we base change to any geometric point of $X^I$ and reduce to the case $I = \{1\}$. The argument of [MV07, § 3] then shows that $\mathrm {CT}{}^{\{1\}}_P(\check {\rho }_P)[2\check {\rho }_P]$ is right $t$-exact and $\mathrm {CT}{}_{P^-}^{\{1\}, !}(\check {\rho }_P)[2\check {\rho }_P]$ is left $t$-exact.

Remark 3.4.5 The tensor functor (3.27) is compatible with compositions. More precisely, for a parabolic subgroup $P_1\subset M$ with Levi quotient $P_1\twoheadrightarrow M_1$, the composition of $\mathrm {CT}{}^I_{P_1}(\check {\rho }_{P_1})[2\check {\rho }_{P_1}]$ with $\mathrm {CT}{}^I_P(\check {\rho }_P)[2\check {\rho }_P]$ is canonically isomorphic to $\mathrm {CT}{}^I_{P_0}(\check {\rho }_{P_0})[2\check {\rho }_{P_0}]$, where $P_0$ denotes the parabolic subgroup $P_0 := P\times _M P_1 \subset G$.

Remark 3.4.6 The Levi quotient of any Borel subgroup $B\subset G$ is identified with the universal Cartan $T$, and the tensor functor $\mathrm {CT}^I_B(\check {\rho })[2\check {\rho }]$ (3.27) is independent of the choice of $B\subset G$. More precisely, given two Borel subgroups $B_1, B_2\subset G$, there is a canonical isomorphism of tensor functors:

(3.29)\begin{equation} CT_{B_1}^I(\check{\rho})[2\check{\rho}] \cong \mathrm{CT}_{B_2}^I(\check{\rho})[2\check{\rho}], \end{equation}

subject to the natural compatibility for three Borel subgroups.

Let us first construct (3.29) subject to the choice of a section $g\in G$ such that the inner automorphism $\mathrm {int}_g$ of $G$ carries $B_1$ to $B_2$.

Indeed, the moduli description of $\mathrm {Hec}_G^I$ shows that $\mathrm {int}_g$ induces the identity automorphism on $\mathrm {Hec}_G^I$. Thus pulling back along $\mathrm {int}_g$ yields the following commutative diagram.

(3.30)

Here, $T_1$ (respectively, $T_2$) denotes the maximal quotient torus of $B_1$ (respectively, $B_2$), so the isomorphism $\mathrm {int}_g : T_1 \cong T_2$ is encoded in the definition of $T$. The commutative diagram (

3.30

) yields an isomorphism $F_g$ of the two functors in (3.29).

It remains to prove that for $B = B_1 = B_2$ and $g \in B$, the isomorphism $F_g$ equals the identity automorphism of $\mathrm {CT}_B^I(\check {\rho })[2\check {\rho }]$, where the endofunctor $(\mathrm {int}_g)^*$ of $\mathrm {Sat}{}^I_{T, \mathscr G_T^I}$ is trivialized as $\mathrm {int}_g$ induces the identity on $T$.

For this statement, we construct the automorphisms $F_g$ as a family over $g\in B$, i.e. they define an automorphism of the functor $\underline E_B\boxtimes \mathrm {CT}_B^I(\check {\rho })[2\check {\rho }]$ valued in $E$-sheaves over $B\times \mathrm {Hec}_T^I$. This automorphism equals the identity, because it restricts to the identity over $e\times \mathrm {Hec}_T^I$ and $B$ is smooth and connected.

3.5 Constant terms: vanishing

3.5.1

In this subsection, we specialize to the case $I = \{1\}$ and study the behavior of the constant term functor (3.26) associated to the Borel subgroup $B\subset G$.

We denote the Weyl group of $G$ by $W$. It canonically acts on $\Lambda$. As in § 3.3, we temporarily drop the superscript $\{1\}$.

3.5.2

Note that for any $\lambda \in \Lambda ^{\sharp }$ and $w\in W$, the difference $\lambda - w(\lambda )$ belongs to $\Lambda ^{\sharp, r}$.

Recall that the $\mathbb {E}_{\infty }$-monoidal morphism $\nu ^{\sharp } : \Lambda ^{\sharp } \rightarrow B^2_X(\underline A)$ is canonically trivialized over $\Lambda ^{\sharp, r}$ (see § 1.6.5). Since $Q$ vanishes over $\Lambda ^{\sharp, r}$, the $\mathbb {Z}$-linear morphism $\omega _X^Q : \Lambda ^{\sharp } \rightarrow B^2_X(\underline A)$ of § 1.6.17 is likewise trivialized over $\Lambda ^{\sharp, r}$.

Therefore, $\nu _{\vartheta }$ is equipped with a canonical $W$-invariance structure.

3.5.3

According to Proposition 3.3.7, the functor $\mathrm {CT}_B(\check {\rho })[2\check {\rho }]$ is the direct sum of functors indexed by pairs of elements $\lambda \in \Lambda ^{\sharp, +}, \lambda _1 \in \Lambda ^{\sharp }$:

(3.31)\begin{align} \mathrm{CT}_B^{\lambda, \lambda_1}(\check{\rho})[2\check{\rho}] : \mathrm{Lis}_{\nu_{\vartheta}(\lambda)}(X) &\subset {}^+\mathrm{Sat}_{G, \mathscr G}\nonumber\\ &\xrightarrow{\mathrm{CT}_B(\check{\rho})[2\check{\rho}]} \mathrm{Sat}_{T, \mathscr G_T} \twoheadrightarrow \mathrm{Lis}_{\nu_{\vartheta}(\lambda_1)}(X). \end{align}

The next proposition describes the behavior of (3.31). Its proof will occupy the remainder of this subsection.

Proposition 3.5.4 Let $\lambda \in \Lambda ^{\sharp, +}$ and $\lambda _1\in \Lambda ^{\sharp }$. The following statements hold:

1. (1) if $\lambda _1 \in W\lambda$, then $\mathrm {CT}_B^{\lambda,\lambda _1}(\check {\rho })[2\check {\rho }]$ is equivalent to the identity functor (in reference to the canonical $W$-invariance structure on $\nu _{\vartheta }$);

2. (2) if $\lambda _1 \notin \lambda + \Lambda ^{\sharp, r}$, then $\mathrm {CT}_B^{\lambda, \lambda _1}(\check {\rho })[2\check {\rho }] = 0$.

3.5.5

In the remainder of this subsection, we fix a splitting of $B\twoheadrightarrow T$ and regard $T$ as a maximal torus in $G$.

For each $\lambda \in \Lambda$, we write $S^{\lambda }$ for the base change of $\mathrm {Gr}_B \rightarrow \mathrm {Gr}_T$ to $X^{\lambda }$. Using the locally closed immersion $\mathrm {Gr}_B \rightarrow \mathrm {Gr}_G$, we may view $S^{\lambda }$ as a locally closed sub-indscheme of $\mathrm {Gr}_G$. In particular, we may form the subschemes $\mathrm {Gr}_G^{\le \lambda }\cap S^{\lambda _1}$, $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}\subset \mathrm {Gr}_G$.

For $\lambda \in \Lambda ^{\sharp, +}$ and $\lambda _1 \in \Lambda ^{\sharp }$, the pair $(\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}, \mathscr G)$ of a scheme equipped with an $A$-gerbe maps to the pairs $(X, \nu _{\vartheta }(\lambda ))$ and $(X, \nu _{\vartheta }(\lambda _1))$, as induced from the two inclusions in the diagram

(3.32)

and the isomorphism of Lemma 3.3.5.

The identifications of $\mathscr G$ over $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}$ with both $p^*\nu _{\vartheta }(\lambda )$ and $p^*\nu _{\vartheta }(\lambda _1)$ compose into an isomorphism:

(3.33)\begin{equation} p^*\nu_{\vartheta}(\lambda) \cong p^*\nu_{\vartheta}(\lambda_1). \end{equation}

If $\lambda _1 \in W\lambda$, the identification (3.33) coincides with that induced from the $W$-invariance structure of $\nu _{\vartheta }$.

Proof of Proposition 3.5.4(1) This part is identical to the non-metaplectic context. Indeed, when $\lambda _1 \in W\cdot \lambda$, the inclusion $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1} \subset \mathrm {Gr}_G^{\le \lambda }\cap S^{\lambda _1}$ is an isomorphism and both schemes are identified with the $L_+N$-orbit of $X^{\lambda }$ (see the proof of [MV07, Theorem 3.2]).

The projection $p$ in (3.32) is thus an affine space bundle of fiber rank $\langle \check {\rho }, \lambda _1 + \lambda \rangle$. Hence, the functor $p_!p^*$ is canonically equivalent to the value of $(-\check {\rho })[-2\check {\rho }]$ at $\lambda + \lambda _1$. As $\mathrm {CT}_B^{\lambda,\lambda _1}$ is identified with $p_!p^*$, the desired conclusion follows.

3.5.6

To prove Proposition 3.5.4(2), we need an additional piece of data associated to the isomorphism (3.33).

To define it, we briefly return to a more abstract setting: $S$ is any base scheme and $A$ is a finite abelian group, $H\rightarrow S$ is a group scheme, and $Y$ is an $S$-scheme equipped with an $H$-action.

Suppose that $\mathscr G_1$, $\mathscr G_2$ are $A$-gerbes over $Y$ equipped with $H$-equivariance structures. Let $f : \mathscr G_1\cong \mathscr G_2$ be an isomorphism of plain $A$-gerbes over $Y$. Consider the following diagram formed by the action and projection maps from $H\times Y$ to $Y$, respectively.

(3.34)

The diagram (3.34) need not commute. The quotient of its upper circuit by its lower circuit defines an $A$-torsor over $H\times Y$ rigidified along $e\times Y$, which we call the obstruction of $f$ to be $H$-equivariant.⁴

3.5.7

We shall apply the above construction to the isomorphism (3.33) of $A$-gerbes over $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}$, equipped with the $T_{\mathrm {ad}}$-action. Here, $T_{\mathrm {ad}}$ is the maximal torus of the adjoint group $G_{\mathrm {ad}}$ induced from $T$, acting by automorphisms of $G$. (We write $\Lambda _{\mathrm {ad}}$ for the sheaf of cocharacters of $T$.) The $A$-gerbes $p^*\nu _{\vartheta }(\lambda )$ and $p^*\nu _{\vartheta }(\lambda _1)$ are $T_{\mathrm {ad}}$-equivariant, as they are pulled back from $X$.

Using the isomorphism (3.5) between rigidified $A$-torsors over $T_{\mathrm {ad}}$ and characters $\Lambda _{\mathrm {ad}} \rightarrow \underline A(-1)$, we may describe the obstruction of (3.33) to be $T_{\mathrm {ad}}$-equivariant as a locally constant section:

(3.35)\begin{equation} \tau^{\lambda, \lambda_1} \in \underline{\operatorname{Hom}}(\Lambda_{\mathrm{ad}}, \underline A(-1))\text{ over }\mathrm{Gr}_G^{\lambda}\cap S^{\lambda_1}. \end{equation}

3.5.8

Note that the equality (1.23) shows that the restriction of $b$ to $\Lambda \otimes \Lambda _{\mathrm {sc}}$, where $\Lambda _{\mathrm {sc}}\subset \Lambda$ is the span of coroots, extends to a bilinear form:

(3.36)\begin{equation} \tilde b : \Lambda_{\mathrm{ad}}\otimes \Lambda_{\mathrm{sc}} \rightarrow \underline A(-1),\quad (\lambda\in\Lambda_{\mathrm{ad}}, \alpha\in\Phi)\mapsto Q(\alpha)\langle\lambda, \check{\alpha}\rangle. \end{equation}

Lemma 3.5.9 Suppose that $\lambda \in \Lambda ^{\sharp, +}$, $\lambda _1\in \Lambda ^{\sharp }$ are such that $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}\neq \varnothing$. (This implies $\lambda - \lambda _1 \in \Lambda _{\mathrm {sc}}$.) Then the obstruction (3.35) of the isomorphism (3.33) to be $T_{\mathrm {ad}}$-equivariant is equal to the constant character $\tilde b(-, \lambda _1 - \lambda )$.

Proof. The calculation can be performed over $\bar {\textsf k}$-points of $X$. From now on, we fix a $\bar {\textsf k}$-point $\bar x$ of $X$ with local uniformizer $\varpi$. The notation $\mathrm {Gr}_G^{\lambda }$ and $S^{\lambda _1}$ now denotes their base changes to $\bar x$. The closed immersion $X^{\lambda }\rightarrow \mathrm {Gr}_G^{\lambda }$ corresponds to the $\bar {\textsf k}$-point $\bar x^{\lambda }$ of $\mathrm {Gr}_G^{\lambda }$. It lifts to a $\bar {\textsf k}$-point $\varpi ^{\lambda }$ of $L(G)$.

Let $B^- \subset G$ denote the Borel subgroup opposite to $B$. It has associated $L(N^-)$-orbit $S^{-, \lambda }$ of $\bar x^{\lambda }$ for each $\lambda \in \Lambda$.

Recall that each irreducible component of $\mathrm {Gr}_G^{\lambda } \cap S^{\lambda _1}$ intersects nontrivially with the ‘Zastava space’ $S^{-, w_0(\lambda )}\cap S^{\lambda _1}$, where $w_0\in W$ stands for the longest element. Indeed, the fact that $\mathrm {Gr}_G^{\lambda } \cap S^{-, w_0(\lambda )}$ is dense in $\mathrm {Gr}_G^{\lambda }$ shows that any irreducible component of $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}$ is contained in $\overline {S^{-, w_0(\lambda )}}\cap S^{\lambda _1}$. If it belonged to the complement of $S^{-, w_0(\lambda )}\cap S^{\lambda _1}$, it would be of dimension strictly less than $\langle \lambda + \lambda _1, \check {\rho }\rangle$ by the dimension calculation of Zastava spaces [BFGM02, 5.10], but $\mathrm {Gr}_G^{\lambda } \cap S^{\lambda _1}$ is pure of dimension $\langle \lambda +\lambda _1, \check {\rho }\rangle$ (see [MV07, Theorem 3.2]).

