2.1.1 Algebraic L-group

In fact, there are at least two conventions for the definition of the L-group. The one which is more traditionally used in the literature is what we shall call the algebraic L-group, denoted ${}^L G^{\operatorname {\mathrm {alg}}}$ , defined as in [Reference Treumann and VenkateshTV16, §2.5]. The root datum $\Psi (G)$ of $G_{\mathbf {F}^s}$ determines a pinning for $\widehat {G}$ , which, in turn, gives a splitting $\mathrm {Out}(\widehat {G}) \rightarrow \operatorname {\mathrm {Aut}}(\widehat {G})$ and an identification $\operatorname {\mathrm {Aut}}(\Psi (G)) \cong \mathrm {Out}(\widehat {G})$ . The $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ -action on $\Psi (G)$ transports to an action $\operatorname {\mathrm {act}}^{\operatorname {\mathrm {alg}}}$ of $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ on $\widehat {G}$ , and we define ${}^L G^{\operatorname {\mathrm {alg}}}$ to be the semidirect product

$$\begin{align*}{}^L G^{\operatorname{\mathrm{alg}}} := \widehat{G} \rtimes_{\operatorname{\mathrm{act}}^{\operatorname{\mathrm{alg}}}} \operatorname{Gal}(\mathbf{F}^s/\mathbf{F}). \end{align*}$$

Since the action $\operatorname {\mathrm {act}}^{\operatorname {\mathrm {alg}}}$ factors through a finite quotient, we may regard ${}^L G^{\operatorname {\mathrm {alg}}}$ as a pro-algebraic group over k.

2.1.2 Geometric L-group

We now make a different construction of the L-group, using the Tannakian theory, following [Reference Richarz and ZhuRZ15, Appendix A] and [Reference ZhuZhu17, §5.5]. We begin with the Geometric Satake equivalence,

$$\begin{align*}\operatorname{\mathrm{P}}_{L^+G_{\mathbf{F}^s}}(\operatorname{\mathrm{Gr}}_{G,\mathbf{F}^s};k) \cong \operatorname{\mathrm{Rep}}_k(\widehat{G}). \end{align*}$$

The Galois group $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ acts on $\operatorname {\mathrm {Gr}}_{G,\mathbf {F}^s}$ , inducing an action on the neutralized Tannakian category $(\operatorname {\mathrm {P}}_{L^+G_{\mathbf {F}^s}}(\operatorname {\mathrm {Gr}}_{G,\mathbf {F}^s};k), \underbrace {H^*(-)}_{\text {fiber functor}})$ . By [Reference Richarz and ZhuRZ15, Lemma A.1], this, in turn, induces an action $\operatorname {\mathrm {act}}^{\operatorname {\mathrm {geom}}}$ of $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ on $\widehat {G}_k$ . We define

$$\begin{align*}{}^L G^{\operatorname{\mathrm{geom}}} := \widehat{G}_k \rtimes_{\operatorname{\mathrm{act}}^{\operatorname{\mathrm{geom}}}} \operatorname{Gal}(\mathbf{F}^s/\mathbf{F}). \end{align*}$$

In the case at hand, we shall see that $\operatorname {\mathrm {act}}^{\operatorname {\mathrm {geom}}}$ also factors through a finite quotient of $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ , so we may also regard ${}^L G^{\operatorname {\mathrm {geom}}}$ as a pro-algebraic group.

2.1.3 Relation between the two L-groups

The relation between these two actions is as follows. We let $\rho $ be the half sum of positive coroots of $\widehat {G}$ , and we denote by $\rho \colon \mathbf {G}_m \rightarrow \widehat {G}_{\operatorname {\mathrm {ad}}}$ the corresponding cocharacter. With $\mathrm {cyc}_p \colon \operatorname {Gal}(\mathbf {F}^s/\mathbf {F}) \rightarrow \mathbf {F}_p^{\times } $ denoting the mod p cyclotomic character, let $\chi $ denote the composite

$$\begin{align*}\operatorname{Gal}(\mathbf{F}^s/\mathbf{F}) \xrightarrow{\mathrm{cyc}_p} \mathbf{F}_p^{\times} \hookrightarrow k^{\times} \xrightarrow{\rho} \widehat{G}_{\operatorname{\mathrm{ad}}}(k). \end{align*}$$

This induces a homomorphism $\operatorname {\mathrm {Ad}}_{\chi } \colon \operatorname {Gal}(\mathbf {F}^s/\mathbf {F}) \rightarrow \operatorname {\mathrm {Aut}}(\widehat {G})$ .

Given a choice of lift $\widetilde {\chi } \colon \operatorname {Gal}(\mathbf {F}^s/\mathbf {F}) \rightarrow \widehat {G}(k)$ of $\chi $ , which could, for example, come from a square root of the mod p cyclotomic character, we get an isomorphism ${}^L G^{\operatorname {\mathrm {alg}}} \xrightarrow {\sim } {}^L G^{\operatorname {\mathrm {geom}}}$ by

(2.1) $$ \begin{align} (g, \gamma) \mapsto (g \widetilde{\chi}(\gamma^{-1}), \gamma). \end{align} $$

By [Reference ZhuZhu17, Remark 5.5.8], a square root of the cyclotomic character exists whenever $\mathrm {char}(\mathbf {F})>0$ . (However, in general, it can happen that ${}^L G^{\operatorname {\mathrm {alg}}}$ and ${}^L G^{\operatorname {\mathrm {geom}}} $ are not isomorphic; for an example, see [Reference ZhuZhu17, Example 5.5.9].)

At different points, we will want to consider both versions of L-groups. If we write ${}^L G$ without a superscript, then by default we mean the algebraic L-group ${}^L G^{\operatorname {\mathrm {alg}}}$ .

2.1.4 Representation categories

For any Galois extension $\mathbf {F}'/\mathbf {F}$ such that $G_{\mathbf {F}'}$ is split, the analogous construction to §2.1.1 gives a ‘finite form’ algebraic L-group $\widehat {G} \rtimes _{\operatorname {\mathrm {act}}^{\operatorname {\mathrm {alg}}}} \operatorname {Gal}(\mathbf {F}'/\mathbf {F})$ . We define the category of (k-linear) algebraic representations of ${}^L G^{\operatorname {\mathrm {alg}}}$ to be

$$\begin{align*}\operatorname{\mathrm{Rep}}_k({}^L G^{\operatorname{\mathrm{alg}}}) := \varinjlim_{\mathbf{F}'} \operatorname{\mathrm{Rep}}_k(\widehat{G} \rtimes_{\operatorname{\mathrm{act}}^{\operatorname{\mathrm{alg}}}} \operatorname{Gal}(\mathbf{F}'/\mathbf{F})). \end{align*}$$

Let $ \operatorname {\mathrm {Rep}}_k({}^L G^{\operatorname {\mathrm {geom}}}) := \operatorname {\mathrm {Rep}}_k(\widehat {G})^{B\operatorname {Gal}(\mathbf {F}^s/\mathbf {F}), \operatorname {\mathrm {geom}}}$ denote the category of continuously $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ -equivariant objects in $\operatorname {\mathrm {Rep}}_k(\widehat {G})$ with respect to the geometric action. The Geometric Satake equivalence induces by descent an equivalence

(2.2) $$ \begin{align} \operatorname{\mathrm{P}}_{L^+G}(\operatorname{\mathrm{Gr}}_G;k) \cong \operatorname{\mathrm{Rep}}_k(\widehat{G})^{B\operatorname{Gal}(\mathbf{F}^s/\mathbf{F}), \operatorname{\mathrm{geom}}}, \end{align} $$

where the action of $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ on $\operatorname {\mathrm {Rep}}_k(\widehat {G})$ on the RHS is via $\operatorname {\mathrm {act}}^{\operatorname {\mathrm {geom}}}$ , and on the LHS, $\operatorname {\mathrm {Gr}}_G$ is considered over $\mathbf {F}$ . By definition, on the right side we take are taking objects on which $\operatorname {Gal}(\mathbf {F}^s/\mathbf {F})$ acts continuously with its Krull topology. Since k is algebraic over $\mathbf {F}_p$ , in this case, $\operatorname {\mathrm {Rep}}_k(\widehat {G})^{B\operatorname {Gal}(\mathbf {F}^s/\mathbf {F}), \operatorname {\mathrm {geom}}}$ can be identified with $\varinjlim _{\mathbf {F}'/\mathbf {F}} \operatorname {\mathrm {Rep}}_k(\widehat {G})^{B\operatorname {Gal}(\mathbf {F}'/\mathbf {F}), \operatorname {\mathrm {geom}}}$ , where the limit runs over finite Galois extensions $\mathbf {F}'/\mathbf {F}$ over which the geometric action factors.