Let us consider the analogue of (3.32) for the Zastava space:

(3.37)

and the induced isomorphism of $A$-gerbes over $S^{-, \lambda }\cap S^{\lambda _1}$:

(3.38)\begin{equation} p^*\nu_{\vartheta}(\lambda) \cong p^*\nu_{\vartheta}(\lambda_1). \end{equation}

It suffices to prove: the obstruction of (3.38) to be $T_{\mathrm {ad}}$-equivariant is equal to $\tilde b(-, \lambda _1 - \lambda )$. Indeed, the desired statement over $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}$ will then follow from the equality $\tilde b(-, \lambda _1 - \lambda ) = \tilde b(-, \lambda _1 - w_0(\lambda ))$, as $w_0(\lambda ) - \lambda \in \Lambda ^{\sharp, r}$ is annihilated by $Q$.

To calculate the obstruction of (3.38) to be $T_{\mathrm {ad}}$-equivariant, we may assume $\lambda _1 = 0$. Indeed, because $\mathscr G$ is induced from an $\mathbb {E}_1$-monoidal morphism $L(G) \rightarrow B^2(\underline A)$, pulling back (3.38) along the isomorphism defined by multiplication by $\varpi ^{\lambda _1}$:

\begin{align*} \varpi^{\lambda_1} : S^{-, \lambda - \lambda_1} \cap S^0 \cong S^{-, \lambda} \cap S^{\lambda_1} \end{align*}

yields the product of the isomorphism $p^*\nu _{\vartheta }(\lambda - \lambda _1) \cong p^*\nu _{\vartheta }(0)$ with the identity automorphism of $p^*\nu _{\vartheta }(\lambda _1)$.

The assumption $\lambda _1 = 0$ forces $\lambda \in \Lambda _{\mathrm {sc}}$, so $S^{-, \lambda }\cap S^0$ is contained in the neutral component of $\mathrm {Gr}_G$.

Let $G_{\mathrm {sc}}\rightarrow G$ denote the simply connected form of $G$, with induced maximal torus $T_{\mathrm {sc}}\subset G_{\mathrm {sc}}$. Write $S_{\mathrm {sc}}^{-, \lambda }$, $S_{\mathrm {sc}}^0$ for the corresponding orbits in $\mathrm {Gr}_{G_{\mathrm {sc}}}$ and $p_{\mathrm {sc}} : S_{\mathrm {sc}}^{-, \lambda }\cap S_{\mathrm {sc}}^0 \rightarrow \bar x$ for the projection. The pullback of (3.38) to $S_{\mathrm {sc}}^{-, \lambda }\cap S_{\mathrm {sc}}^0$ is the composition of isomorphisms:

(3.39)\begin{equation} (p_{\mathrm{sc}})^*\nu_{\vartheta}(\lambda) \cong \mathscr G_{\mathrm{sc}} \cong (p_{\mathrm{sc}})^*\nu_{\vartheta}(0), \end{equation}

where $\mathscr G_{\mathrm {sc}}$ denotes (the restriction of) the $A$-gerbe over $\mathrm {Gr}_{G_{\mathrm {sc}}}$ defined by the pullback $\mu _{\mathrm {sc}}$ of $\mu$ along $B(G_{\mathrm {sc}})\rightarrow B(G)$. The two isomorphisms in (3.39) are induced from the inclusion of $S_{\mathrm {sc}}^{-, \lambda }\cap S_{\mathrm {sc}}^0$ in $S_{\mathrm {sc}}^{-, \lambda }$ and $S_{\mathrm {sc}}^0$, respectively.

Since the $\bar {\textsf k}$-points of $\mathrm {Gr}_{G_{\mathrm {sc}}}$ map bijectively to those of the neutral component of $\mathrm {Gr}_G$, it suffices to calculate the obstruction of (3.39) to be $T_{\mathrm {ad}}$-equivariant. (Note that $T_{\mathrm {ad}}$ acts by automorphisms of $G_{\mathrm {sc}}$, as the latter is functorially attached to $G$.)

We shall now appeal to the canonical $T_{\mathrm {ad}}$-equivariance structure of $\mathscr G_{\mathrm {sc}}$. Indeed, the rigidified morphism $\mu _{\mathrm {sc}} : B(G_{\mathrm {sc}}) \rightarrow B^4\underline A(1)$ is $T_{\mathrm {ad}}$-equivariant, and the ‘($T_{\mathrm {ad}}$, $T_{\mathrm {sc}}$)-commutator’ of the induced $\mathbb {E}_1$-monoidal morphism $T_{\mathrm {sc}}\rightarrow B^3\underline A(1)$ is the bi-rigidified morphism:

\begin{align*} T_{\mathrm{ad}}\times T_{\mathrm{sc}} \rightarrow B^2\underline A(1), \end{align*}

defined by the pairing $\tilde b$ (see [Zha22, § 5.5]).

As in the proof of Lemma 3.1.6 (task (2)), the $A$-gerbe $\mathscr G_{\mathrm {sc}} \otimes (p_{\mathrm {sc}})^*\nu _{\vartheta }(\lambda )^{\otimes -1}$ over $S_{\mathrm {sc}}^{-, \lambda }$ descends to the rigidified $A$-gerbe $\Psi ^{\tilde b(-, \lambda )}$ over $B_{\bar x^{\lambda }}(T_{\mathrm {ad}})$. Similarly, the $A$-gerbe $\mathscr G_{\mathrm {sc}}\otimes (p_{\mathrm {sc}})^*\nu _{\vartheta }(0)^{\otimes -1}$ over $S_{\mathrm {sc}}^0$ descends to the (trivial) rigidified $A$-gerbe $\Psi ^{\tilde b(-, 0)}$ over $B_{\bar x^0}(T_{\mathrm {ad}})$. The obstruction of (3.39) to be $T_{\mathrm {ad}}$-equivariant is thus the difference of obstructions:

\begin{align*} -\tilde b(-, 0) + \tilde b(-, \lambda) = \tilde b(-, \lambda). \end{align*}

This establishes the desired equality.

Remark 3.5.10 The proof of Lemma 3.5.9 also establishes its variant where $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}$ is replaced by the Zastava space $S^{-, \lambda }\cap S^{\lambda _1}$.

Proof of Proposition 3.5.4(2) The $\lambda$-summand of $\mathrm {Sat}_{G, \mathscr G}$ consists of intermediate extensions of $\mathscr G$-twisted $E$-local systems along $\mathrm {Gr}_G^{\lambda } \subset \mathrm {Gr}_G^{\le \lambda }$, so their restrictions to any boundary stratum $\mathrm {Gr}_G^{\lambda _2}$ lie in perverse cohomological degrees $\le -\langle 2\check {\rho }, \lambda _2\rangle - 1$. The $t$-exactness of $\mathrm {CT}^{\lambda, \lambda _1}(\check {\rho })[2\check {\rho }]$ implies that only the open stratum $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}$ contributes. In other words, it is isomorphic to the degree-$0$ piece of the functor:

(3.40)\begin{equation} p_!p^*(\langle\check{\rho}, \lambda + \lambda_1\rangle)[\langle 2\check{\rho}, \lambda + \lambda_1\rangle] : \mathrm{Lis}_{\nu_{\vartheta}(\lambda)}(X) \rightarrow \mathrm{Shv}_{\nu_{\vartheta}(\lambda_1)}(X), \end{equation}

defined by the isomorphism (3.33) of $A$-gerbes.

We shall prove that (3.40) vanishes. This statement can be verified over $\bar {\textsf k}$-points, so we fix a $\bar {\textsf k}$-point $\bar x$ of $X$ and trivialize the $A$-gerbes $\nu _{\vartheta }(\lambda )_{\bar x}$ and $\nu _{\vartheta }(\lambda _1)_{\bar x}$ over $\bar x$.

The isomorphism (3.33) thus defines an $A$-torsor $\tau$ over $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1}$, and the image of $\underline E$ under (3.40) is isomorphic, up to cohomological shift and Tate twist, to the complex

(3.41)\begin{equation} \Gamma_c(\mathrm{Gr}_G^{\lambda}\cap S^{\lambda_1}, \mathscr L), \end{equation}

where $\mathscr L$ is the rank-$1$ $E$-local system induced from $\tau$ along $A\subset E^{\times }$.

If $\lambda _1 - \lambda \notin \Lambda _{\mathrm {sc}}$, then $\mathrm {Gr}_G^{\lambda }\cap S^{\lambda _1} = \varnothing$ and (3.41) clearly vanishes.

Suppose that $\lambda _1 - \lambda \in \Lambda _{\mathrm {sc}}$. We write $\lambda _1 - \lambda = \sum _{\alpha } d_{\alpha }\alpha$ for $\alpha \in \Delta$ and $d_{\alpha }\in \mathbb {Z}$. The hypothesis $\lambda _1 - \lambda \notin \Lambda ^{\sharp, r}$ means that $d_{\alpha }$ is indivisible by $\mathrm {ord}(Q(\alpha ))$ for some $\alpha \in \Delta$.

By Lemma 3.5.9, the $A$-torsor $\tau$ is $T_{\mathrm {ad}}$-equivariant against the multiplicative $A$-torsor $\Psi ^{\tilde b(-, \lambda _1 - \lambda )}$. Using Lemma 1.3.5, we see that (3.41) vanishes as long as $H^0(T_{\mathrm {ad}}, \Psi ^{\tilde b(-, \lambda _1 - \lambda )}) = 0$. The latter vanishing follows from the hypothesis $\lambda _1 - \lambda \notin \Lambda ^{\sharp, r}$, as the pullback of $\Psi ^{\tilde b(-, \lambda _1 - \lambda )}$ along the fundamental coweight $\delta _{\alpha } : \mathbb {G}_m \rightarrow T_{\mathrm {ad}}$ dual to $\check {\alpha }\in \check {\Delta }$ yields $\Psi ^{d_{\alpha }Q(\alpha )}$.

3.6 Constant terms: fiber functor

3.6.1

We return to the context of § 3.4.1. The goal of this subsection is to use the constant term functor associated to $B$ to construct a ‘fiber functor’ for $\mathrm {Sat}_{G, \mathscr G^I}^I$.

Our first task is to record a corollary of Proposition 3.5.4(1), which concerns those properties reflected by the constant term functor.

Lemma 3.6.2 For any nonempty finite set $I$, the functor $\mathrm {CT}{}^I_P(\check {\rho }_P)[2\check {\rho }_P]$ (3.27) satisfies the following properties:

1. (1) it is conservative;

2. (2) an object of $\mathrm {Shv}_{\mathscr G^I}(\mathrm {Hec}_G^I)$ is universally locally acyclic relative to $X^I$ if and only if its image is.

Proof. Since the functors (3.27) are compatible with compositions (Remark 3.4.5), we may assume $P = B$.

Statement (1) is reduced to its analogue over $\bar {\textsf k}$-points of $X^I$, thus to the case $I = \{1\}$ after possibly replacing $G$ by a product of copies of $G$. Then it follows from the decomposition (3.24) and the special case of Proposition 3.5.4(1) for $\lambda _1 = \lambda \in \Lambda ^{\sharp, +}$.

Statement (2) follows from the criterion [HS23, Theorem 4.4(iv)] of universal local acyclicity and the conservativity of $\mathrm {CT}_B^I(\check {\rho })[2\check {\rho }]$.

3.6.3

Our second task is to construct a decomposition of $E$-linear abelian categories:

(3.42)\begin{equation} {}^+\mathrm{Sat}{}_{G, \mathscr G^I}^I \cong \bigoplus_{\lambda^I \in (\hat Z_H)^I} {}^+\mathrm{Sat}{}^{I, \lambda^I}_{G, \mathscr G^I}, \end{equation}

which is compatible with the tensor structure on ${}^+\mathrm {Sat}{}_{G, \mathscr G^I}^I$, i.e.:

1. (1) the unit $e_!(\underline E)$ belongs to ${}^+\mathrm {Sat}{}_{G, \mathscr G^I}^{I, 0}$;

2. (2) the monoidal product of $\mathscr F_1 \in {}^+\mathrm {Sat}{}_{G, \mathscr G^I}^{I, \lambda _1^I}$ and $\mathscr F_2\in {}^+\mathrm {Sat}{}^{I, \lambda _2^I}_{G, \mathscr G^I}$ belongs to ${}^+\mathrm {Sat}{}^{I, \lambda _1^I + \lambda _2^I}_{G, \mathscr G^I}$.

Furthermore, the decomposition (3.42) is of étale local nature over $X^I$.

Proof Construction of (3.42)

Let $\pi : \Lambda ^{\sharp } \twoheadrightarrow \hat Z_H$ denote the projection map. (Recall that $\hat Z_H$ is the quotient of $\Lambda ^{\sharp }$ by $\Lambda ^{\sharp, r}$.) We proceed in increasing generality.

Case: $I = \{1\}$. For each $\lambda \in \hat Z_H$, we define the direct summand:

(3.43)\begin{equation} {}^+\mathrm{Sat}{}^{\{1\}, \lambda}_{G, \mathscr G^{\{1\}}} := \bigoplus_{\substack{\lambda^+ \in \Lambda^{\sharp, +} \\ \pi(\lambda^+) = \lambda}} \mathrm{Lis}_{\nu_{\vartheta}(\lambda^+)}(X) \subset {}^+\mathrm{Sat}{}^{\{1\}}_{G, \mathscr G^{\{1\}}}, \end{equation}

according to the decomposition (3.24).