An isomorphism (2.1) gives an embedding $\operatorname {\mathrm {Rep}}_k({}^L G^{\operatorname {\mathrm {alg}}}) \hookrightarrow \operatorname {\mathrm {Rep}}_k(\widehat {G})^{\operatorname {Gal}(\mathbf {F}^s/\mathbf {F}), \operatorname {\mathrm {geom}}}$ , which, as just remarked, is an equivalence for our choice of k. See [Reference Richarz and ZhuRZ15, Proposition A.10] for a description of the essential image in general.

2.1.5 L-parameters

2.2.1 Generators

We define $\mathcal {O}({}^L G_k^{\operatorname {\mathrm {alg}}}) := \varinjlim _{F'/F} \mathcal {O}(\widehat {G}_k \rtimes \operatorname {Gal}(F'/F))$ , where the limit runs over finite extensions $F'/F$ over which the $\Gamma $ -action on $\widehat {G}_k$ factors.

Generators of $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G^{\operatorname {\mathrm {alg}}})$ will be denoted $S_{I, f, (\gamma _i)_{i \in I}}$ , where the indexing set $(I, f, (\gamma _i)_{i \in I})$ consists of

1. (i) I is a finite (possibly empty) set,

2. (ii) $f \in \mathcal {O}(\widehat {G}_k \backslash ({}^L G_k^{\operatorname {\mathrm {alg}}})^I / \widehat {G}_k) := \mathcal {O}(({}^L G_k^{\operatorname {\mathrm {alg}}})^I )^{\widehat {G}_k \times \widehat {G}_k }$ , where the quotient is for the actions of $\widehat {G}_k$ by diagonal left and right translation, respectively, and

3. (iii) $\gamma _i \in \Gamma $ for each $i \in I$ .

2.2.2 Relations

Next, we describe the relations. (Compare [Reference LafforgueLaf18, §10].)

1. (i) $S_{\emptyset , f, *} = f(1_G)$ , an element of $k \subset \operatorname {\mathrm {Exc}}(\Gamma , {}^L G^{\operatorname {\mathrm {alg}}})$ .

2. (ii) The map $f \mapsto S_{I, f, (\gamma _i)_{i \in I}}$ is a k-algebra homomorphism in f; that is,

   $$ \begin{align*} S_{I, f+f', (\gamma_i)_{i \in I}} &= S_{I, f, (\gamma_i)_{i \in I}} + S_{I, f', (\gamma_i)_{i \in I}}, \\ S_{I, ff', (\gamma_i)_{i \in I}} &= S_{I, f, (\gamma_i)_{i \in I}} \cdot S_{I, f', (\gamma_i)_{i \in I}}, \end{align*} $$
   and
   $$\begin{align*}S_{I, \lambda f, (\gamma_i)_{i \in I}} = \lambda S_{I, f, (\gamma_i)_{i \in I}} \text{ for all }\lambda \in k. \end{align*}$$

3. (iii) For all maps of finite sets $\zeta \colon I \rightarrow J$ , all $f \in \mathcal {O}(\widehat {G}_k \backslash ({}^L G_k^{\operatorname {\mathrm {alg}}})^I / \widehat {G}_k)$ , and all $(\gamma _j)_{j \in J} \in \Gamma ^J$ , we have

   $$\begin{align*}S_{J, f^{\zeta}, (\gamma_j)_{j \in J}} = S_{I, f, (\gamma_{\zeta(i)})_{i \in I}}, \end{align*}$$
   where $f^{\zeta } \in \mathcal {O}(\widehat {G}_k \backslash ({}^L G_k^{\operatorname {\mathrm {alg}}})^J / \widehat {G}_k)$ is defined by $f^{\zeta } ((g_j)_{j \in J}) := f((g_{\zeta (i)})_{i \in I})$ .

4. (iv) For all $f \in \mathcal {O}(\widehat {G}_k \backslash ({}^L G_k^{\operatorname {\mathrm {alg}}})^I / \widehat {G}_k)$ and $(\gamma _i)_{i \in I}, (\gamma _i')_{i \in I}, (\gamma _i")_{i \in I} \in \Gamma ^I$ , we have

   $$\begin{align*}S_{I \sqcup I \sqcup I, \widetilde{f}, (\gamma_i)_{i \in I} \times (\gamma_i')_{i \in I} \times (\gamma_i")_{i \in I} } = S_{I, f, (\gamma_i (\gamma_i')^{-1} \gamma_i")_{i \in I} }, \end{align*}$$
   where $\widetilde {f} \in \mathcal {O}(\widehat {G}_k \backslash ({}^L G_k^{\operatorname {\mathrm {alg}}})^{I \sqcup I \sqcup I}/ \widehat {G}_k)$ is defined by
   $$\begin{align*}\widetilde{f}((g_i)_{i \in I} \times (g_i')_{i \in I} \times (g_i")_{i \in I}) = f((g_i (g_i')^{-1} g_i")_{i \in I}). \end{align*}$$

5. (v) If f is inflated from a function on $\Gamma ^I$ , then $S_{I , f, (\gamma _i)_{i \in I}}$ equals the scalar $f((\gamma _i)_{i \in I})$ . More generally, if J is a subset of I and f is inflated from a function on $(\widehat {G}_k \backslash ({}^L G_k^{\operatorname {\mathrm {alg}}})^J / \widehat {G}_k) \times \Gamma ^{I \setminus J}$ , then we have

   $$\begin{align*}S_{I, f, (\gamma_i)_{i \in I}} = S_{J, \check{f}, (\gamma_j)_{j \in J}}, \end{align*}$$
   where $\check {f}((g_j)_{j \in J}) := f((g_j)_{j \in J}, (\gamma _i)_{i \in I \setminus J})$ . (Compare [Reference LafforgueLaf18, p. 164].)

2.4.1 Generators

We take a set of generators indexed by tuples of data of the form $(I, W, x, \xi , (\gamma _i)_{i \in I})$ , where

1. (i) I is a finite set,

2. (ii) $W \in \operatorname {\mathrm {Rep}}_k(({}^L G^{\operatorname {\mathrm {alg}}})^I)$ (cf. §2.1.4),

3. (iii) $x \in W$ is a vector invariant under the diagonal $\widehat {G}_k$ -action,

4. (iv) $\xi \in W^*$ is a functional invariant under the diagonal $\widehat {G}_k$ -action,

5. (v) $\gamma _i \in \Gamma $ for each i.

The corresponding generator of $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G^{\operatorname {\mathrm {alg}}})$ will be denoted by $S_{I, W, x, \xi , (\gamma _i)_{i \in I}} \in \operatorname {\mathrm {Exc}}(\Gamma , {}^L G^{\operatorname {\mathrm {alg}}})$ .

2.4.2 Relations

Next, we describe the relations.

1. (i) $S_{\emptyset , x, \xi , \emptyset } = \langle x, \xi \rangle $ , an element of $k \subset \operatorname {\mathrm {Exc}}(\Gamma , {}^L G^{\operatorname {\mathrm {alg}}})$ .

2. (ii) For any morphism of $({}^L G_k^{\operatorname {\mathrm {alg}}})^I$ -representations $u \colon W \rightarrow W'$ and functional $\xi ' \in (W')^*$ invariant under the diagonal $\widehat {G}_k$ -action, we have

   (2.4) $$ \begin{align} S_{I, W, x, u^* (\xi'), (\gamma_i)_{i \in I} } = S_{I, W', u(x), \xi', (\gamma_i)_{i \in I}}, \end{align} $$
   where $u^* \colon (W')^* \rightarrow W^*$ denotes the dual to u.