It is clear that $e_!(\underline E)$ belongs to ${}^+\mathrm {Sat}{}_{G, \mathscr G^{\{1\}}}^{\{1\}, 0}$.

To prove the compatibility with monoidal product, we note that by Proposition 3.5.4, the summand (3.43) consists precisely of objects in ${}^+\mathrm {Sat}{}^{\{1\}}_{G, \mathscr G^{\{1\}}}$ whose images under $\mathrm {CT}{}^{\{1\}}_B(\check {\rho })[2\check {\rho }]$ are supported on the strata $X^{\lambda _1}$, for $\lambda _1\in \Lambda ^{\sharp }$ with $\pi (\lambda _1) = \lambda$. However, for the torus $T$, the decomposition (3.24) is compatible with the monoidal product. Thus, the same holds for the decomposition of ${}^+\mathrm {Sat}{}^{\{1\}}_{G, \mathscr G^{\{1\}}}$ defined by the summands (3.43).

Case: disjoint locus. For a nonempty finite set $I$, we consider the identity map $p = \mathrm {id}_I$. The open subscheme $X^p\subset X^I$ is the pairwise disjoint locus.

For each $\lambda ^I \in (\hat Z_H)^I$, we set

(3.44)\begin{equation} {}^+\mathrm{Sat}^{p, \lambda^I}_{G, \mathscr G^I} \subset {}^+\mathrm{Sat}{}^p_{G, \mathscr G^I} \end{equation}

to be the full subcategory consisting of objects whose images under $\mathrm {CT}{}_B^I(\check {\rho })[2\check {\rho }]$ are supported on the strata $X^{\lambda _1^I}$, for $\lambda _1^I \in (\Lambda ^{\sharp })^I$ with $\pi (\lambda _1^I) = \lambda ^I$.

The fact that the full subcategories (3.44) induce a direct sum decomposition of ${}^+\mathrm {Sat}{}^p_{G, \mathscr G^I}$ compatible with its tensor structure follows from the case for $I = \{1\}$.

Case: general. Let $I$, $p$, $\lambda ^I$ be as above. We set

(3.45)\begin{equation} {}^+\mathrm{Sat}{}^{I, \lambda^I}_{G, \mathscr G^I} \subset {}^+\mathrm{Sat}{}^I_{G, \mathscr G^I} \end{equation}

to be the full subcategory consisting of objects whose restrictions along $X^p\subset X^I$ belong to the full subcategory (3.44).

The fact that (3.45) induces a direct sum decomposition of ${}^+\mathrm {Sat}{}^I_{G, \mathscr G^I}$ follows from the closedness of the full subcategory ${}^+\mathrm {Sat}{}^I_{G, \mathscr G^I} \subset {}^+\mathrm {Sat}{}^p_{G, \mathscr G^I}$ under direct summands. Its compatibility with tensor structure follows from the case for the disjoint locus.

Remark 3.6.4 For $I = \{1\}$, the decomposition (3.42) coarsens the decomposition (3.24). However, the latter decomposition is incompatible with the tensor structure unless $G = T$ (cf. Remark 3.3.8).

3.6.5

Using the decomposition (3.42), being of étale local nature over $X^I$, we may twist the tensor category ${}^+\mathrm {Sat}{}_{G, \mathscr G^I}^I$ by the $\mathbb {E}_{\infty }$-monoidal morphism negative to $\nu _{\vartheta }^I$ (i.e. the formation (1.38) applied to $\nu _{\vartheta }^{\otimes -1}$):

\begin{align*} (\nu_{\vartheta}^I)^{\otimes -1} : (\hat Z_H)^I \rightarrow B^2_{X^I}(A),\quad (\lambda_i)_{i\in I}\mapsto \boxtimes_{i\in I}\nu_{\vartheta}(\lambda_i)^{\otimes -1}. \end{align*}

This process yields a tensor category $({}^+\mathrm {Sat}{}^I_{G, \mathscr G^I})_{(\nu _{\vartheta }^I)^{\otimes -1}}$.

Combining $\mathrm {CT}{}^I_B(\check {\rho })[2\check {\rho }]$ with the geometric Satake equivalence for split tori (§ 3.2.5), we find a tensor functor:

(3.46)\begin{align} \omega^I : ({}^+\mathrm{Sat}{}^I_{G, \mathscr G^I})_{(\nu_{\vartheta}^I)^{\otimes -1}} &\xrightarrow{\mathrm{CT}_B^I(\check{\rho})[2\check{\rho}]}(\mathrm{Sat}_{T, \mathscr G^I_T}^I)_{(\nu_{\vartheta}^I)^{\otimes -1}}\nonumber\\ &\cong (\mathrm{Rep}_{T_H^I, \nu_{\vartheta}^I}^I)_{(\nu_{\vartheta}^I)^{\otimes -1}} \cong \mathrm{Rep}_{T_H^I}^I \rightarrow \mathrm{Lis}(X^I), \end{align}

where $T_H \subset H$ is the maximal torus (with character lattice $\Lambda ^{\sharp }$), and the last functor is that forgetting the $T_H^I$-action.

Remark 3.6.6 The tensor functor $\omega ^I$ satisfies the following additional properties:

1. (1) it is conservative (Lemma 3.6.2(1));

2. (2) it is exact (Proposition 3.4.4);

3. (3) it is independent of the choice of $B$ (Remark 3.4.6);

3.7 Tannakian reconstruction

3.7.1

Consider the $2$-category whose objects are tensor (i.e. $E$-linear symmetric monoidal abelian) categories, whose morphisms are tensor functors, and whose $2$-morphisms are natural transformations compatible with the tensor structures.

A tensor category is called rigid if all of its objects are dualizable. Note that in a rigid tensor category, the monoidal product with any object is exact.

3.7.2

Let $\mathbf {A}$ be a rigid tensor category. The following categories are related by a pair of adjoint functors:

1. (1) commutative Hopf algebras in $\mathrm {Ind}(\mathbf {A})$;

2. (2) pairs $(\mathbf {C}, \omega )$ where $\mathbf {C}$ is a rigid tensor category under $\mathbf {A}$ and $\omega : \mathbf {C} \rightarrow \mathbf {A}$ is a conservative, exact, $\mathbf {A}$-linear tensor functor.

The functor (1) $\Rightarrow$ (2) associates to a Hopf algebra $\mathscr A\in \mathrm {Ind}(\mathbf {A})$ its tensor category of comodules $\mathbf {C} := \mathrm {Comod}_{\mathscr A}(\mathbf {A})$ on objects of $\mathbf {A}$ and the forgetful functor $\omega$.

Its left adjoint (2) $\Rightarrow$ (1) associates to a pair $(\mathbf {C}, \omega )$ the object $\mathrm {Ind}(\omega )\cdot \omega ^R(1) \in \mathrm {Ind}(\mathbf {A})$, where $1\in \mathbf {A}$ is the tensor unit. Here, the ind-extension $\mathrm {Ind}(\omega )$ preserves all colimits, and thus admits a right adjoint $\omega ^R$. Since $\mathrm {Ind}(\omega )$ is a tensor functor, $\omega ^R$ preserves commutative algebras. The coalgebra structure is induced from the comonad $\mathrm {Ind}(\omega )\cdot \omega ^R$. The existence of inverse follows from the rigidity of $\mathbf {C}$.

Lemma 3.7.3 The adjunction in § 3.7.2 is an equivalence.

Proof. We show that the counit and unit are isomorphisms.

Counit. Let $\mathscr A \in \mathbf {A}$ be a bi-algebra. The right adjoint $\omega ^R$ is given by $(-)\otimes \mathscr A$, where $\mathscr A$ is viewed as an $\mathscr A$-comodule by its coalgebra structure. The counit map $\mathrm {Ind}(\omega )\cdot \omega ^R(1)\rightarrow \mathscr A$ is thus an isomorphism.

Unit. Let $(\mathbf {C}, \omega )$ be a pair as in § 3.7.2(2). The functor $\mathrm {Ind}(\omega )$ remains conservative, so the adjunction $(\mathrm {Ind}(\omega ), \omega ^R)$ satisfies the hypothesis of the Barr–Beck theorem, i.e. it is comonadic. We see that $\mathrm {Ind}(\mathbf {C})$ is equivalent to the category of $(\mathrm {Ind}(\omega )\cdot \omega ^R)$-comodules in $\mathrm {Ind}(\mathbf {A})$, compatibly with the forgetful functors.

Since $\mathrm {Ind}(\omega )$ is $\mathrm {Ind}(\mathbf {A})$-linear, so is the functor $\omega ^R$. This shows that the comonad $\mathrm {Ind}(\omega )\cdot \omega ^R$ is identified with tensoring by the coalgebra $\mathscr A := \mathrm {Ind}(\omega )\cdot \omega ^R(1)$.

Finally, we must show that $\mathbf {C}\subset \mathrm {Ind}(\mathbf {C})$ coincides with the full subcategory of objects whose image under $\mathrm {Ind}(\omega )$ belongs to $\mathbf {A}$. Since $\mathbf {C}$ (respectively, $\mathbf {A}$) is abelian, hence idempotent-complete, it is the full subcategory of compact objects of $\mathrm {Ind}(\mathbf {C})$ (respectively, $\mathrm {Ind}(\mathbf {A})$). We conclude using the observation that $\mathrm {Ind}(\omega )$ respects and reflects compactness.

3.7.4

For a nonempty finite set $I$, the tensor category ${}^+\mathrm {Sat}{}^I_{G, \mathscr G^I}$ admits a $\mathrm {Lis}(X^I)$-structure supplied by $e_!$, for $e : X^I \rightarrow \mathrm {Hec}_G^I$ being the unit section.

Moreover, ${}^+\mathrm {Sat}{}^I_{G, \mathscr G^I}$ is rigid. Indeed, by the definition of the fusion product, we can check the dualizability of an object in ${}^+\mathrm {Sat}{}^I_{G, \mathscr G^I}$ over the pairwise disjoint locus of $X^I$. There, the statement reduces to $\bar {\textsf k}$-points and the case $I = \{1\}$, where it follows from the argument of [Zhu17, Theorem 5.2.9].

The same assertions are inherited by the twisted tensor category $({}^+\mathrm {Sat}{}^I_{G, \mathscr G^I})_{(\nu _{\vartheta }^I)^{\otimes -1}}$, so Lemma 3.7.3 produces a commutative Hopf algebra $\mathscr A^I \in \mathrm {Ind}(\mathrm {Lis}(X^I))$.

Proposition 3.7.5 There is a canonical isomorphism of Hopf algebras:

(3.47)\begin{align} \mathscr A^I \cong \mathscr O_{H^I}\in \mathrm{Ind}(\mathrm{Lis}(X^I)). \end{align}

3.7.6

Let $H_1$ denote the locally constant étale sheaf over $X$ of affine group schemes over $E$ corresponding to $A^{\{1\}}$.

The isomorphism (3.47) for $G = T$ is constructed in § 3.2.5. The composition of all but the last functor in the definition of $\omega ^{\{1\}}$ (3.46) yields a homomorphism from the maximal torus $T_H\subset H$ to $H_1$.

The isomorphism $H_1 \cong H$ supplied by (3.47) will extend the identity map on $T_H$.

Lemma 3.7.7 The following statements hold:

1. (1) $H_1$ is a sheaf of reductive group schemes;

2. (2) the map $T_H \rightarrow H_1$ realizes $T_H$ as a maximal torus of $H_1$.

Proof. Both statements may be verified over $\bar {\textsf k}$-points of $X$. We shall now fix a $\bar {\textsf k}$-point $\bar x$ and write $H_{1, \bar x}$ (respectively, $T_{H, \bar x}$) for the stalk of $H_1$ (respectively, $T_H$) at $\bar x$.

By Proposition 3.3.7, the underlying $E$-linear abelian category of finite-dimensional representations of $H_{1, \bar x}$ decomposes as a sum of copies of the category of finite-dimensional $E$-vector spaces, indexed by $\Lambda ^{\sharp, +}$. We write $\mathscr E^{\lambda } \in \mathrm {Rep}_{H_1, \bar x}$ for the object corresponding to the $1$-dimensional $E$-vector space $E$ and index $\lambda \in \Lambda ^{\sharp, +}$.

Observe that $H_{1, \bar x}$ is of finite type. Indeed, $\Lambda ^{\sharp, +}$ is finitely generated as a monoid and $\mathscr E^{\lambda _1 + \lambda _2}$ appears as a summand of $\mathscr E^{\lambda _1}\otimes \mathscr E^{\lambda _2}$ by Proposition 3.5.4(1). Thus, $\mathrm {Rep}_{H_1, \bar x}$ has a finite number of tensor generators, so [DM82, Proposition 2.20] applies.

Next, $H_{1, \bar x}$ is connected because it does not have nontrivial finite tensor subcategories [DM82, Corollary 2.22].

Finally, $H_{1, \bar x}$ is reductive because $\mathrm {Rep}_{H_1, \bar x}$ is semisimple [DM82, Proposition 2.23] according to its aforementioned decomposition. This proves statement (1).

By Proposition 3.5.4(1), any object of $\mathrm {Rep}_{T_H, \bar x}$ is a subobject of an object coming from $\mathrm {Rep}_{H_1, \bar x}$. The morphism $T_{H, \bar x}\rightarrow H_{1, \bar x}$ is thus a closed immersion by [DM82, Proposition 2.21(b)]. The argument of [Zhu17, Lemma 5.3.17], substituting Proposition 3.5.4 for [Zhu17, Theorem 5.3.9], shows that $T_{H, \bar x}$ is a maximal torus.

3.7.8

Let us fix a $\bar {\textsf k}$-point $\bar x$ of $X$. We shall upgrade the maximal torus $T_{H, \bar x} \subset H_{1, \bar x}$ to a pinning of the reductive group scheme $H_{1, \bar x}$.

Denote by $2\rho _H \in \check {\Lambda }^{\sharp }$ the sum of positive coroots of $H_{\bar x}$. As in [BR18, § 9.2], we choose a Borel subgroup $B_{H_1, \bar x} \subset H_{1, \bar x}$ containing $T_{H, \bar x}$ such that $2\rho _H$ is a dominant cocharacter with respect to $B_{H_1, \bar x}$. This choice has the property that the dominant characters of $T_{H, \bar x}$ it defines agree with those defined by $B_{H, \bar x} \subset H_{\bar x}$ (see [BR18, Lemma 9.5]).

We record the data which have been constructed:

(3.48)

Lemma 3.7.9 The two rows of (3.48) induce the same based root data on the character lattice of $T_{H, \bar x}$.

Proof. The construction of $B_{H_1, \bar x}$ being compatible with the constant term functor (3.27), we reduce to the case where $G$ is of semisimple rank one. There, we must show that $B_{H_1, \bar x}\subset H_{1, \bar x}$ has a unique simple root given by $\mathrm {ord}(Q(\alpha ))\alpha \in \Lambda ^{\sharp }$, and its associated coroot is given by $\mathrm {ord}(Q(\alpha ))^{-1}\check {\alpha }$.

To identify the simple roots, it suffices to show that the $T_{H,\bar x}$-weights of the simple object $\mathscr E^{\lambda } \in \mathrm {Rep}_{H_1, \bar x}$ for $\lambda \in \Lambda ^{\sharp, +}$ (notation as in the proof of Lemma 3.7.7) is given by

(3.49)\begin{equation} \lambda - d\cdot \mathrm{ord}(Q(\alpha))\alpha, \quad 0 \le d\le \mathrm{ord}(Q(\alpha))^{-1}\langle\check{\alpha}, \lambda\rangle. \end{equation}

We argue as in [FL10, § 4.4]. Proposition 3.5.4(2) implies that the $T_{H, \bar x}$-weights of $\mathscr E^{\lambda }$ are contained in the set (3.49). To show that they exhaust the latter, it suffices to prove that the $A$-torsor $\tau ^{\lambda, \lambda _1}$ over $\mathrm {Gr}_{G, \bar x}^{\lambda }\cap S_{\bar x}^{\lambda _1}$ defined by (3.33) and arbitrary trivializations of $\nu _{\vartheta }(\lambda )_{\bar x}$, $\nu _{\vartheta }(\lambda _1)_{\bar x}$ is (non-canonically) trivial whenever $\lambda _1$ belongs to (3.49).

The statement for the extremal cases $\lambda _1 = \lambda$, $\lambda _1 = s_{\alpha }(\lambda )$ follows from Proposition 3.5.4(1). It remains to treat the intermediate cases:

\begin{align*} \lambda_1 = \lambda - d\cdot\mathrm{ord}(Q(\alpha))\alpha,\quad 1\le d \le \mathrm{ord}(Q(\alpha))^{-1}\langle\check{\alpha}, \lambda\rangle - 1. \end{align*}

Write $a := \langle \check {\rho }, \lambda \rangle$ and $a_1 := \langle \check {\rho }, \lambda _1\rangle$ where $\check {\rho } := \check {\alpha }/2$, so $a > 0$ and $|a_1| < a$. Identifying $N$ with $\mathbb {G}_a$, the $L(N)_{\bar x}$-action on $\bar x^{\lambda _1}$ induces an isomorphism:

(3.50)\begin{equation} \mathbb{A}^{a + a_1} \cong \varpi^{-a + a_1} L_+(\mathbb{G}_a)_{\bar x}/\varpi^{2a_1}L_+(\mathbb{G}_a)_{\bar x} \cong \mathrm{Gr}_{G, \bar x}^{\le\lambda}\cap S_{\bar x}^{\lambda_1}, \end{equation}

Under (3.50), the open subscheme $\mathrm {Gr}_{G, \bar x}^{\lambda }\cap S_{\bar x}^{\lambda _1}$ corresponds to the locus with invertible leading coefficient in $\varpi ^{-a + a_1}L_+(\mathbb {G}_a)_{\bar x}$, hence to $\mathbb {G}_m\times \mathbb {A}^{a+a_1-1}\subset \mathbb {A}^{a + a_1}$. The projection of $\mathrm {Gr}_{G, \bar x}^{\lambda }\cap S^{\lambda _1}_{\bar x}$ onto $\mathbb {G}_m$ intertwines the $T_{\mathrm {ad}}$-action with $\mathbb {G}_m$-multiplication.

The $A$-torsor $\tau ^{\lambda, \lambda _1}$ canonically descends to $\mathbb {G}_m$, where it is (non-canonically) trivial because $\tau ^{\lambda, \lambda _1}$ is $T_{\mathrm {ad}}$-equivariant by Lemma 3.5.9.

The identification of simple roots of $B_{H_1, \bar x}\subset H_{1, \bar x}$ being complete, the identification of simple coroots follows, because $\mathrm {ord}(Q(\alpha ))^{-1}\check {\alpha }$ is the unique cocharacter pairing non-negatively with all dominant characters of $T_{H, \bar x}$ and yields $2$ when paired with $\mathrm {ord}(Q(\alpha ))\alpha$.

3.7.10

Let us now upgrade the top row of (3.48) to a pinning on $H_{1, \bar x}$.

Proof Construction

The decomposition (3.42) gives rise to two tensor functors:

(3.51)\begin{equation} \mathrm{Rep}_{H_1, \bar x}^0 \subset \mathrm{Rep}_{H_1, \bar x} \rightarrow \bigoplus_{\lambda\in\hat Z_H}\mathrm{Mod}_E, \end{equation}

where $\mathrm {Rep}_{H_1, \bar x}^0$ denotes the summand corresponding to $0 \in \hat Z_H$, and the second functor is the decomposition of $\omega ^{\{1\}}$ according to $\hat Z_H$-weights.

Both functors in (3.51) commute with fiber functors to $\mathrm {Mod}_E$. Thus, they define morphisms of affine groups schemes over $E$:

(3.52)\begin{equation} 1 \rightarrow Z_{H, \bar x} \rightarrow H_{1, \bar x} \rightarrow H^0_{1, \bar x} \rightarrow 1. \end{equation}

The criterion [EHS07, Theorem A.1(iii)] shows that (3.52) is a short exact sequence. The identification of root data (Lemma 3.7.9) shows that $H^0_{1, \bar x}$ is the adjoint group of $H_{1, \bar x}$. It is equipped with an induced maximal torus and a Borel subgroup:

(3.53)\begin{equation} T^0_{H_1, \bar x} \subset B^0_{H_1, \bar x} \subset H^0_{1, \bar x}. \end{equation}

Suppose first that $G$ is of semisimple rank one. Then the same holds for $H_{1, \bar x}$, and $H^0_{1, \bar x}$ has the root data of $\mathrm {PGL}_2$. Any two isomorphisms $H^0_{1, \bar x} \cong \mathrm {PGL}_2$ matching (3.53) with the standard triple $\mathbb {G}_m \subset B\subset \mathrm {PGL}_2$ differ by the inner automorphism of $\mathrm {PGL}_2$ defined by a unique element $a\in \mathbb {G}_m$. The adjoint action of $\mathrm {PGL}_2$ on the vector space $E^{\oplus 3}$ differs from its twist by any nontrivial element $a\in \mathbb {G}_m$.

Consider the simple object $\mathscr E^{\mathrm {ord}(Q(\alpha ))\alpha } \in \mathrm {Rep}_{H_1, \bar x}^0$. Its image under $\omega ^{\{1\}}$ is canonically equivalent to $E^{\oplus 3}$, using the Tate twist introduced in our constant term functor. Thus, there is a unique isomorphism between the triple (3.53) and the standard triple $\mathbb {G}_m\subset B\subset \mathrm {PGL}_2$ under which $\mathscr E^{\mathrm {ord}(Q(\alpha ))\alpha }$ corresponds to the adjoint action of $\mathrm {PGL}_2$ on $E^{\oplus 3}$.

Under this isomorphism, the pinning of $\mathrm {PGL}_2$ transfers to $H^0_{1, \bar x}$. Since $H_{1, \bar x}\rightarrow H^0_{1, \bar x}$ is an isomorphism on root subgroups, $H_{1, \bar x}$ inherits a pinning.

For a general reductive group scheme $G$, the pinning on $H_{1, \bar x}$ is constructed from the constant term functors and the semisimple rank-one case.

3.7.11

Combining Lemma 3.7.9 and the construction in § 3.7.10 of a pinning on $H_{1, \bar x}$, we obtain a canonical isomorphism of pinned reductive group schemes over $E$:

(3.54)\begin{equation} H_{1, \bar x} \cong H_{\bar x}, \end{equation}

for every $\bar {\textsf k}$-point $\bar x$ of $X$.

Finally, we shall lift (3.54) to the desired isomorphism (3.47).

Proof of Proposition 3.7.5 We construct the isomorphism (3.47) in increasing generality.

Case: $I = \{1\}$. Claim: $H_1$ is constant as an étale sheaf over $X$. Once this claim is established, the desired isomorphism $H_1\cong H$ is supplied by (3.54) at any $\bar {\textsf k}$-point $\bar x$.

To prove that $H_1$ is constant, we may assume that $X$ is connected with a fixed $\bar {\textsf k}$-point $\bar x$. Then $H_1$ is recovered from its fiber $H_{1, \bar x}$ equipped with the $\pi _1(X, \bar x)$-action. It remains to show that this $\pi _1(X, \bar x)$-action is trivial.

Since the tensor functor $\mathrm {Rep}_{H_1, \bar x} \rightarrow \mathrm {Rep}_{T_H, \bar x}$ arises as the fiber at $\bar x$ of a tensor functor whose target $\underline {\mathrm {Rep}}{}^{\{1\}}_{T_H}$ is constant, the $\pi _1(X, \bar x)$-action on the maximal torus $T_{H, \bar x}\subset H_{1, \bar x}$ is trivial. This shows that $\pi _1(X, \bar x)$ acts trivially on the based root data of $H_{1, \bar x}$.

It remains to show that the $\pi _1(X, \bar x)$-action preserves the pinning on $H_{1, \bar x}$ constructed in § 3.7.10. We may do so under the additional assumption that $G$ is of semisimple rank one, and it suffices to show that $\pi _1(X, \bar x)$ acts trivially on the adjoint group $H_{1, \bar x}^0$.

By functoriality of the intermediate extension, the simple object $\mathscr E^{\mathrm {ord}(Q(\alpha ))\alpha } \in \mathrm {Rep}_{H_1, \bar x}^0$ is $\pi _1(X, \bar x)$-equivariant compatibly with the fiber functor. Thus, the corresponding action of $H_{1, \bar x}^0$ on $E^{\oplus 3}$ is $\pi _1(X, \bar x)$-equivariant. However, the induced homomorphism $H^0_{1, \bar x} \rightarrow \mathrm {GL}(E^{\oplus 3})$ is injective, as it coincides with the adjoint representation of $\mathrm {PGL}_2$. It follows that $\pi _1(X, \bar x)$ acts trivially on $H^0_{1, \bar x}$.

Case: general. Let $I$ be any nonempty finite set. The external fusion product of $\mathrm {Sat}{}_{G,\mathscr G^I}^I$ induces a morphism of Hopf algebras:

(3.55)\begin{equation} \boxtimes_{i\in I} \mathscr A^{\{i\}} \rightarrow \mathscr A^I\in \mathrm{Ind}(\mathrm{Lis}(X^I)). \end{equation}

It suffices to prove that (3.55) is an isomorphism.

This assertion can be proved over the pairwise disjoint locus in $X^I$ and furthermore over any $\bar {\textsf k}$-point $\bar x$. There, it follows the compatibility between the isomorphism (3.54) and finite product of reductive groups.

3.7.12

We now prove the geometric Satake equivalence (Theorem 2.4.4) for $(G, \mu )$.

Proof Construction of (2.23)

Suppose first that $G$ is split.

For any nonempty finite set $I$, the tensor category $({}^+\mathrm {Sat}{}^I_{G, \mathscr G^I})_{(\nu _{\vartheta }^I)^{\otimes -1}}$ is identified with $\mathrm {Rep}{}^I_{H^I}$ by Proposition 3.7.5.

This identification being of étale local nature over $X^I$, we obtain the desired equivalence of tensor categories (2.23) after twisting both sides by $\nu _{\vartheta }^I$.

The nonsplit case follows via étale descent.

Remark 3.7.13 As an addendum to the proof of Theorem 2.4.4, we note that (2.23) is compatible with constant term functors, i.e. the following square commutes:

where the right vertical arrow is the restriction along the maximal torus $T_H^I\subset H^I$. (Recall that $\mathrm {CT}_B^I(\check {\rho })[2\check {\rho }]$ is independent of the choice of $B$ according to Remark 3.4.6.)

The compatibility statements in § 2.4.5 follow directly from the construction of the equivalence.

4. Global function fields

This section contains our results particular to smooth curves over a finite field.

In § 4.1, we propagate the $A$-gerbe $\mathscr G^I$ defined in § 2.2 to various moduli spaces associated to a global curve. The crucial observation is that $\mathscr G^I$ is canonically trivialized over the moduli stack of Shtukas, allowing us to obtain the space of genuine automorphic forms from its cohomology. This geometric origin of genuine automorphic forms is already indicated by V. Lafforgue in [Laf18, § 14] in a narrower context.

In § 4.2, we prove a lemma concerning Artin reciprocity. It implies that in the function field context, Weissman's meta-Galois group [Wei18, § 4] is the central extension associated to the $\{\pm 1\}$-gerbe of theta characteristics. This is a slightly surprising fact, but it follows from very natural considerations.

Finally, we explain in §§ 4.3 and 4.4 how to extend the arguments of V. Lafforgue [Laf18] (using improvements by Xue [Xue20a, Xue20b]) to obtain the spectral decomposition of genuine cusp forms defined on covering groups.