3. (iii) For two tuples $(I_1, W_1, x_1, \xi _1, (\gamma _i^1)_{i \in I_1})$ and $(I_2, W_2, x_2, \xi _2, (\gamma _i^2)_{i \in I_2})$ as in §2.4.1, we have

   (2.5) $$ \begin{align} S_{I_1 \sqcup I_2, W_1 \boxtimes W_2, x_1 \boxtimes x_2, \xi_1 \boxtimes \xi_2, (\gamma_i^1)_{i \in I_1} \times (\gamma_i^2)_{i \in I_2}} = S_{I_1, W_1, x_1, \xi_1, (\gamma_i^1)_{i \in I_1}} \circ S_{I_2, W_2, x_2, \xi_2, (\gamma_i^2)_{i \in I_2}}. \end{align} $$
   Also,
   (2.6) $$ \begin{align} S_{I_1 \sqcup I_2, W_1 \oplus W_2, (x_1, x_2) , \xi_1 \oplus \xi_2, (\gamma_i^1)_{i \in I_1} \times (\gamma_i^2)_{i \in I_2}} = S_{I_1, W_1, x_1, \xi_1, (\gamma_i^1)_{i \in I_1}} + S_{I_2, W_2, x_2, \xi_2, (\gamma_i^2)_{i \in I_2}}. \end{align} $$
   Furthermore, the assignment $(I,W, x, \xi , (\gamma _i)_{i \in I}) \mapsto S_{I, W, x, \xi , (\gamma _i)_{i \in I}} \in \operatorname {\mathrm {Exc}}(\Gamma , {}^L G^{\operatorname {\mathrm {alg}}})$ is k-linear in x and $\xi $ .

4. (iv) Let $\zeta \colon I \rightarrow J$ be a map of finite sets. Suppose $W \in \operatorname {\mathrm {Rep}}(({}^L G)^I)$ , $x \in W^{\Delta (\widehat {G})}$ , $\xi \colon (W^*)^{\Delta (\widehat {G})}$ , and $(\gamma _j)_{j \in J} \in \Gamma ^J$ . Letting $W^{\zeta }$ be the restriction of W under the functor $\operatorname {\mathrm {Rep}}(({}^L G)^I) \rightarrow \operatorname {\mathrm {Rep}}(({}^L G)^J)$ induced by $\zeta $ , we have

   (2.7) $$ \begin{align} S_{J, W^{\zeta}, x, \xi, (\gamma_j)_{j \in J}}= S_{I, W, x, \xi, (\gamma_{\zeta(i)})_{i \in I}}. \end{align} $$

5. (v) Let and be the natural counit and unit. Conflating x with a $\widehat {G}$ -invariant map and similarly for $\xi $ , we have

   (2.8) $$ \begin{align} S_{I, W, x, \xi, (\gamma_i (\gamma^{\prime}_i)^{-1} \gamma_i")_{i \in I}} = S_{I \sqcup I \sqcup I, W \boxtimes W^* \boxtimes W, \delta_W \boxtimes x, \xi \boxtimes \operatorname{\mathrm{ev}}_W, (\gamma_i)_{i \in I} \times (\gamma_i')_{i \in I} \times (\gamma_i")_{i \in I}}. \end{align} $$

6. (vi) For $J \subset I$ , if W is inflated from a representation of $({}^L G^{\operatorname {\mathrm {alg}}})^J \times \Gamma ^{I \setminus J}$ , then we have

   $$\begin{align*}S_{I, W, x, \xi, (\gamma_i)_{i \in I}} = S_{J, W|_{({}^L G^{\operatorname{\mathrm{alg}}})^J }, ((1_j)_{j \in J}, (\gamma_i)_{i \in I \setminus J}) \cdot x , \xi, (\gamma_j)_{j \in J}}. \end{align*}$$

2.4.3 Relation between the presentations

The two presentations in §2.2 and §2.4 are related as follows. The generator $S_{I, W, x, \xi , (\gamma _i)_{i \in I}}$ corresponds to $S_{I, f_{x, \xi }, (\gamma _i)_{i \in I}}$ , where $f_{x, \xi }$ is the function on $({}^L G_k)^I$ given by $(g_i)_{i \in I} \mapsto \langle \xi , (g_i)_{i \in I} \cdot x \rangle $ . The assumptions on $\xi $ and x imply that $f_{x, \xi }$ is invariant under the left and right diagonal $\widehat {G}_k$ -actions. The relations in §2.4.2 imply that $S_{I, W, x, \xi , (\gamma _i)_{i\in I}}$ depends only on $f_{x, \xi }$ (and not on the choice of $x, \xi $ ) by [Reference LafforgueLaf18, Lemme 10.6].

3.3.1 Pullback

Since $f^* \colon D^b_{\sigma }(S^{\sigma }; k) \rightarrow D^b_{\sigma }(Y^{\sigma }; k)$ preserves stalks, it preserves flat and perfect complexes and so descends to the Tate category to induce $f^* \colon \operatorname {\mathrm {Shv}}(S^{\sigma }; \mathcal {T}_{\Lambda }) \rightarrow \operatorname {\mathrm {Shv}}(Y^{\sigma }; \mathcal {T}_{\Lambda })$ and $f^* \colon \operatorname {\mathrm {Perf}}(S^{\sigma }; \mathcal {T}_{\Lambda }) \rightarrow \operatorname {\mathrm {Perf}}(Y^{\sigma }; \mathcal {T}_{\Lambda })$ .

3.3.2 Proper pushforward

By Lemma 3.9, $Rf_!\colon D^b(Y^{\sigma }; \Lambda [\sigma ]) \rightarrow D^b(S^{\sigma }; \Lambda [\sigma ])$ descends to

$$\begin{align*}Rf_! \colon \operatorname{\mathrm{Shv}}(Y^{\sigma}; \mathcal{T}_{\Lambda}) \rightarrow \operatorname{\mathrm{Shv}}(S^{\sigma}; \mathcal{T}_{\Lambda}). \end{align*}$$

Proposition 3.12. The following diagram commutes:

Proof. We may as well replace S by $S^{\sigma }$ and thus assume that the $\sigma $ -action on S is trivial. Let $\mathcal {F} \in D_\sigma ^b(Y;\Lambda )$ . Denoting $i \colon Y^{\sigma } \hookrightarrow Y$ and j the inclusion of the open complement, we have a distinguished triangle in $D^b_{\sigma }(Y; \Lambda )$ :

$$\begin{align*}j_! j^* \mathcal{F} \rightarrow \mathcal{F} \rightarrow i_* i^* \mathcal{F}. \end{align*}$$

Abbreviate $f^{\sigma } := f \circ i \colon Y^{\sigma } \rightarrow S$ . By definition, $\sigma $ acts freely on U, which implies that $Rf_! \circ (j_! j^* \mathcal {F}) \in \operatorname {\mathrm {Flat}}^b(S; \Lambda [\sigma ])$ by Corollary 3.11. Hence, the cone of $Rf_! \mathcal {F} \rightarrow Rf^{\sigma }_! (i^* \mathcal {F})$ lies in $\operatorname {\mathrm {Flat}}^b(S; \Lambda [\sigma ])$ and therefore becomes $0$ in $\operatorname {\mathrm {Shv}}(S; \mathcal {T}_{\Lambda })$ . Hence, we have

$$\begin{align*}\mathbb{T}^*(Rf_! \mathcal{F}) \cong \mathbb{T}^* ( Rf^{\sigma}_! (i^* \mathcal{F})) \cong Rf_! \operatorname{\mathrm{Psm}}(\mathcal{F}) \in \operatorname{\mathrm{Shv}}(S; \mathcal{T}_{\Lambda}), \end{align*}$$

which exactly expresses the desired commutativity.

3.4.1 Tate cohomology of complexes

The exact sequence of $\Lambda [\sigma ]$ -modules

$$\begin{align*}0 \rightarrow \Lambda \rightarrow \Lambda[\sigma] \xrightarrow{1-\sigma} \Lambda[\sigma] \rightarrow \Lambda \rightarrow 0 \end{align*}$$

induces a morphism

(3.5) $$ \begin{align} \Lambda \rightarrow \Lambda[2] \in D^b(\Lambda[\sigma]). \end{align} $$

Given a bounded-below complex of $\Lambda [\sigma ]$ -modules $C^{\bullet }$ , we define its Tate cohomology as

$$ \begin{align*} T^i(C^{\bullet}) &= \varinjlim_{n \rightarrow \infty} \operatorname{\mathrm{Hom}}_{D(\Lambda[\sigma])}(\Lambda, C^{\bullet}[i+2n]), \end{align*} $$

where the transition maps are those induced by (3.5).