4.0.1

Let $\textsf k$ be a finite field of cardinality $q$. For any $\textsf k$-scheme $S$, we write $\mathrm {Fr}_S$ for the $q$th power Frobenius endomorphism of $S$.

Let $X$ be a smooth, proper, and geometrically connected curve over $\textsf k$. Denote by $F$ its field of fractions and $\mathbb {A}_F$ (respectively, $\mathbb {O}_F$) its ring of (respectively, integral) adèles.

The coefficient field $E$ is as in § 2.0.1. We assume that $q$ has a square root in $E$ which will be fixed: this corresponds to the choice of $\underline E( {1}/{2})$ used in the geometric Satake equivalence (see § 2.4).

Let $D \subset X$ be a $k$-finite closed subscheme and write $\mathring X:= X - D$ for its open complement.

Let $G\rightarrow X$ be a smooth affine group scheme with connected geometric fibers, equipped with an étale metaplectic cover $\mu$ defined over $\mathring X$.

4.1 The global $A$-gerbe

4.1.1

Let $\mathrm {Bun}_{G, D}$ denote the stack whose $S$-points, for any affine $\textsf k$-scheme $S$, consist of pairs $(P, \phi )$ where $P$ is a $G$-torsor over $S\times X$ and $\phi$ is a rigidification of $P$ along $S\times D$.

For each nonempty finite set $I$, let $\mathrm {Hec}_{G, D}^I$ denote the stack whose $S$-points consist of an $S$-point $x^I$ of $\mathring X^I$, pairs $(P_0, \phi _0)$, $(P_1, \phi _1)$ of $G$-torsors over $S\times X$ rigidified along $D$, and an isomorphism of them off the union of graphs $\Gamma _{x^I}\subset S\times X$.

We refer to such an isomorphisms as a ‘modification’ at $x^I$ and denote it by

\begin{align*} (P_0, \phi_0) \overset{x^I}{\sim} (P_1, \phi_1). \end{align*}

4.1.2

If $I$ is equipped with an ordered partition into nonempty finite sets $I \cong I_1\sqcup \cdots \sqcup I_k$, we write $\widetilde {\mathrm {Hec}}{}_{G, D}^{I_1,\ldots, I_k}$ for the stack whose $S$-points consist of an $S$-point $x^I$ of $\mathring X^I$ and modifications:

(4.1)\begin{equation} (P_0, \phi_0) \overset{x^{I_1}}{\sim} (P_1, \phi_1) \overset{x^{I_2}}{\sim} \cdots \overset{x^{I_k}}{\sim} (P_k, \phi_k), \end{equation}

where each $x^{I_a}$ denotes the corresponding $S$-point of $\mathring X^{I_a}$ (for $1\le a\le k$).

Restricting the data (4.1) to the formal disk $D_{x^I}$ defines a morphism from $\widetilde {\mathrm {Hec}}{}_{G, D}^{I_1, \ldots, I_k}$ to the local iterated Hecke stack $\widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_k}$ of § 2.1. For each $0\le a \le k$, remembering $(P_a, \phi _a)$ defines a morphism $p_a : \widetilde {\mathrm {Hec}}{}_{G, D}^{I_1,\ldots, I_k} \rightarrow \mathrm {Bun}_{G, D}$.

Finally, an $S$-point of the moduli stack of iterated Shtukas $\mathrm {Sht}{}^{I_1,\ldots, I_k}_{G, D}$ consists of an $S$-point (4.1) of $\widetilde {\mathrm {Hec}}{}_{G, D}^{I_1,\ldots, I_k}$ together with an isomorphism:

(4.2)\begin{equation} (P_k, \phi_k) \cong {}^{\tau}(P_0, \phi_0) := (\mathrm{Fr}_S \!\times \mathrm{id}_X)^*(P_0, \phi_0). \end{equation}

Some of the relevant morphisms are recorded in the following diagram, where the square is Cartesian by definition.

(4.3)

4.1.3

We shall functorially assign an étale $A$-gerbe $\mathscr G_D$ over $\mathrm {Bun}_{G, D}$ to $\mu$.

Proof Construction

The projection map $p : \mathrm {Bun}_{G, D} \times X\rightarrow \mathrm {Bun}_{G, D}$, being proper and smooth of relative dimension one, defines a morphism of complexes $p_*(\underline A(1)[4]) \rightarrow \underline A[2]$. Its global section over $\mathrm {Bun}_{G, D}$ yields the ‘transgression’ map:

\begin{align*} [X] : \Gamma(\mathrm{Bun}_{G, D}\times X, \underline A(1)[4]) \rightarrow \Gamma(\mathrm{Bun}_{G, D}, \underline A[2]). \end{align*}

Let us view the universal $G$-torsor as a morphism of $X$-stacks $P : \mathrm {Bun}_{G, D}\times X \rightarrow B_X(G)$ whose base change $P_D$ along $D\subset X$ is rigidified.

Consider the following commutative diagram:

(4.4)

where $\Gamma _e$ denotes the complex of rigidified sections, and the top row is the triangle induced from the Cousin triangles associated to $D\rightarrow X$ and $B_D(G)\rightarrow B_X(G)$.

The rigidification of $P_D$ induces a trivialization of the restriction of $[X] \cdot P^*$ in (4.4) to $\Gamma _e(B_D(G), \underline A[2])$. Hence, $[X]\cdot P^*$ factors through a morphism:

(4.5)\begin{equation} \Gamma_e(B_{\mathring X}G, \underline A(1)[4]) \rightarrow \Gamma(\mathrm{Bun}_{G, D}, \underline A[2]). \end{equation}

The desired functor $\mu \mapsto \mathscr G_{\mu }$ is obtained from (4.5) upon taking connective truncations and passing to the underlying $\infty$-groupoids.

Remark 4.1.4 If $D = \varnothing$, then the rigidification of $\mu$ along $e : \mathring X\rightarrow B_{\mathring X}(G)$ is not needed for the construction of $\mathscr G_D$.

Remark 4.1.5 Suppose that $G$ is split reductive.

Inspecting the top row in (4.4) and using the computation of étale cohomology of $B(G)$ in degrees $\le 3$ (see [Zha22, § 5.1]), we see that $\mu$ may not extend across $D$, and when $\mu$ extends across some point $x\in D$, the choice of possible extensions is not unique.

In particular, the étale metaplectic cover $\mu$ generally contains more data than its restriction to the generic point $\eta \in X$.

4.1.6

Denote by $K_D$ the kernel of the projection $G(\mathbb {O}_F) \rightarrow G(\mathscr O_D)$. The gluing maps yield an inclusion of groupoids:

(4.6)\begin{equation} G(F) \backslash G(\mathbb{A}_F)/K_D \subset \mathrm{Bun}_{G, D}(\textsf k), \end{equation}

whose essential image consists of pairs $(P, \phi )$ where $P$ is generically trivial. (This uses the vanishing of $H^1(\mathscr O_x, G)$ for a closed point $x\in X$, which follows from Lang's isogeny.)

The additional pieces of $\mathrm {Bun}_{G, D}(\textsf k)$ are labeled by the Shafarevich set:

\begin{align*} {\unicode{x0428}}^1(F, G) := \operatorname{Ker}\bigg(H^1(F, G) \rightarrow \prod_{x\in X}H^1(F_x, G)\bigg). \end{align*}

Namely, restriction of a $G$-torsor to the generic point defines a surjective map of pointed groupoids $\mathrm {Bun}_{G, D}(\textsf k) \rightarrow {\unicode{x0428}} ^1(F, G)$ and (4.6) coincides with its kernel.

4.1.7

Recall that $\mu$ defines a central extension $\widetilde G_F$ of $G(\mathbb {A}_F)$ by $A$, equipped with canonical splittings over $G(F)$ and $K_D$ (see §§ 1.5.4 and 2.2.9).

On the other hand, $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G_D)(\textsf k)$ is a set-theoretic $A$-torsor $\widetilde {\mathrm {Bun}}_{G, D}$ over $\mathrm {Bun}_{G, D}(\textsf k)$. Its restriction along (4.6) is identified with the set-theoretic $A$-torsor $G(F)\backslash \widetilde G_F/K_D$.

To explain this identification, we note that for each closed point $x\in X$ with residue field $\textsf k_1\supset \textsf k$, the gluing map $G(F_x) \rightarrow \mathrm {Bun}_{G, D}(\textsf k)$ arises as the $\textsf k$-points of a map:

(4.7)\begin{equation} \mathrm{res}(L(G)_x) \rightarrow \mathrm{Bun}_{G, D}, \end{equation}

where $\mathrm {res}(L(G)_x)$ is the Weil restriction of $L(G)_x$ along $\textsf k_1\supset \textsf k$. Indeed, an $S$-point of $\mathrm {res}(L(G)_x)$ is equivalent to a section of $G$ over the punctured formal disk around $S\times x \subset S\times X$ (because the map $(S\times X)_{\textsf k_1} \rightarrow S\times X$ is étale). This section may be used to glue the trivial bundles on $(S\times X)- (S\times x)$ and the formal disk around $S\times x$ using the Beauville–Laszlo theorem, defining (4.7).

Comparing the constructions of §§ 2.2.2 and 4.1.3, we see that $\mathscr G_D$ pulls back to $\operatorname {Nm}(\mathscr G_x)$ along (4.7), using the notation of Remark 2.2.10. The same remark implies that $\widetilde {\mathrm {Bun}}_{G, D}(\textsf k)$ pulls back to $\widetilde G_x$ along the gluing map $G(F_x) \rightarrow \mathrm {Bun}_{G, D}(\textsf k)$. If $x\in \mathring X$, this identification is compatible with the sections over $G(\mathscr O_x)$, and if $x \in D$, it is compatible with the sections over the first congruence subgroup $K_x := \ker (G(\mathscr O_x)\rightarrow G(\textsf k_1))$.

To see that $\widetilde {\mathrm {Bun}}_{G, D}(\textsf k)$ pulls back to $G(F)\backslash \widetilde G_F/K_D$ along (4.6), we perform the same construction for a finite collection of closed points $\{x_i\}$ ($i\in I$), compare the sections of $\prod _{i\in I}\widetilde G_{x_i}$ over the subgroup $G\big (X - \bigcup _{i\in I}x_i\big ) \subset \prod _{i\in I}G(F_{x_i})$, and pass to the colimit as in the definition of $\widetilde G_F$ (see [Zha22, § 2.2]).

4.1.8

Let $I$ be a nonempty finite set equipped with an ordered partition into nonempty finite subsets $I \cong I_1\sqcup \cdots \sqcup I_k$.

Recall that $\mu$ defines an étale $A$-gerbe $\mathscr G^I$ over $\mathrm {Hec}{}_G^I$. We may form its pullback $\mathscr G^{I_1,\ldots, I_k} := m^*(\mathscr G^I)$ along the composition map $m : \widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_k} \rightarrow \mathrm {Hec}_G^I$ (see § 2.1.5). Its further pullback along the restriction map in (4.3) defines an $A$-gerbe $\mathscr G_D^{I_1,\ldots, I_k}$ over $\widetilde {\mathrm {Hec}}{}_{G, D}^{I_1,\ldots, I_k}$.

Let us construct an isomorphism of $A$-gerbes over $\widetilde {\mathrm {Hec}}{}_{G, D}^{I_1,\ldots, I_k}$:

(4.8)\begin{equation} p_0^*(\mathscr G_D)\otimes p_k^*(\mathscr G_D)^{\otimes -1} \cong \mathscr G_D^{I_1, \ldots, I_k}. \end{equation}

Proof Construction

In view of the isomorphism (2.14), it suffices to construct (4.8) in the special case $k = 1$, as the general case will be a product of the isomorphisms associated to each $I_a$ (for $1\le a\le k$).

Consider now an $S$-point of $\mathrm {Hec}_{G, D}^I$ given by the modification datum $(P_0, \phi _0)\overset {x^I}{\sim }(P_1, \phi _1)$.

Let $i : \Gamma _{x^I} \subset S\times X$ and $\hat i : \Gamma _{x^I} \subset D_{x^I}$ denote the closed immersions. The construction of § 2.2.2 involves a morphism of complexes:

(4.9)\begin{equation} \Gamma(\Gamma_{x^I}, i^!\underline A(1)[4]) \rightarrow \Gamma(S, \underline A[2]). \end{equation}

The restriction of $p_0^*(\mathscr G_D) \otimes p_1^*(\mathscr G_D)^{\otimes -1}$ to $S$ is defined by the image of the section $P_0^*(\mu ) - P_1^*(\mu )$ under (4.9), where each $P_0$, $P_1$ is viewed as a morphism $S\times X \rightarrow B(G)$. It defines a section of $i^!A(1)[4]$ using the isomorphism $P_0\cong P_1$ off $\Gamma _{x^I}$.

Under the identification $i^!\underline A(1)[4] \cong \hat i^!\underline A(1)[4]$, this section is also defined by the restrictions of $P_0$, $P_1$ to $D_{x^I}$ and their identification over $\mathring D_{x^I}$. The image of this section under (4.9) is precisely the restriction of $\mathscr G_D^I$ to $S$.

4.1.9

Let $I \cong I_1\sqcup \cdots \sqcup I_k$ be as above.

We shall trivialize the restriction of the $A$-gerbe $\mathscr G^{I_1,\ldots, I_k}$ to $\mathrm {Sht}{}^{I_1,\ldots, I_k}_{G, D}$.

Proof Construction

Indeed, the isomorphism (4.8) exhibits this restriction as the pullback of $\mathscr G_D \otimes \mathrm {Fr}_{\mathrm {Bun}_{G, D}}^*(\mathscr G_D)^{\otimes -1}$ along the morphism $p_0 : \mathrm {Sht}{}^{I_1,\ldots, I_k}_{G, D} \rightarrow \mathrm {Bun}_{G, D}$, but there is an isomorphism $\mathrm {Fr}_{\mathrm {Bun}_{G, D}}^*(\mathscr G_D) \cong \mathscr G_D$ supplied by (1.15).

4.1.10

In particular, direct image with compact support along the projection $\mathrm {Sht}{}^{I_1,\ldots, I_k}_{G, D} \rightarrow \mathring X^I$ defines a functor of $\infty$-categories:

(4.10)\begin{equation} \mathrm{Shv}_{\mathscr G^{I_1,\ldots, I_k}}\big(\widetilde{\mathrm{Hec}}{}_G^{I_1,\ldots, I_k}\big) \rightarrow \mathrm{Ind}(\mathrm{Shv}(\mathring X^I)),\quad \mathscr F \mapsto \Gamma_c\big(\mathrm{Sht}_{G, D}^{I_1,\ldots, I_k}, \mathscr F\big). \end{equation}

Here, the functor of compactly supported cohomology of a constructible $E$-sheaf is well-defined because $\mathrm {Sht}{}^{I_1,\ldots,I_k}_{G, D}$ is a union of quasi-compact open substacks which are ind-algebraic stacks of ind-finite type [Laf18, Lemme 12.19].

4.1.11

For a nonempty finite set $I$, consider the unit $e_!(\underline E)$ of the Satake category defined using the trivialization of $\mathscr G^I$ along the unit section of $\mathrm {Hec}{}_G^I$ (see § 2.3).

Recall also the $E$-vector space $\operatorname {Fun}_c(\widetilde {\mathrm {Bun}}{}_{G, D}, A\subset E^{\times })$ of genuine functions of compact support on $\widetilde {\mathrm {Bun}}{}_{G, D}$.

We shall construct a canonical isomorphism of ind-constructible sheaves over $\mathring X^I$:

(4.11)\begin{equation} \Gamma_c(\mathrm{Sht}_{G, D}^I, e_!(\underline E)) \cong \operatorname{Fun}_c(\widetilde{\mathrm{Bun}}{}_{G, D}, A\subset E^{\times})\otimes\underline E. \end{equation}

Proof Construction

The base change of $\mathrm {Sht}{}_{G, D}^I$ along the unit section $\mathring X^I \rightarrow \mathrm {Hec}_G^I$ is identified with the following fiber product:

We shall obtain (4.11) by playing with two distinct trivializations of the restriction of $\mathscr G^I$ to $(\mathrm {Bun}_{G, D})^{\mathrm {Fr}}\times \mathring X^I$, coming from § 4.1.9 and the unit section of $\mathrm {Hec}_G^I$, respectively.

To wit, the image of $e_!(\underline E)$ under (4.10) is calculated as follows: we start with the constant sheaf $\underline E$ over $\mathrm {Bun}_{G, D}\times \mathring X^I$, view it as twisted by the trivial $A$-gerbe $\Delta ^*(\mathscr G_{D}\boxtimes \mathscr G_D^{\otimes -1})$, pull it back to $(\mathrm {Bun}_{G, D})^{\mathrm {Fr}}\times \mathring X^I$ and view it as twisted by the equivalent $A$-gerbe $(\mathrm {id}, \mathrm {Fr}_{\mathrm {Bun}_{G, D}})^*(\mathscr G_D \boxtimes \mathscr G_D^{\otimes -1})$ but trivialized by (1.15), and finally take its $!$-direct image towards $\mathring X^I$.

The fact that this procedure yields $\operatorname {Fun}_c(\widetilde {\mathrm {Bun}}{}_{G, D}, A\subset E^{\times })\otimes \underline E$ is observed in § 1.4.10.

4.2 A lemma for $\mathbb {G}_m$

4.2.1

We assume $D\neq \varnothing$ in this subsection.

Let $\infty D$ denote the formal completion of $X$ along $D$. Write $\eta = \operatorname {Spec}(F)$ for the generic point of $X$ and choose an algebraic closure $F\subset \bar F$, with $\bar {\eta } := \operatorname {Spec}(\bar F)$.

An $S$-point of the stack $\mathrm {Bun}_{G, \infty D}$ consists of a $G$-torsor over $S\times X$ equipped with a trivialization over the formal disk around $S\times D\subset S\times X$, or equivalently a $G$-torsor over $S\times \mathring X$ equipped with a trivialization over the punctured formal disk around $S\times D\subset S\times X$. In particular, it is well-defined even when $G$ is only a smooth affine group scheme over $\mathring X$.

4.2.2

The Artin reciprocity map is an isomorphism of topological abelian groups:

(4.12)\begin{equation} \mathrm{Art} : \pi_1(\mathring X, \bar{\eta})^{\mathrm{ab}} \cong \mathrm{Bun}_{\mathbb{G}_m, \infty D}(\textsf k)^{\mathrm{profin}}, \end{equation}

where the target denotes the profinite completion of $\mathrm {Bun}_{\mathbb {G}_m, \infty D}(\textsf k)$. It is normalized so that the geometric Frobenius element $\varphi _x \in \mathrm {Gal}(\bar {\textsf k}_x/\textsf k_x)$, for each closed point $x\in \mathring X$ with residue field $\textsf k_x$, maps to $\mathscr O(x)$. (Note that $\mathrm {Bun}_{\mathbb {G}_m, \infty D}$ is a scheme when $D\neq \varnothing$.)

On the other hand, we have the Abel–Jacobi morphism:

\begin{align*} \mathrm{AJ} : \mathring X \rightarrow \mathrm{Bun}_{\mathbb{G}_m, \infty D},\quad x\mapsto\mathscr O(x), \end{align*}

where $\mathscr O(x)$ is equipped with its canonical trivialization over $\infty D$, as $x\notin D$.

4.2.3

Recall the notion of the trace of Frobenius of an $A$-gerbe from § 1.4.

The following lemma shows that the Abel–Jacobi morphism geometrizes Artin reciprocity on the level of ‘character $A$-gerbes’.

Lemma 4.2.4 The following diagram is canonically commutative.

(4.13)

(Since $A$ is finite, every multiplicative $A$-torsor over $\mathrm {Bun}_{\mathbb {G}_m, \infty D}( {\textsf k})$ descends along its profinite completion, so $\mathrm {Art}^*$ is well-defined.)

Proof. We divide the proof into two claims.

Claim (4.13) is commutative over the neutral component.

To prove this assertion, it suffices to consider the loop spaces of (4.13) and show that the resulting diagram is commutative.

Namely, given a $\mathbb {Z}$-linear morphism $\mathrm {Bun}_{\mathbb {G}_m, \infty D} \rightarrow B(\underline A)$, or equivalently a commutative multiplicative $A$-torsor $t$ over $\mathrm {Bun}_{\mathbb {G}_m, \infty D}$, we need to compare the character $\pi _1(\mathring X, \bar {\eta })\rightarrow A$ associated to $\mathrm {AJ}^*(t)$ with the pullback of $\operatorname {Tr}(\mathrm {Fr}\mid t)(\textsf k)$ along (4.12).

Their equality is a familiar fact in geometric class field theory and follows immediately from the Chebotarev density theorem.

Claim Any $\mathbb {Z}$-linear morphism $\mathrm {Bun}_{\mathbb {G}_m, \infty D} \rightarrow B^2(\underline A)$ is (non-canonically) trivial over a finite extension of ${{\textsf k}}$.

Indeed, the commutativity of (4.13) can be verified étale locally over $\operatorname {Spec}(\textsf k)$, so it will follow from combining the two claims.

To prove Claim 2, it suffices to establish the vanishing:

(4.14)\begin{align} \underline{\operatorname{Ext}}^2(\mathrm{Bun}_{\mathbb{G}_m, \infty D}, \underline A) = 0, \end{align}

where $\underline {\operatorname {Ext}}$ denotes the internal Ext-group for étale sheaves over $\operatorname {Spec}(\textsf k)$.

Replacing $\textsf k$ by a finite extension if necessary, the reduced subscheme of $D$ becomes a finite nonempty collection of $\textsf k$-points $x^I$ of $X$. We choose an element $i_0\in I$ and fit $\mathrm {Bun}_{\mathbb {G}_m, \infty D}$ into the following system of three short exact sequences.

Here, $\mathrm {Pic} \cong \mathrm {Bun}_{\mathbb {G}_m, x^{i_0}}$ is the Picard scheme of $X$ and $\mathrm {Pic}^0$ is its neutral component.

It thus suffices to prove that $\underline {\operatorname {Ext}}^2(M, \underline A) = 0$, for $M$ a pro-unipotent group scheme, $\mathbb {G}_m$, an abelian variety, and $\underline {\mathbb {Z}}$.

The pro-unipotent case and the case $M = \underline {\mathbb {Z}}$ are clear. For $M = \mathbb {G}_m$ or an abelian variety, we may assume that $\underline A = \mu _n$, where $n$ is invertible in $\textsf k$. Morphisms $M \rightarrow \mu _n[2]$ of complexes are equivalent to morphisms $M_{[n]} \rightarrow \mathbb {G}_m[1]$, where $M_{[n]}\subset M$ is the subgroup scheme of $n$-torsion elements. However, $\underline {\operatorname {Ext}}^1(M_{[n]}, \mathbb {G}_m) = 0$ because $M_{[n]}$ is finite (locally) free [GRR72, VIII, Proposition 3.3.1].

4.2.5

Recall that for a sheaf of abelian groups $M$ over $\operatorname {Spec}(\textsf k)$, the groupoid of $\mathbb {E}_{\infty }$-monoidal morphisms $M \rightarrow B^2(\underline A)$ fits into the split fiber sequence (1.28).

For $M = \mathrm {Bun}_{\mathbb {G}_m, \infty D}$, the splitting supplies us with a retract:

(4.15)\begin{equation} \mathrm{Maps}_{\mathbb{E}_{\infty}}(\mathrm{Bun}_{\mathbb{G}_m, \infty D}, B^2(\underline A)) \rightarrow \mathrm{Maps}_{\mathbb{Z}}(\mathrm{Bun}_{\mathbb{G}_m, \infty D}, B^2(\underline A)). \end{equation}

Observe that the functors denoted by $\mathrm {AJ}^*$ and $\operatorname {Tr}(\mathrm {Fr}\mid -)(\textsf k)$ in (4.13) are naturally defined on $\mathrm {Maps}_{\mathbb {E}_{\infty }}(\mathrm {Bun}_{\mathbb {G}_m, \infty D}, B^2(\underline A))$, but they both factor through the retract (4.15). Indeed, they both depend only on the underlying $\mathbb {E}_1$-monoidal structure.

Combining this observation, the compatibility in § 4.1.7, and Lemma 4.2.4, we find two commutative squares as follows.

(4.16)

4.2.6

For the moment, let us assume that $\mathrm {char}(\textsf k) \neq 2$ and $A = \{\pm 1\} \subset E^{\times }$. We shall use diagram (4.16) to relate Weissman's meta-Galois group (see [Wei18, § 4]) with the $\{\pm 1\}$-gerbe of theta characteristics.

Consider the étale metaplectic cover $\mu : B_X(\mathbb {G}_m) \rightarrow B^4_X(\{\pm 1\}^{\otimes 2})$ defined by the cocycle:

(4.17)\begin{equation} \Lambda\otimes\Lambda \rightarrow \mathbb{Z}/2,\quad (1, 1)\mapsto 1, \end{equation}

where $\Lambda \cong \mathbb {Z}$ is the cocharacter lattice of $\mathbb {G}_m$ (see [Zha22, § 4.4]).

Its induced topological cover of $\mathbb {A}_F^{\times }$ is the central extension defined by the cocycle:

\begin{align*} \mathbb{A}_F^{\times} \otimes \mathbb{A}_F^{\times} \rightarrow \{\pm 1\},\quad (a, b)\mapsto \prod_{x\in X} \operatorname{Hilb}_x(a,b), \end{align*}

where $\operatorname {Hilb}_x : F_x\otimes F_x \rightarrow \{\pm 1\}$ denotes the quadratic Hilbert symbol at $x\in X$, equipped with canonical splittings over $F^{\times }$ and $\mathbb {O}_F^{\times }$.

By construction, the image of $\mu$ under the lower circuit of (4.16) is the meta-Galois group of $\mathring X$:

(4.18)\begin{align} 1 \rightarrow \{\pm 1\} \rightarrow \widetilde{\pi}_1(\mathring X, \bar{\eta}) \rightarrow \pi_1(\mathring X, \bar{\eta}) \rightarrow 1. \end{align}

Corollary 4.2.7 If $\mathrm {char}( {\textsf k})\neq 2$, then (4.18) is canonically identified with the central extension associated to the $\{\pm 1\}$-gerbe $\omega _{\mathring X}^{1/2}$ under (1.7).

Proof. Let us trace the image of $\mu$ under the upper circuit of (4.16).

Indeed, $\mu$ induces the $A$-gerbe $\mathscr G_D$ over $\mathrm {Bun}_{\mathbb {G}_m, \infty D}$ and it suffices to make the identification $\mathrm {AJ}^*(\mathscr G_D) \cong \omega _{\mathring X}^{1/2}$. This isomorphism is supplied by Lemma 3.1.6 for $\lambda = 1 \in \mathbb {Z}$.

Remark 4.2.8 It follows from Corollary 4.2.7 that the meta-Galois group (4.18) for function fields (global, local, and local integral) is non-canonically split, and is functorial with respect to finite separable extensions.

These facts are established by Weissman by different means, see [Wei18, §§ 4.2, 4.4].

The stipulation that, for $F$ of equal characteristic $2$, the meta-Galois group is the split extension [Wei18, § 4.1] appears to align with the classical fact that a canonical theta characteristic exists over $X$ when $\mathrm {char}(\textsf k) = 2$.