Evidently, $T^i(C^{\bullet })$ is 2-periodic in i. It is clear that this construction descends to the derived category, so we can view Tate cohomology as a collection of functors

$$\begin{align*}T^i \colon D^b(\Lambda[\sigma]) \rightarrow \mathrm{Mod}_{/T^0(\Lambda)}. \end{align*}$$

Now we specialize to the case where $\Lambda = k $ . Note that by [Sta20, Tag 051E], a module over $k[\sigma ]$ is flat if and only if it is free. Since Tate cohomology of free $k[\sigma ]$ -complexes vanishes (by inspection), this construction further factors through the Tate category, inducing

$$\begin{align*}T^i \colon \operatorname{\mathrm{Shv}}(\operatorname{\mathrm{pt}}; \mathcal{T}_{k}) \rightarrow \mathrm{Vect}_{/k}. \end{align*}$$

Remark 3.13. If $C^{\bullet }$ is a bounded $k[\sigma ]$ -module, which will always be the case for us in practice, then we may regard $C^{\bullet } \in D(\operatorname {\mathrm {pt}}; k[\sigma ])$ and the argument of [Reference Leslie and LonerganLL21, Proposition 4.5.1] gives a natural isomorphism

$$\begin{align*}T^i(C^{\bullet}) \cong \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{Shv}}(\operatorname{\mathrm{pt}}; \mathcal{T}_k)}(k, \mathbb{T}^* C^{\bullet}[i]). \end{align*}$$

The following is obvious from the definition but important enough to record.

3.4.2 Tate cohomology sheaves

If S has the trivial $\sigma $ -action, then $\operatorname {\mathrm {Shv}}(S; \mathcal {T}_\Lambda )$ is defined. Given $\mathcal {F} \in D^b(S; \Lambda [\sigma ])$ , we define Tate cohomology sheaves

$$\begin{align*}T^i \mathcal{F} := \varinjlim_{n \rightarrow \infty} \mathcal{H}om_{D(S; \Lambda[\sigma])}(\Lambda, \mathcal{F}[i+2n]), \end{align*}$$

where the transition maps are induced by the map $\Lambda \rightarrow \Lambda [2] \in D(S; \Lambda [\sigma ])$ pulled back from (3.5). The $T^i\mathcal {F}$ are étale sheaves of $T^0\Lambda $ -modules, where $T^0\Lambda $ is the 0th Tate cohomology of $\Lambda $ viewed as a trivial $\sigma $ -module.

For $\Lambda = k$ , we also have the description

$$\begin{align*}T^i \mathcal{F} \cong \mathcal{H}om_{\operatorname{\mathrm{Shv}}(S; \mathcal{T}_k)}(k, \mathbb{T}^* \mathcal{F}[i]) \end{align*}$$

on S, which is an étale sheaf of $T^0(k) = k$ -modules.

3.4.3 Tate cohomology for a morphism

For $\mathcal {F} \in D^b_{\sigma }(Y; k)$ , we have $Rf_! \mathcal {F} \in D^b_{\sigma }(S; k)$ . If S has the trivial $\sigma $ -action, then we can form $T^i Rf_! \mathcal {F}$ , which we call the ‘relative Tate cohomology of $\mathcal {F}$ ’.

If S is the spectrum of a separably closed field equipped with the trivial $\sigma $ -action, then we will abbreviate $T^i(Y;\mathcal {F}) := T^i (Rf_! \mathcal {F})$ and call it the ‘Tate cohomology of Y with coefficients in $\mathcal {F}$ ’.

3.4.4 The long exact sequence for Tate cohomology

Given a distinguished triangle $\mathcal {F}' \rightarrow \mathcal {F} \rightarrow \mathcal {F}" \in D^b_\sigma (Y;k)$ , we have a long exact sequence

3.4.5 Equivariant localization

We explain that Proposition 3.12 encompasses the classical equivariant localization theorems of ‘Smith theory’ (e.g., [Reference QuillenQui71, Theorem 4.2]). Proposition 3.12 says that for $\mathcal {F} \in D_{\sigma }^b(Y; k)$ , we have

$$\begin{align*}\operatorname{\mathrm{Psm}} (Rf_! \mathcal{F}) \cong (Rf|_{Y^{\sigma}})_! \operatorname{\mathrm{Psm}}(\mathcal{F}) \in \operatorname{\mathrm{Shv}}(S^\sigma;\mathcal{T}_{k}). \end{align*}$$

In particular, if L is the spectrum of a separably closed with the trivial action of $\sigma $ , we get

(3.6) $$ \begin{align} \mathbb{T}^* R\Gamma_c(Y; \mathcal{F}) \cong R\Gamma_c (Y^{\sigma}; \operatorname{\mathrm{Psm}}(\mathcal{F})). \end{align} $$

4.6.1 The base change setup

We now specialize the situation a bit further: H is any reductive group over $\mathbf {F}$ and $G = H^p$ . We let $\sigma $ act on G by cyclic rotation, sending the ith factor to the $(i+1)$ st (mod p) factor. Then it is clear that the stratification on $\operatorname {\mathrm {Gr}}_G$ by $L^+G$ -orbits induces by restriction the stratification on $\operatorname {\mathrm {Gr}}_H$ by $L^+H$ -orbits.

Evidently the ‘diagonal’ embedding $H \hookrightarrow G$ realizes H as the fixed points of G under the automorphism $\sigma $ . This map $H \hookrightarrow G$ also induces a diagonal map $\operatorname {\mathrm {Gr}}_H \rightarrow \operatorname {\mathrm {Gr}}_G$ .

Henceforth, we assume that p is odd and good for $\widehat {G}$ so that the results of §4.5 apply.

We aim to give a ‘geometric’ description of the corresponding functor under the Geometric Satake equivalence, $\operatorname {\mathrm {Parity}}_{L^+G}(\operatorname {\mathrm {Gr}}_G; k) \rightarrow \operatorname {\mathrm {Parity}}_{L^+H}(\operatorname {\mathrm {Gr}}_H;k)$ , in terms of Smith theory. (Of course, one could give an ‘ad hoc’ description using that $G = H^p$ . The point is to define a functor that does not make reference to this, which will then generalize well, using descent, to the situation where $G = \operatorname {\mathrm {Res}}_{\mathbf {E}/\mathbf {F}}(H)$ for a nontrivial field extension $\mathbf {E}/\mathbf {F}$ .)

Definition 4.12. Given $\mathcal {F} \in \operatorname {\mathrm {P}}_{L^+G}(\operatorname {\mathrm {Gr}}_G; \Lambda )$ , we define

$$\begin{align*}\operatorname{\mathrm{Nm}}(\mathcal{F}) := \mathcal{F} \star {}^{\sigma} \mathcal{F} \star \ldots \star {}^{\sigma^{p-1}} \mathcal{F} \in \operatorname{\mathrm{P}}_{L^+G \rtimes \sigma}(\operatorname{\mathrm{Gr}}_G; \Lambda), \end{align*}$$

equipped with the $\sigma $ -equivariant structure coming from the commutativity constraint for $(\operatorname {\mathrm {P}}_{L^+G}(\operatorname {\mathrm {Gr}}_G;\Lambda ), \star )$ :

(4.4) $$ \begin{align} {}^{\sigma}\operatorname{\mathrm{Nm}}(\mathcal{F}) = {}^{\sigma} \mathcal{F} \star \ldots \star {}^{\sigma^{p-1}} \mathcal{F} \star \mathcal{F} \xrightarrow{\sim} \mathcal{F} \star {}^{\sigma} \mathcal{F} \star \ldots \star {}^{\sigma^{p-1}} \mathcal{F} = \operatorname{\mathrm{Nm}}(\mathcal{F}). \end{align} $$

Using the realization functor $\operatorname {\mathrm {P}}_{L^+G \rtimes \sigma }(\operatorname {\mathrm {Gr}}_G; \Lambda ) \rightarrow D_{L^+G \rtimes \sigma }(\operatorname {\mathrm {Gr}}_G; \Lambda )$ , we view $\operatorname {\mathrm {Nm}}(\mathcal {F}) \in D_{L^+G \rtimes \sigma }(\operatorname {\mathrm {Gr}}_G;\Lambda )$ (so that we may apply the Smith functor, for example). Equipping a general object of $D_{L^+G}(\operatorname {\mathrm {Gr}}_G;\Lambda )$ with a $\sigma $ -equivariant structure is much more involved than just specifying isomorphisms (4.4) (satisfying cocycle conditions), so we emphasize that we construct $\operatorname {\mathrm {Nm}}(\mathcal {F})$ first as a $\sigma $ -equivariant perverse sheaf and then apply the realization functor to get a $\sigma $ -equivariant object of $D_{L^+G}(\operatorname {\mathrm {Gr}}_G;\Lambda )$ .

Proof. If $\mathcal {F}$ has a finite $L^+G$ -equivariant filtration whose associated graded satisfies the hypotheses of the Lemma, then the statement of Lemma for $\mathcal {F}$ can be checked on the associated graded. Since $\operatorname {\mathrm {Gr}}_G^{\underline {\lambda }}$ is a product of homogeneous spaces for (a finite type quotient of) $L^+ H$ , there is a finite $L^+G$ -equivariant filtration of $\mathcal {F}$ with associated graded of being a direct sum of sheaves of the form $\mathcal {F}_1 \boxtimes \ldots \boxtimes \mathcal {F}_p$ , where each $\mathcal {F}_i \in \operatorname {\mathrm {P}}_{L^+H}(\operatorname {\mathrm {Gr}}_H; \mathbb {O})$ . Hence, we reduce to the case where $\mathcal {F}$ is itself of this form. Then

$$\begin{align*}\operatorname{\mathrm{Nm}}(\mathcal{F}) \approx (\mathcal{F}_1 \star \mathcal{F}_2 \star \ldots \star \mathcal{F}_p) \boxtimes (\mathcal{F}_2 \star \ldots \star \mathcal{F}_p \star \mathcal{F}_1) \boxtimes \ldots \boxtimes (\mathcal{F}_p \star \mathcal{F}_1 \star \ldots \star \mathcal{F}_{p-1}), \end{align*}$$

with $\sigma $ acting by rotating the tensor factors and the $\sigma $ -equivariant structure coming from the commutativity constraint.

Write $\mathcal {F}' := \mathcal {F}_1 \star \mathcal {F}_2 \star \ldots \star \mathcal {F}_p \in \operatorname {\mathrm {P}}_{L^+H}(\operatorname {\mathrm {Gr}}_H; \mathbb {O})$ . Since i may be identified with the diagonal embedding $\operatorname {\mathrm {Gr}}_H \hookrightarrow \operatorname {\mathrm {Gr}}_H^p$ , we have $i^*(\operatorname {\mathrm {Nm}} \mathcal {F}) \approx (\mathcal {F}')^{\otimes p}$ , with $\sigma $ -equivariant structure given by cyclic rotation of the tensor factors. In particular, the stalk of $i^* (\operatorname {\mathrm {Nm}} \mathcal {F})$ at $x \in \operatorname {\mathrm {Gr}}_H$ is the tensor-induction of the stalk of $\mathcal {F}^{\prime }_x$ from $\mathbb {O}$ to $\mathbb {O}[\sigma ]$ . Hence, it suffices to prove that any cohomology sheaf of such a tensor induction has an $\mathbb {O}[\sigma ]$ -equivariant filtration by either trivial or free $\mathbb {O}[\sigma ]$ -modules. This is verified by explicit inspection: choosing a basis for $\mathcal {H}^j(\mathcal {F}^{\prime }_x)$ , the induced basis of $\mathcal {H}^j(\mathcal {F}^{\prime }_x)^{\otimes p}$ is grouped into either trivial or free orbits under the $\sigma $ -action.

The argument for (ii) is completely analogous (alternatively, we could deduce it simply by applying Verdier duality to (i)).

4.6.2 Smith theory for parity sheaves

We return momentarily to the general setup for Smith theory: Y is a variety over $\mathbf {F}$ with an admissible $\sigma $ -action and $Z = Y^{\sigma }$ . We assume that $\mathbf {F}$ is separably closed and the stratification S on Y satisfies the JMW condition.

Proof. This theorem is closely related to [Reference Leslie and LonerganLL21, Theorem 6.1.1], but loc. cit. imposes the stronger condition that the $\sigma $ -action on all stalks is trivial. This is satisfied in their application (to the loop-rotation action), but not in ours, so we need to redo the argument in the requisite generality.

Let $Z = Y^{\sigma }$ and take the induced stratification on Z. Let $i \colon Z \rightarrow Y$ , $i_{\lambda }^Y \colon Y_{\lambda } \hookrightarrow Y$ , $i_{\lambda }^Z \colon Z_{\lambda } \hookrightarrow Z$ , $i^{\lambda } \colon Z_{\lambda } \hookrightarrow Y_{\lambda }$ . Without loss of generality, suppose $\mathcal {E}$ is an even complex on Y. We are given that $(i_{\lambda }^Y)^{?} \mathcal {E}$ has $\mathbb {O}$ -free cohomology sheaves supported in degrees congruent to $\dagger _Y(\lambda )$ mod $2$ , where $? \in \{*, !\}$ ; we want to show that $(i_{\lambda }^Z)^? \operatorname {\mathrm {Psm}}(\mathcal {E})$ has Tate-cohomology sheaves supported in degrees congruent to $\dagger _Z(\lambda )$ mod $2$ . Unraveling the definitions, we have

$$ \begin{align*} (i_{\lambda}^Z)^* \operatorname{\mathrm{Psm}}(\mathcal{E}) & =(i_{\lambda}^Z)^* \mathbb{T}^* i^* \mathcal{E} \cong \mathbb{T}^* (i_{\lambda}^Z)^* i^* \mathcal{E} \cong \mathbb{T}^* (i^{\lambda})^* (i_{\lambda}^Y)^*\mathcal{E}. \end{align*} $$

Similarly, using Lemma 3.8, we have

(4.5) $$ \begin{align} (i_{\lambda}^Z)^! \operatorname{\mathrm{Psm}}(\mathcal{E}) \cong \mathbb{T}^* (i^{\lambda})^! (i_{\lambda}^Y)^! \mathcal{E}. \end{align} $$

By hypothesis, $(i_{\lambda }^Y)^* \mathcal {E}$ has its cohomology sheaves supported in degrees congruent to $\dagger _Y(\lambda ) \pmod {2}$ . So the stalks of $(i^{\lambda })^* (i_{\lambda }^Y)^* \mathcal {E}$ are supported in degrees congruent to $\dagger _Y(\lambda ) \pmod {2}$ , and we must verify that their Tate cohomology groups are also supported in degrees of a single parity.

By assumption (*), all the stalks have an $\mathbb {O}[\sigma ]$ -stable filtration with associated graded being a direct sum of trivial or free $\mathbb {O}[\sigma ]$ -modules. For trivial $\mathbb {O}[\sigma ]$ -modules, the odd Tate cohomology groups vanish by (4.1), while for free $\mathbb {O}[\sigma ]$ -modules, all the Tate cohomology groups vanish. Hence, all odd Tate cohomology groups vanish by the long exact sequence for Tate cohomology (§3.4.4). This shows that the Tate cohomology sheaves of $(i^{\lambda })^* (i_{\lambda }^Y)^* \mathcal {E}$ are supported in degrees congruent to $\dagger _Y(Y_\lambda ) \pmod {2}$ .

To show that $(i^{\lambda })^! (i_{\lambda }^Y)^! \mathcal {E}$ also has Tate cohomology sheaves supported in degrees congruent to $\dagger _Y(\lambda ) \pmod {2}$ , we make a similar analogous argument using (4.5) instead. This shows that $ \mathbb {T}^* (i^{\lambda })^* (i_{\lambda }^Y)^! \mathcal {E}$ lies in degrees congruent to $\dagger _Y(Y_\lambda )\ \pmod {2}$ , and then we conclude by observing that $(i^{\lambda })^* (i_{\lambda }^Y)^! \mathcal {E}$ differs from $(i^{\lambda })^! (i_{\lambda }^Y)^! \mathcal {E}$ by an even shift (and twist) by the Gysin isomorphism, which applies because the strata are assumed to be smooth (noting that the smoothness of $Z_\lambda $ follows from the smoothness of $Y_\lambda $ by [Reference Conrad, Gabber and PrasadCGP15, Proposition A.8.11], so $Z_\lambda \hookrightarrow Y_\lambda $ is a regular embedding).

For an $\mathbb {O}$ -linear abelian category $\mathsf {C}$ , with all Hom-spaces being free $\mathbb {O}$ -modules, we abbreviate

$$\begin{align*}\mathsf{C} \otimes_{\mathbb{O}} k := \mathsf{C} \otimes_{\mathbb{O}-\operatorname{\mathrm{Mod}}} (k-\operatorname{\mathrm{Mod}}). \end{align*}$$

Lemma 4.16. Suppose that all the strata $Y_{\lambda }$ are simply connected and all parity sheaves $\mathcal {E}(\lambda , \mathcal {L})$ exist for all $Y_\lambda \in S$ . Then we have that

$$\begin{align*}\mathrm{Parity}^0_{S, \sigma}(Y; \mathbb{O}) \otimes_{\mathbb{O}} k \xrightarrow{\sim} \mathrm{Parity}^0_{S, \sigma}(Y; k). \end{align*}$$

4.6.3 The base change functor

We return now to the base change setup of §4.6.1, with $\mathbf {F}$ separably closed. Let $\mathcal {F} \in \operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_G; \mathbb {O})$ . Then $\mathcal {F} \in \operatorname {\mathrm {P}}_{L^+G}(\operatorname {\mathrm {Gr}}_G; \mathbb {O})$ is perverse since p is good for $\widehat {G}$ (this is a part of Theorem 4.8), and $\operatorname {\mathrm {Nm}}(\mathcal {F}) \in \operatorname {\mathrm {Parity}}^0_{L^+G \rtimes \sigma }(\operatorname {\mathrm {Gr}}_G; \mathbb {O})$ is a parity sheaf by [Reference Juteau, Mautner and WilliamsonJMW16, Theorem 1.5]. Furthermore, the $\sigma $ -equivariant structure on $\operatorname {\mathrm {Nm}}(\mathcal {F})$ satisfies the assumption (*) of Proposition 4.15 by Lemma 4.14.