4.2.9

Let $T$ be a split torus and $T^{\sharp } \rightarrow T$ be the isogeny defined in § 3.1.1.

Since the restriction of $\mu$ to $B_{\mathring X}(T^{\sharp })$ acquires an $\mathbb {E}_{\infty }$-monoidal structure [Zha22, § 4.6], the restriction of $\mathscr G_D$ to $\mathrm {Bun}_{T^{\sharp }, D}$ has the structure of an $\mathbb {E}_{\infty }$-monoidal morphism $\mathrm {Bun}_{T^{\sharp }, D} \rightarrow B^2(\underline A)$. In particular, its trace of Frobenius defines a $\mathbb {Z}$-linear morphism:

(4.19)\begin{equation} \operatorname{Tr}(\mathrm{Fr}\mid\mathscr G_D) : \mathrm{Bun}_{T^{\sharp}, \infty D}(\textsf k) \rightarrow B(E^{\times}). \end{equation}

The Artin reciprocity map (4.12) induces an equivalence of groupoids between $\mathbb {Z}$-linear morphisms $\mathrm {Bun}_{T^{\sharp }, \infty D}(\textsf k) \rightarrow B(E^{\times })$ and $\mathbb {Z}$-linear morphisms $\mathrm {Weil}(\mathring X, \bar {\eta })^{\mathrm {ab}} \rightarrow B(H(E))$, where $H$ is identified with the $E$-torus dual to $T^{\sharp }$. The latter groupoid admits a functor to central extensions of $\mathrm {Weil}(\mathring X, \bar {\eta })$ by $H(E)$.

Using an argument similar to the proof of Corollary 4.2.7, we identify the image of (4.19) under this functor with (the Weil form of) the L-group ${}^LH_{\mathring X, \vartheta }$.

Thus, we obtain a canonical bijection:

(4.20)\begin{equation} \begin{Bmatrix} \text{genuine characters} \\ \widetilde{\mathrm{Bun}}_{T^{\sharp}, \infty D} \rightarrow E^{\times} \end{Bmatrix} \cong \begin{Bmatrix} \text{sections of} \\ {}^LH_{\mathring X, \vartheta} \rightarrow \mathrm{Weil}(\mathring X, \bar{\eta}) \end{Bmatrix}, \end{equation}

as both sides correspond to null-homotopies of (4.19).

Taking colimit of the bijection (4.20) over increasing $D$, we obtain a bijection between genuine characters $T^{\sharp }(F)\backslash \widetilde T^{\sharp } \rightarrow E^{\times }$ and sections of ${}^LH_{\eta, \vartheta } \rightarrow \mathrm {Weil}(\eta, \bar {\eta })$. We thus recover a part of the Langlands correspondence for covering groups of split tori [Wei18, Part 3].

4.3 Cusp forms

4.3.1

Suppose that the restriction of $G$ to $\mathring X$ is reductive and its restriction to $\operatorname {Spec}(\mathscr O_x)$ for each closed point $x\in D$ is parahoric.

4.3.2

Let $Z$ denote the radical of $G_{\mathring X}$, viewed as an affine group scheme (in fact, a torus) over $\mathring X$. The symmetric form $b$ associated to $\mu$ restricts to a bilinear form $\Lambda _Z \otimes \Lambda \rightarrow \underline A(-1)$, and we set $\Lambda ^{\sharp }_Z \subset \Lambda _Z$ to be its kernel. It corresponds to an isogeny of tori $Z^{\sharp } \rightarrow Z$. The group scheme $Z^{\sharp }$ plays the role of the ‘center’ in the metaplectic context.

The stack $\mathrm {Bun}_{Z^{\sharp }, \infty D}$ is defined as in § 4.2.1.

Remark 4.3.3 The $E$-torus dual to $Z^{\sharp }$ is canonically identified with the maximal abelian quotient $H^{\mathrm {ab}}$ of $H$.

Indeed, a character of $H^{\mathrm {ab}}$ is by definition a section of $\Lambda ^{\sharp }$ which pairs to zero with $\mathrm {ord}(Q(\alpha ))^{-1}\check {\alpha }$ for each $\alpha \in \Delta$. This is precisely a section of $\Lambda _Z^{\sharp } \cong \Lambda ^{\sharp } \cap \Lambda _Z$.

4.3.4

Recall that the restriction $\mu _{Z^{\sharp }}$ of $\mu$ to $B(Z^{\sharp })$ acquires a canonical $\mathbb {E}_{\infty }$-monoidal structure ([Zha22, § 4.6]).

Furthermore, $\mu$ is $B(Z^{\sharp })$-equivariant against $\mu _{Z^{\sharp }}$ with respect to the action of $B(Z^{\sharp })$ on $B(G)$ (see [Zha22, § 5.4]).

These observations imply that $\widetilde {\mathrm {Bun}}_{Z^{\sharp }, \infty D}$ has a commutative multiplicative structure and acts naturally on $\widetilde {\mathrm {Bun}}_{G, D}$.

4.3.5

Let us now choose a subgroup $\Xi \subset \widetilde {\mathrm {Bun}}_{Z^{\sharp }, D}$ which maps isomorphically onto its image in $\mathrm {Bun}_{Z^{\sharp }, D}(\textsf k)$ and such that $\Xi \backslash \mathrm {Bun}_{Z^{\sharp }, D}(\textsf k)$ is finite. (One may think of $\Xi$ as the kernel of a genuine character on $\widetilde {\mathrm {Bun}}_{Z^{\sharp }, D}$ with finite image.)

Then $\Xi$ is the $\textsf k$-points of a discrete subscheme to be denoted with the same letter:

\begin{align*} \Xi \subset \operatorname{Tr}(\mathrm{Fr}\mid\mathscr G_{Z^{\sharp}, D}), \end{align*}

which maps isomorphically onto its image in $(\mathrm {Bun}_{Z^{\sharp }, D})^{\mathrm {Fr}}$.

4.3.6

Consider the $E$-vector space of compactly supported functions $f : \Xi \backslash \widetilde {\mathrm {Bun}}_{G, D} \rightarrow E$ such that $f(x\cdot a) = f(x)\cdot a$ for each $x\in \Xi \backslash \widetilde {\mathrm {Bun}}_{G, D}$ and $a\in A$:

(4.21)\begin{equation} \operatorname{Fun}_c(\Xi\backslash\widetilde{\mathrm{Bun}}{}_{G, D}, A \subset E^{\times}). \end{equation}

We shall call such functions genuine automorphic forms.

They define genuine automorphic forms in the sense of § 1.5.5 as follows: using the compatibility of § 4.1.7, we may restrict along (4.6) to obtain compactly supported genuine functions over $G(F)\Xi \backslash \widetilde G_F/K_D$.

4.3.7

Suppose that $P_{\eta }$ is a parabolic subgroup of the restriction $G_{\eta }$ of $G$ to the generic point $\eta \subset X$. Then $P_{\eta }$ extends uniquely to a parabolic subgroup $P$ of $G_{\mathring X}$. Let $P\twoheadrightarrow M$ denote its Levi quotient.

The restriction of $\mu$ to $B_{\mathring X}(P)$ canonically descends to a rigidified section of $B^4A(1)$ over $B_{\mathring X}(M)$. In particular, the canonical maps $G \leftarrow P \rightarrow M$ induce maps of stacks as follows.

(4.22)

The $E$-vectors space of genuine cusp forms:

(4.23)\begin{equation} \operatorname{Fun}_{\mathrm{cusp}}(\Xi\backslash\widetilde{\mathrm{Bun}}_{G, D}, A\subset E^{\times}) \end{equation}

is defined to be the subspace of (4.21) consisting of elements which vanish under the integral transform along (4.22) for all proper parabolic subgroups $P_{\eta }\subset G_{\eta }$. (The definition of the integral transform requires fixing a Haar measure on the appropriate unipotent groups, but its vanishing is independent of this choice.)

Remark 4.3.8 Over the subspace of (4.21) of functions supported on $G(F)\Xi \backslash \widetilde G(\mathbb {A}_F)/K_D$, the cuspidality condition coincides with that from [BJ79, § 3.3].

To check their agreement, one needs the vanishing of ${\unicode{x0428}} ^1(F, N_P)$ for the unipotent radical $N_P\subset P$, which follows from [ABD⁺66, Exposé XXVI, Corollaire 2.2].

4.3.9

Recall the definition of Hecke operators in § 2.4.8: for each closed point $x\in \mathring X$ and $V \in \mathrm {Rep}^{\mathrm {alg}}({}^LH_{x, \vartheta })$, we have associated an element,

\begin{align*} h_{V, x} \in \operatorname{Fun}_c(G(\mathscr O_x)\backslash \widetilde G_x/G(\mathscr O_x), A\subset E^{\times}). \end{align*}

It acts on the vector space (4.23) via convolution $(f, h_{V, x})\mapsto f\star h_{V, x}$ along the multiplication map $\widetilde G_F\times \widetilde G_x \rightarrow \widetilde G_F$.

4.3.10

We are now ready to state the Langlands parametrization of genuine cusp forms on $\widetilde {\mathrm {Bun}}{}_{G, D}/\Xi$.

The coefficient field $E$ is taken to be $\overline {\mathbb {Q}}{}_{\ell }$. We recall that that data $(G, \mu )$ over $\mathring X$ define an L-group as a short exact sequence (see (1.34)):

\begin{align*} 1 \rightarrow H_{\bar{\eta}}(\overline{\mathbb{Q}}{}_{\ell}) \rightarrow {}^LH_{\mathring X, \vartheta} \rightarrow \pi_1(\mathring X, \bar{\eta}) \rightarrow 1. \end{align*}

Theorem 4.3.11 Assume $D \neq \varnothing$. There is a canonical decomposition:

(4.24)\begin{equation} \operatorname{Fun}_{\mathrm{cusp}}(\Xi\backslash\widetilde{\mathrm{Bun}}_{G, D}, A\subset \overline{\mathbb{Q}}{}_{\ell}^{\times}) \cong \bigoplus_{[\sigma]} \mathbf{H}_{D, [\sigma]}, \end{equation}

where $[\sigma ]$ ranges over $H_{\bar {\eta }}(\overline {\mathbb {Q}}{}_{\ell })$-conjugacy classes of sections of ${}^LH_{\mathring X, \vartheta } \rightarrow \pi _1(\mathring X, \bar {\eta })$.

For each $x\in \mathring X$ and $V \in \mathrm {Rep}^{\mathrm {alg}}({}^LH_{x,\vartheta })$, the summand $\mathbf {H}_{D, [\sigma ]}$ is an eigenspace for $h_{V, x}$ with eigenvalue $\operatorname {Tr}([\sigma _x]\cdot \varphi _x\mid V)$, where $\varphi _x \in \pi _1(x, \bar x)$ denotes the geometric Frobenius and $[\sigma _x]$ denotes the $H_{\bar {\eta }}(\overline {\mathbb {Q}}_{\ell })$-conjugacy class of sections of ${}^LH_{x, \vartheta } \rightarrow \pi _1(x, \bar x)$ induced from $[\sigma ]$.

4.4 Excursion

4.4.1

In this subsection, we prove Theorem 4.3.11. The notation is as in § 4.3.

In the course of the proof, we work with a sufficiently large subfield $E\subset \overline {\mathbb {Q}}{}_{\ell }$ finite over $\mathbb {Q}_{\ell }$ and make our choice $q^{1/2} \in E^{\times }$.

4.4.2

For a nonempty finite set $I$ equipped with an ordered partition $I \cong I_1\sqcup \cdots \sqcup I_k$, we specialize the commutative diagram (4.3) to $G = Z^{\sharp }$ and restrict along the unit section of the Hecke stacks.

We record this commutative diagram of stacks, along with the $A$-gerbes defined by $\mu _{Z^{\sharp }}$ on some of them as follows.

(4.25)

The left column consists of strictly commutative Picard stacks over $\mathring X^I$, and the right column consists of those over $\textsf k$. Morphisms in (4.25) are compatible with these structures, and the $A$-gerbes admit $\mathbb {E}_{\infty }$-monoidal structures.

The action of $Z^{\sharp }$ on $G$ induces an action of each term in (4.25) on the corresponding term in (4.3), the morphisms among them being equivariant.

4.4.3

Recall that $e^*(\mathscr G^I_{Z^{\sharp }})$ is canonically trivialized as an $A$-gerbe over $B_{\mathring X^I}(L_+^I(Z^{\sharp }))$ equipped with an $\mathbb {E}_{\infty }$-monoidal structure.

Lemma 4.4.4 The forgetful functor out of $B_{\mathring X^I}(L^I_+(Z^{\sharp }))$-equivariant objects in $\widetilde {\mathrm {Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k}$ is an equivalence of categories:

(4.26)\begin{equation} \big(\widetilde{\mathrm{Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k}\big)^{B_{\mathring X^I}(L_+^I(Z^{\sharp}))} \cong \widetilde{\mathrm{Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k}. \end{equation}

Proof. The forgetful functor is fully faithful, as $B_{\mathring X^I}(L_+^I(Z^{\sharp }))$ may be written as an inverse limit of connected smooth algebraic stacks.

To show that it is essentially surjective, it suffices to prove that for each $\mathscr F \in \widetilde {\mathrm {Sat}}{}^{I_1,\ldots, I_k}_{G, \mathscr G^I}$, there is an isomorphism relating its pullback under the action and projection maps:

(4.27)\begin{equation} \mathrm{act}^*(\mathscr F) \cong \mathrm{pr}^*(\mathscr F),\quad \text{over }B_{\mathring X^I}(L_+^I(Z^{\sharp})) \times_{\mathring X^I} \widetilde{\mathrm{Hec}}{}_G^{I_1,\ldots, I_k}, \end{equation}

extending the natural one over $e \times \widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots,I_k}$, the extension being necessarily unique.