The Schubert cells of $\operatorname {\mathrm {Gr}}_G$ are indexed by tuples $\lambda := (\lambda _1, \ldots , \lambda _p) \in X_*(G)^+$ , with each $\lambda _i \in X_*(H)^+$ , and we have

$$\begin{align*}\begin{cases} \operatorname{\mathrm{Gr}}_G^{\lambda} \cap \operatorname{\mathrm{Gr}}_H = \operatorname{\mathrm{Gr}}_H^{\lambda_1} & \lambda = (\lambda_1, \ldots, \lambda_1), \\ \operatorname{\mathrm{Gr}}_G^{\lambda} \cap \operatorname{\mathrm{Gr}}_H = \emptyset & \text{otherwise.} \end{cases} \end{align*}$$

We claim that as long as $p>2$ , the induced pariversity coincides with the dimension pariversity on $\operatorname {\mathrm {Gr}}_H$ – that is, for $\lambda = (\lambda _1, \ldots , \lambda _p) \in X_*(G)^+$ , we have

$$\begin{align*}\dim \operatorname{\mathrm{Gr}}_H^{\lambda_1} \equiv \dim \operatorname{\mathrm{Gr}}_G^{\lambda}\quad \pmod{2}. \end{align*}$$

This will imply the following:

1. 1. We may apply Proposition 4.15 to deduce that $\operatorname {\mathrm {Psm}}(\operatorname {\mathrm {Nm}}(\mathcal {F})) \in \operatorname {\mathrm {Parity}}_{(L^+H)}(\operatorname {\mathrm {Gr}}_H; \mathcal {T}_{\mathbb {O}})$ is Tate-parity with respect to the dimension pariversity on $\operatorname {\mathrm {Gr}}_H$ .

2. 2. $\operatorname {\mathrm {Psm}}(\operatorname {\mathrm {Nm}}(\mathcal {F})) \in \operatorname {\mathrm {Parity}}^0_{(L^+H)}(\operatorname {\mathrm {Gr}}_H; \mathcal {T}_{\mathbb {O}})$ (i.e., is normalized).

To prove the claim, we may focus on the case where $\lambda _1 = \ldots = \lambda _p$ or else the statement is vacuous. By [Reference ZhuZhu17, Proposition 2.1.5], we have $\dim \operatorname {\mathrm {Gr}}_G^{\lambda } = \langle 2\rho _G, \lambda \rangle $ . So we just have to verify that $\langle 2 \rho _G , (\lambda _1, \ldots , \lambda _1) \rangle \equiv \langle 2\rho _H, \lambda _1 \rangle\ \pmod {2}$ . Indeed, $\rho _G = (\rho _H, \ldots , \rho _H)$ , so $\langle 2 \rho _G , (\lambda _1, \ldots , \lambda _1) \rangle = p \langle 2 \rho _H, \lambda _1 \rangle $ , and p is odd.

Thanks to points (1) and (2) above, we can apply the lifting functor L to $\operatorname {\mathrm {Psm}}(\operatorname {\mathrm {Nm}}(\mathcal {F}))$ . By Lemma 4.16, the composite functor $L \circ \operatorname {\mathrm {Psm}} \circ \operatorname {\mathrm {Nm}}$ factors uniquely through a functor $\operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_G;k) \rightarrow \mathrm {Parity}^0_{L^+H}(\operatorname {\mathrm {Gr}}_H; k)$ .

Construction 4.17 (Frobenius twist of categories).

Let $\operatorname {\mathrm {Frob}}$ be the absolute Frobenius of k. Given a k-linear category $\mathsf {C}$ , there is another k-linear category $\mathsf {C}^{(p)} := \mathsf {C} \otimes _{k, \operatorname {\mathrm {Frob}}} k$ . Concretely, it is equivalent to the category which has the same objects as $\mathsf {C}$ , and morphisms

$$\begin{align*}\operatorname{\mathrm{Hom}}_{\mathsf{C}^{(p)}}(x,y) = \operatorname{\mathrm{Hom}}_{\mathsf{C}}(x,y)^{(p)} := \operatorname{\mathrm{Hom}}_{\mathsf{C}}(x,y) \otimes_{k, \operatorname{\mathrm{Frob}}} k. \end{align*}$$

The tautological map $\operatorname {\mathrm {Hom}}_{\mathsf {C}}(x,y) \rightarrow \operatorname {\mathrm {Hom}}_{\mathsf {C}}(x,y)^{(p)}$ is $\operatorname {\mathrm {Frob}}$ -semilinear over k and induces an equivalence $\operatorname {\mathrm {Frob}}_{\mathsf {C}} \colon \mathsf {C} \xrightarrow {\sim } \mathsf {C}^{(p)}$ , which is $\operatorname {\mathrm {Frob}}$ -semilinear. The functor $\operatorname {\mathrm {Frob}}_{\mathsf {C}} \colon \mathsf {C} \rightarrow \mathsf {C}^{(p)}$ is characterized by the universal property that any $\operatorname {\mathrm {Frob}}$ -semilinear functor $F \colon \mathsf {C} \rightarrow \mathsf {D}$ (meaning a functor between k-linear categories that is $\operatorname {\mathrm {Frob}}$ -semilinear over k on morphisms) factors uniquely through a k-linear functor $\mathsf {C}^{(p)} \rightarrow \mathsf {D}$ .

Now, given a presentation

(4.6) $$ \begin{align} F_0 \colon \mathsf{C} \cong \mathsf{C}_0 \otimes_{\mathbf{F}_p} k := \mathsf{C}_{0} \otimes_{\operatorname{\mathrm{Vect}}_{/\mathbf{F}_p}} \operatorname{\mathrm{Vect}}_{/k} \end{align} $$

for some $\mathbf {F}_p$ -linear category $\mathsf {C}_0$ , then there is another k-linear equivalence $ \mathsf {C} \xrightarrow {\sim } \mathsf {C}^{(p)}$ , which with reference to (4.6) is the tensor product of $\operatorname {\mathrm {Id}}_{\mathsf {C}_0}$ with the k-linear equivalence $\operatorname {\mathrm {Vect}}_{/k}^{(p)} \cong \operatorname {\mathrm {Vect}}_{/k}$ induced by the k-linear isomorphism $k \otimes _{\operatorname {\mathrm {Frob}}, k} k \cong k$ . Therefore, (4.6) induces a $\operatorname {\mathrm {Frob}}$ -semilinear equivalence $\operatorname {\mathrm {Frob}}_{F_0} \colon \mathsf {C} \xrightarrow {\sim } \mathsf {C}$ .

Definition 4.18. We define

$$\begin{align*}\mathsf{BC}^{(p)} \colon \operatorname{\mathrm{Parity}}^0_{L^+G}(\operatorname{\mathrm{Gr}}_G; k) \rightarrow \operatorname{\mathrm{Parity}}_{L^+H}^0(\operatorname{\mathrm{Gr}}_H; k) \end{align*}$$

to be the functor unique filling in the commutative diagram

(4.7)

One more step is required to define what we call the base change functor $\mathsf {BC}$ . Note that $\mathsf {BC}^{(p)}$ is $\operatorname {\mathrm {Frob}}$ -semilinear over k; we wish to linearize it. It is evident from the definitions that the equivalence $D(\operatorname {\mathrm {Gr}}_G; k) \cong D(\operatorname {\mathrm {Gr}}_G; \mathbf {F}_p) \otimes _{\mathbf {F}_p} k$ induces

$$\begin{align*}F_0 \colon \operatorname{\mathrm{Parity}}^0_{L^+G}(\operatorname{\mathrm{Gr}}_G; k) \cong \operatorname{\mathrm{Parity}}^0_{L^+G}(\operatorname{\mathrm{Gr}}_G; \mathbf{F}_p) \otimes_{\mathbf{F}_p} k. \end{align*}$$

Let $\operatorname {\mathrm {Frob}}_p := \operatorname {\mathrm {Frob}}_{F_0} \colon \operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_G; k) \xrightarrow {\sim } \operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_G; k)$ be the k-semilinear equivalence induced by $F_0$ , as explained in Construction 4.17. We define

$$\begin{align*}\mathsf{BC} := \mathsf{BC}^{(p)} \circ \operatorname{\mathrm{Frob}}_p^{-1}: \operatorname{\mathrm{Parity}}^0_{L^+G}(\operatorname{\mathrm{Gr}}_G; k) \rightarrow \mathrm{Parity}^0_{L^+H}(\operatorname{\mathrm{Gr}}_H; k). \end{align*}$$

Theorem 4.20. Let $\operatorname {\mathrm {Res}}_{\mathsf {BC}} \colon \operatorname {\mathrm {Rep}}_k(\widehat {G}) \rightarrow \operatorname {\mathrm {Rep}}_k(\widehat {H})$ be a restriction along the diagonal embedding. We also denote by $\operatorname {\mathrm {Res}}_{\mathsf {BC}}$ the same functor restricted to the subcategories of tilting modules. Footnote ¹¹ The following diagram commutes:

The proof is given in Appendix A.

The triangulated structure on $D_{L^+H}(\operatorname {\mathrm {Gr}}_H; k[\sigma ])$ equips $\operatorname {\mathrm {Perf}}_{L^+H}(\operatorname {\mathrm {Gr}}_G; \mathcal {T}_k)$ with the notion of cone (namely, the image of a cone in $D_{L^+H}(\operatorname {\mathrm {Gr}}_H; k[\sigma ])$ ). We say that a sequence $\mathcal {A} \rightarrow \mathcal {B} \rightarrow \mathcal {C}$ in $D_{L^+H}(\operatorname {\mathrm {Gr}}_H; \mathcal {T}_k)$ is exact if the induced map $\mathrm {Cone}(\mathcal {A} \rightarrow \mathcal {B}) \rightarrow \mathcal {C} $ in $\operatorname {\mathrm {Perf}}_{L^+H}(\operatorname {\mathrm {Gr}}_H; \mathcal {T}_k)$ is an isomorphism.

Lemma 4.21. The composite functor

$$\begin{align*}\operatorname{\mathrm{Rep}}_k(\widehat{G}) \xrightarrow{\operatorname{\mathrm{Frob}}_p^{-1}} \operatorname{\mathrm{Rep}}_k(\widehat{G}) \xrightarrow{\operatorname{\mathrm{Sat}}} \operatorname{\mathrm{P}}_{L^+G}(\operatorname{\mathrm{Gr}}_G; k) \xrightarrow{\operatorname{\mathrm{Nm}}} \operatorname{\mathrm{P}}_{L^+G \rtimes \sigma}(\operatorname{\mathrm{Gr}}_G; k) \xrightarrow{\operatorname{\mathrm{Psm}}} \operatorname{\mathrm{Perf}}_{(L^+H)}(\operatorname{\mathrm{Gr}}_H; \mathcal{T}_k) \end{align*}$$

is exact (i.e., sends exact sequences to exact sequences in the above sense).

The proof requires some notions from Appendix A and will be postponed to §A.6.

4.6.4 Equivariantization and Galois descent

Assuming Theorem 4.20, let us give a few variants related to descent to a ground field which is not separably closed. Suppose H base changed from some subfield $\mathbf {F}_0 \subset \mathbf {F}$ , and $G = \operatorname {\mathrm {Res}}_{\mathbf {E}_0/\mathbf {F}_0} (H_{\mathbf {E}_0})$ for some Galois extension $\mathbf {E}_0/\mathbf {F}_0$ with Galois group $\mathbf {Z}/p\mathbf {Z}$ . Then $G_{\mathbf {F}} \approx (H_{\mathbf {F}})^p$ and $\operatorname {\mathrm {Aut}}(\mathbf {F}/\mathbf {F}_0)$ acts on $H_{\mathbf {F}},G_{\mathbf {F}}$ and therefore also on $\operatorname {\mathrm {Gr}}_{H, {\mathbf {F}}}, \operatorname {\mathrm {Gr}}_{G, {\mathbf {F}}}$ .

Lemma 4.22 (Galois equivariance).

In the situation above, the functor

$$\begin{align*}\mathsf{BC} \colon \operatorname{\mathrm{Parity}}^0_{L^+G}(\operatorname{\mathrm{Gr}}_{G, \mathbf{F}}; k) \rightarrow \operatorname{\mathrm{Parity}}^0_{L^+H}(\operatorname{\mathrm{Gr}}_{H, \mathbf{F}}; k) \end{align*}$$

is equivariant with respect to the action of $\operatorname {\mathrm {Aut}}(\mathbf {F}/\mathbf {F}_0)$ .

We refer to [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10] for the theory of ‘equivariantization’ and ‘de-equivariantization’ of categories. Given a group $\Gamma $ acting on categories $\mathsf {C}, \mathsf {D}$ and a $\Gamma $ -equivariant functor $F \colon \mathsf {C} \rightarrow \mathsf {D}$ , the $\Gamma $ -equivariantization of F is the functor $F^{B \Gamma } \colon \mathsf {C}^{B \Gamma } \rightarrow \mathsf {D}^{B \Gamma }$ . If $\mathsf {C}$ and $\mathsf {D}$ are derived categories of sheaves and F is induced by geometric operations that are $\Gamma $ -equivariant, then the equivariantization construction exists for equivariant derived categories. (We make this remark because if the $\Gamma $ -equivariantization of a derived category is not the same as the $\Gamma $ -equivariant derived category; it is the latter that we want to consider.)

Thanks to Lemma 4.22, the equivariantization of $\mathsf {BC}$ induces

$$\begin{align*}\mathsf{BC}^{B\operatorname{\mathrm{Aut}}(\mathbf{F}/\mathbf{F}_0)} \colon \operatorname{\mathrm{Parity}}^0_{L^+G}(\operatorname{\mathrm{Gr}}_{G, \mathbf{F}}; k)^{B\operatorname{\mathrm{Aut}}(\mathbf{F}/\mathbf{F}_0)} \rightarrow \operatorname{\mathrm{Parity}}^0_{L^+H}(\operatorname{\mathrm{Gr}}_{H, \mathbf{F}}; k)^{B\operatorname{\mathrm{Aut}}(\mathbf{F}/\mathbf{F}_0)}. \end{align*}$$

We define $\operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_{G, \mathbf {F}_0};k) := \operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_{G, \mathbf {F}}; k)^{B\operatorname {\mathrm {Aut}}(\mathbf {F}/\mathbf {F}_0)} $ and similarly for H (note that in §4.1, parity sheaves were only defined for varieties over separably closed fields since the axioms of a JMW stratification would not otherwise be satisfied). We define $\operatorname {\mathrm {Tilt}}_k({}^L G)$ to be the subcategory of $\operatorname {\mathrm {Rep}}_k({}^L G)$ consisting of representations whose restriction to $\widehat {G}$ is tilting and $\mathrm {Tilt}_k({}^L G^{\operatorname {\mathrm {geom}}})$ to be the full subcategory of $\operatorname {\mathrm {Rep}}_k({}^L G^{\operatorname {\mathrm {geom}}})$ consisting of representations whose restriction to $\widehat {G}$ is tilting; then $\mathrm {Tilt}_k({}^L G^{\operatorname {\mathrm {geom}}}) \cong \operatorname {\mathrm {Tilt}}_k(\widehat {G})^{\operatorname {\mathrm {Aut}}(\mathbf {F}/\mathbf {F}_0), \operatorname {\mathrm {geom}}}$ , and similarly for ${}^L H$ .

Then applying $\operatorname {\mathrm {Aut}}(\mathbf {F}/\mathbf {F}_0)$ -equivariantization to Theorem 4.20 yields:

Corollary 4.23. The following diagram is commutative.

4.6.5

Let $H/\mathbf {F}_0$ and $G/\mathbf {F}_0$ be as before. The following compatibility statement will be needed later.

Lemma 4.24. The cube

(4.8)

commutes, where L is lifting functor $L \colon \operatorname {\mathrm {Parity}}^0_{L^+H}(\operatorname {\mathrm {Gr}}_H; \mathcal {T}_{\mathbb {O}}) \rightarrow \operatorname {\mathrm {Parity}}^0_{L^+H}(\operatorname {\mathrm {Gr}}_H; k)$ from §4.4, and the unlabeled arrow is defined as the one that makes the top face commute. (It exists by the universal property of the categorical tensor product.)