By universal local acyclicity, it is enough to show that such an extension exists over the pairwise disjoint locus of $\mathring X^I$ (see [HS23, Theorem 6.8]), and furthermore over any $\bar {\textsf k}$-point of the latter. The pointwise statement is a consequence of Proposition 3.3.7.

4.4.5

Using Lemma 4.4.4, we see that the pullback of any object $\mathscr F\in \widetilde {\mathrm {Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k}$ to $\mathrm {Sht}_{G, D}^{I_1,\ldots, I_k}$ along the vertical map in (4.3) acquires a $(\mathrm {Bun}_{Z^{\sharp }, D})^{\mathrm {Fr}}$-equivariance structure.

In particular, taking direct image with compact support along the projection $\Xi \backslash \mathrm {Sht}{}^{I_1,\ldots, I_k}_{G, D} \rightarrow \mathring X^I$ defines a functor:

(4.28)\begin{equation} \widetilde{\mathrm{Sat}}{}_{G, \mathscr G^I}^{I_1,\ldots, I_k} \rightarrow \mathrm{Ind}(\mathrm{Shv}(\mathring X^I)),\quad \mathscr F \mapsto \Gamma_c\big(\Xi\backslash\mathrm{Sht}_{G, D}^{I_1,\ldots, I_k}, \mathscr F\big). \end{equation}

(Note that $\Xi \backslash \mathrm {Sht}{}^{I_1,\ldots, I_k}_{G, D}$ is an ind-algebraic stack of ind-finite type.)

The isomorphism (4.11) identifies the image of $e_!(\underline E)$ under (4.28) (for $k = 1$) with the constant sheaf over $\mathring X^I$ with values in $\operatorname {Fun}_c(\Xi \backslash \widetilde {\mathrm {Bun}}{}_{G, D}, A\subset E^{\times })$.

4.4.6

Finally, we summarize the arguments of V. Lafforgue [Laf18] and Xue [Xue20b], which establish the spectral decomposition (4.24), taking as sole input the Satake functors $\mathscr S^{I_1,\ldots, I_k}$ (2.32) and the cohomology of Shtukas (4.28).

The notion of Shtukas and the method to construct representations of copies of $\pi _1(\mathring X, \bar {\eta })$ using partial Frobenii originated in Drinfeld's work on $\mathrm {GL}_2$ (see [Dri87a, Dri87b, Dri88]).

This summary is only included to give detailed references to the results of [Laf18] and [Xue20b]. It contains no originality whatsoever.

Proof of Theorem 4.3.11 For a nonempty finite set $I$, composing the Satake functor $\mathscr S^I$ with (4.28) (for $k = 1$) and taking the middle cohomology group $H^0$ yields a functor:

(4.29)\begin{equation} \mathrm{Rep}^{\mathrm{alg}}(({}^LH_{\mathring X, \vartheta})^I) \rightarrow \mathrm{Ind}(\mathrm{Shv}(\mathring X^I)),\quad W\mapsto \mathscr H_{I, W}:= H^0_c(\Xi\backslash\mathrm{Sht}_{G, D}^I, \mathscr S^I(W)). \end{equation}

Note that there are forgetful functors:

\begin{align*} \mathrm{Ind}(\mathrm{Rep}(\pi_1(\mathring X, \bar {\eta})^I)) \rightarrow \mathrm{Ind}(\mathrm{Lis}(\mathring X^I)) \subset \mathrm{Ind}(\mathrm{Shv}(\mathring X^I)), \end{align*}

where $\mathrm {Rep}(\pi _1(\mathring X, \bar {\eta })^I)$ denotes the category of finite-dimensional continuous $E$-linear representations of $\pi _1(\mathring X, \bar {\eta })^I$.

Claim $\mathscr H_{I, W}$ canonically lifts to $\mathrm {Ind}(\mathrm {Rep}(\pi _1(\mathring X, \bar {\eta })^I))$.

The claim is proved using the action of partial Frobenii on $\mathscr H_{I, W}$. To wit, for an ordered partition $I \cong I_1\sqcup \cdots \sqcup I_k$ into nonempty finite subsets, we have a commutative diagram

(4.30)

where the top horizontal morphism sends the data (4.1), (4.2) to

\begin{align*} (P_1, \phi_1) \overset{x^{I_2}}{\sim} \cdots \overset{x^{I_k}}{\sim} {}^{\tau}(P_0, \phi_0) \overset{{}^{\tau}x^{I_1}}{\sim} {}^{\tau}(P_1, \phi_1), \end{align*}

and the middle and lower horizontal morphisms are the partial Frobenii, acting as the Frobenius on the factor corresponding to $I_1$ and the identity on the remaining factors.

We view $\mathscr S^{I_1,\ldots, I_k}$ as valued in the category of untwisted perverse sheaves over $\Xi \backslash \mathrm {Sht}_{G, D}^{I_1,\ldots, I_k}$, by pulling back along the restriction maps to the local Hecke stack and using the trivialization of the $A$-gerbe $\mathscr G^{I_1,\ldots, I_k}$ over $\mathrm {Sht}{}_{G, D}^{I_1,\ldots, I_k}$ constructed in § 4.1.9. Perversity of the pullback follows from the smoothness and dimension count in [Laf18, § 2].

The trivialization of $\mathscr G^{I_1,\ldots, I_k}$ over $\mathrm {Sht}_{G, D}^{I_1,\ldots, I_k}$ is $\mathrm {Fr}_{I_1}$-equivariant in the sense that it commutes with the canonical isomorphism $\mathrm {Fr}_{I_1}^*(\mathscr G^{I_1,\ldots, I_k}) \cong \mathscr G^{I_1,\ldots, I_k}$. By the construction of $\mathscr S^{I_1,\ldots, I_k}$ and the commutative diagram (4.30), we obtain a natural isomorphism:

(4.31)\begin{equation} (\mathrm{Fr}_{I_1})^*\mathscr S^{I_2,\ldots, I_1}(W) \cong \mathscr S^{I_1,\ldots, I_k}(W). \end{equation}

Since the outer square of (4.30) is Cartesian up to universal homeomorphisms, (4.31) induces an isomorphism:

(4.32)\begin{equation} F_{I_1} : (\mathrm{Fr}_{I_1}\!)^*\mathscr H_{I, W} \cong \mathscr H_{I, W}. \end{equation}

By re-ordering the partition of $I$, we obtain similar isomorphisms $F_{I_2}, \ldots, F_{I_k}$. One sees as in [Laf18, §§ 3 and 4] that the isomorphisms $F_{I_1}, \ldots, F_{I_k}$ pairwise commute and their composition equals the canonical identification of $\mathscr H_{I, W}$ with its Frobenius pullback.

Using (4.32), we construct the following endomorphism as in [Laf18, § 12.3.3]:

\begin{align*} S_{V, x} \in \operatorname{End}(\mathscr H_{I, W}),\quad\text{for } x\in\mathring X \text{ and } V\in \mathrm{Rep}^{\mathrm{alg}}({}^LH_{x, \vartheta}). \end{align*}

Note that $V$ (unlike $W$) is a representation of the local L-group, so the associated ‘creation’ and ‘annihilation’ operators, corresponding to the unit and counit of $V$, are only defined over the subscheme $\mathring X^I \times x^{\{1,2\}} \subset \mathring X^{I\sqcup \{1, 2\}}$. (Informally, $S_{V, x}$ is the trace of the Frobenius endomorphism on ‘a copy of $V$ inserted at $x$’.)

The fact that $S_{V, x}$ restricts to the action of the Hecke operator $h_{V, x}$ over $(\mathring X - x)^I$ is proved as in [Laf18, § 6].

Furthermore, the $E$-sheaf $\mathscr H_{I\sqcup \{0\}, W\boxtimes V}$ is well defined over $\mathring X^I\times x$, and (4.32) induces an endomorphism $(F_{\{0\}})^{\deg (x)}$ of it. The argument of [Laf18, § 7] gives the following identity (to be thought of as a Cayley–Hamilton theorem for $(F_{\{0\}})^{\deg (x)}$):

(4.33)\begin{equation} \sum_{i = 0}^{\dim(V)} (-1)^i S_{\wedge^{\dim(V)-i}(V), x} \cdot ((F_{\{0\}})^{\deg(x)})^i = 0. \end{equation}

The isomorphisms (4.32), together with (4.33), imply the claim by [Xue20b, Proposition 1.3.4, Theorem 4.2.3].

Next, we define the subsheaf of cuspidal cohomology $\mathscr H_{I, W, \mathrm {cusp}} \subset \mathscr H_{I, W}$ either by [Laf18, § 12.3.4] or by a generalization of [Xue20b, § 7] (which is stated for split reductive groups). It belongs to $\mathrm {Rep}(\pi _1(\mathring X, \bar {\eta })^I)$. Thus, (4.29) induces a system of functors:

(4.34)\begin{equation} \mathrm{Rep}^{\mathrm{alg}}(({}^LH_{\mathring X, \vartheta})^I) \rightarrow \mathrm{Rep}(\pi_1(\mathring X, \bar{\eta})^I),\quad W\mapsto \mathscr H_{I, W, \mathrm{cusp}}, \end{equation}

indexed by nonempty finite sets $I$, which are compatible with surjections of such. As noted in § 4.4.5, the object $\mathscr H_{I, \mathbf 1, \mathrm {cusp}}$ associated to the trivial representation $\mathbf 1$ is isomorphic to the (finite-dimensional) $E$-vector space:

(4.35)\begin{equation} \operatorname{Fun}_{\mathrm{cusp}}(\Xi\backslash\widetilde{\mathrm{Bun}}{}_{G, D}, A\subset E^{\times}), \end{equation}

equipped with the trivial $\pi _1(\mathring X, \bar {\eta })^I$-action.

Using the construction of [Laf18, §§ 9 and 10], the system of functors (4.34) equips the $E$-vector space (4.35) with the action of a commutative $E$-algebra $\mathscr B$ (of ‘excursion operators’), and [Laf18, § 11] associates to each $E$-point of $\operatorname {Spec}(\mathscr B)$ an $H(E_1)$-conjugacy class of sections of ${}^LH_{\mathring X, \vartheta } \rightarrow \pi _1(\mathring X, \bar {\eta })$ for some finite extension $E\subset E_1$ in $\overline {\mathbb {Q}}{}_{\ell }$.

Finally, the desired decomposition (4.24) is the decomposition of (4.35) according to its support in $\operatorname {Spec}(\mathscr B\otimes _E \overline {\mathbb {Q}}{}_{\ell })$. The action of the Hecke operator $h_{V, x}$ on the summand $\mathbf {H}_{D, [\sigma ]}$ is calculated as in [Laf18, § 11].

Remark 4.4.7 It follows from the construction that each conjugacy class $[\sigma ]$ appearing in (4.24) is associated to a section $\sigma : \Gamma \rightarrow {}^LH_{\Gamma, \vartheta }$ of the finite form of the L-group (see Remark 1.6.16) and a finite extension $\mathbb {Q}_{\ell }\subset E$ in $\overline {\mathbb {Q}}{}_{\ell }$:

\begin{align*} 1 \rightarrow H_{\bar{\eta}}(E) \rightarrow {}^LH_{\Gamma, \vartheta} \rightarrow \Gamma \rightarrow 1. \end{align*}

Furthermore, the Zariski closure of $\sigma (\Gamma ) \subset {}^LH_{\Gamma, \vartheta }$ is a (possibly disconnected) reductive group, when ${}^LH_{\Gamma, \vartheta }$ is equipped with the algebraic structure induced from $H_{\bar {\eta }}$.

4.4.8

The spectral decomposition (4.24) is compatible with inclusions of nonempty $\textsf k$-finite closed subschemes $D\subset D_1$ of $X$, i.e. the following diagram is commutative:

(4.36)

where the left vertical arrow is induced from $\pi _1(\mathring X_1, \bar {\eta }) \rightarrow \pi _1(\mathring X, \bar {\eta })$, for $\mathring X_1 := X - D_1$, and the right vertical arrow is the restriction along $\widetilde {\mathrm {Bun}}_{G, D_1} \rightarrow \widetilde {\mathrm {Bun}}_{G, D}$.

The generic version of the spectral decomposition (0.3) asserted in Theorem A is a formal consequence of (4.24) and the compatibility (4.36).

Namely, the existence of parahoric models [BT84], combined with [Zha22, Lemma 2.2.5], shows that the $2$-groupoid of pairs $(G, \mu )$, where $G\rightarrow \operatorname {Spec}(F)$ is a reductive group and $\mu$ is an étale metaplectic cover of $G$, is the filtered colimit over nonempty $\textsf k$-finite subschemes $D\subset X$ of the $2$-groupoid of pairs $(G_1, \mu _1)$, where $G_1\rightarrow X$ is a smooth affine group scheme, reductive over $X - D$ and parahoric along $D$, and $\mu _1$ is an étale metaplectic cover of the restriction $G_{1, X- D}$.

We thus obtain (0.3) as the filtered colimit of (4.24), applied to each $(G_1, \mu _1)$ as above, over nonempty $\textsf k$-finite subschemes $D\subset X$.

Remark 4.4.9 In contrast to the non-metaplectic context, the generic version (0.3) does not allow us to state its compatibility with Satake isomorphism in a canonical manner.

Namely, for $\mu$ to be ‘unramified’ at a closed point $x\in X$ involves the datum of an extension across $x$ which is generally not unique (Remark 4.1.5). Distinct choices of extensions give rise to distinct sections of $\widetilde G_x \rightarrow G(F_x)$ over $G(\mathscr O_x)$, and correspondingly distinct splittings of the L-group ${}^LH_{F, \vartheta } \rightarrow \mathrm {Gal}(\bar F/F)$ over the inertia subgroup $I_x \subset \mathrm {Gal}(\bar F_x/F_x)$.