Proof. The left face commutes by definition of $\mathbb {F}$ . It is obvious from the definition that the front face commutes. The bottom face commutes by compatibility of $\operatorname {\mathrm {Psm}}$ with tensoring coefficients. The top face commutes by definition.

To see that the right face commutes, consider the diagram

(4.9)

The right square of (4.9) is the right square of the cube (4.8), which we want to show commutes. Since the middle column is a Verdier quotient of the left column, it suffices to show that the outer square of (4.9) commutes. Next, note that the composite of the upper horizontal arrows in (4.9) is the modular reduction functor $\mathbb {F}$ by definition (4.3), so we can factor the outer square of (4.9) as the outer quadrilateral in the diagram below.

Obviously the commutativity of the outer quadilateral follows from the commutativity of the inner rectangle, which is then immediate from the definition of $\mathbb {F}$ .

It remains to show that the back face commutes. Consider juxtaposing the top and back faces of cube (4.8) to get

(4.10)

We want to show that the lower square commutes. The composite vertical arrows on left and right columns are both the modular reduction functor $\mathbb {F}$ from $\mathbb {O}$ -coefficients to k coefficients, so the outer square commutes. The dashed arrow is defined as the k-linearization of $L \circ \operatorname {\mathrm {Psm}}$ , noting that $\operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_G; k) \cong \operatorname {\mathrm {Parity}}^0_{L^+G}(\operatorname {\mathrm {Gr}}_G; \mathbb {O}) \otimes _{\mathbb {O}} k$ . Therefore, the right-then-down (resp. down-then-right) composite functor in the lower square is the k-linearization of the right-then-down (resp. down-then-right) composite functor in the upper square, so the commutativity of the lower square follows from that of the upper square, completing the proof.

5.1.1 Shtukas

Fix a smooth projective curve X over a finite field $\mathbf {F}_{\ell }$ of characteristic $\neq p$ . Let G be a smooth algebraic group scheme over X. We assume that G is generically reductive. Let $\stackrel {\circ }X \hookrightarrow X$ be the locus where G is reductive. For each finite set I, the stack $\operatorname {\mathrm {Sht}}_{G,I}$ has the following functor of points on $\mathbf {F}_{\ell }$ -schemes S:

$$\begin{align*}\operatorname{\mathrm{Sht}}_{G,I} \colon S \mapsto \left\{ \begin{array}{@{}c@{}} (x_i)_{i \in I} \in X^I(S) \\ \mathcal{E} = \acute{\text{e}}\text{tale }G\text{-torsor over }X \times S \\ \varphi \colon \mathcal{E}|_{X \times S - \bigcup_{i \in I} \Gamma_{x_i}} \xrightarrow{\sim} {}^{\tau} \mathcal{E}|_{X \times S - \bigcup_{i \in I} \Gamma_{x_i}} \end{array} \right\}, \end{align*}$$

where $\tau $ is the Frobenius $\operatorname {\mathrm {Frob}}_{\ell }$ on the S factor in $X \times S$ , and ${}^{\tau } \mathcal {E}$ is the pullback of $\mathcal {E}$ under the map $1 \times \tau \colon X \times S \rightarrow X \times S$ .

Geometrically, $\operatorname {\mathrm {Sht}}_{G,I}$ has a Schubert stratification whose strata are Deligne-Mumford stacks locally of finite type. We regard it as an ind-(locally finite type) Deligne-Mumford stack.

There is a map

$$\begin{align*}\pi_I \colon \operatorname{\mathrm{Sht}}_{G,I} \rightarrow X^I \end{align*}$$

projecting a tuple $(\{x_i\}_{i \in I}, \mathcal {E}, \varphi _i)$ to $\{x_i\}_{i \in I}$ . Let $\stackrel {\circ } {\mathrm {Sht}}_{G,I} := {\mathrm {Sht}}_{G,I} \times _{X^I} (\stackrel {\circ }{X})^I$ .

5.1.2 Hecke stack

The Hecke stack $\operatorname {\mathrm {Hk}}_{G,I}$ classifies

$$\begin{align*}\operatorname{\mathrm{Hk}}_{G,I} \colon S \mapsto \left\{ \begin{array}{@{}c@{}} (x_i)_{i \in I} \in X^I(S) \ \\ \mathcal{E} , \mathcal{E}' = \acute{\text{e}}\text{tale }G\text{-torsors over }X \times S \\ \varphi \colon \mathcal{E}|_{X \times S - \bigcup \Gamma_{x_i}} \xrightarrow{\sim} \mathcal{E}'|_{X \times S - \bigcup \Gamma_{x_i}} \end{array} \right\}. \end{align*}$$

Recall that $G \rightarrow X$ is reductive over $\stackrel {\circ }X$ . Let $\stackrel {\circ }{\mathrm {Hk}}_{G,I} := {\mathrm {Hk}}_{G,I} \times _{X^I} (\stackrel {\circ }{X})^I$ . The Geometric Satake equivalence provides a functor $\operatorname {\mathrm {Rep}}_k(({}^L G)^I) \rightarrow D(\stackrel {\circ }{\mathrm {Hk}}_{G,I};k)$ , which we normalize as in [Reference LafforgueLaf18, Theorem 0.9].

5.1.3 Satake sheaves

There is a map $\operatorname {\mathrm {Sht}}_{G,I} \rightarrow \operatorname {\mathrm {Hk}}_{G,I}$ sending $(\{x_i\}_{i \in I}, \mathcal {E}, \varphi )$ to $(\{x_i\}_{i \in I}, \mathcal {E}, {}^{\tau } \mathcal {E}, \varphi )$ . Composing with the $*$ -pullback through $\operatorname {\mathrm {Sht}}_{G,I} \rightarrow \operatorname {\mathrm {Hk}}_{G,I}$ induces a functor

$$\begin{align*}\operatorname{\mathrm{Sat}}^{\operatorname{\mathrm{geom}}} \colon \operatorname{\mathrm{Rep}}_k(\widehat{G}^I)^{B\operatorname{Gal}(F^s/F), \operatorname{\mathrm{geom}}}\rightarrow D^b(\stackrel{\circ}{\mathrm{Sht}}_{G,I};k). \end{align*}$$

Finally, we may identify $\operatorname {\mathrm {Rep}}_k(({}^L G^{\operatorname {\mathrm {alg}}})^I) \xrightarrow {\sim } \operatorname {\mathrm {Rep}}_k(\widehat {G}^I)^{B\operatorname {Gal}(F^s/F), \operatorname {\mathrm {geom}}}$ as in §2.1.4, giving a functor (cf. [Reference LafforgueLaf18, Theorem 0.11])

$$\begin{align*}\operatorname{\mathrm{Sat}} \colon \operatorname{\mathrm{Rep}}_k(({}^L G^{\operatorname{\mathrm{alg}}})^I) \rightarrow D^b(\stackrel{\circ}{\mathrm{Sht}}_{G,I};k). \end{align*}$$

The Schubert stratification is defined by the support of the sheaves in the image of $\operatorname {\mathrm {Sat}}$ , with the closure relations corresponding to the Bruhat order. (In particular, $\operatorname {\mathrm {Sat}}$ lands in the derived category of sheaves constructible with respect to the Schubert stratification on $\stackrel {\circ }{\mathrm {Sht}}_{G,I}$ .)

5.1.4 Level structures

For $D \subset X$ a closed finite subscheme, there are level covers $\operatorname {\mathrm {Sht}}_{G,D,I} \rightarrow \operatorname {\mathrm {Sht}}_{G,I}|_{(X \setminus D)^I}$ which parametrize the additional datum of a trivialization of $\mathcal {E}$ over $S \times D$ compatible with $\tau $ and $\varphi $ . Note that by definition, the ‘legs’ $\{x_i\}_{i \in I} \in (X \setminus D)(S)^I$ avoid D.

5.1.5 Iterated shtukas

Let $I_1, \ldots , I_r$ be a partition of I. We define $\operatorname {\mathrm {Sht}}_{G,D,I}^{(I_1, \ldots , I_r)}$ (sometimes called a moduli stack of iterated shtukas) to be the stack

$$\begin{align*}\operatorname{\mathrm{Sht}}_{G,D,I}^{(I_1, \ldots, I_r)} \colon S \mapsto \left\{ \begin{array}{@{}c@{}} (x_i)_{i \in I} \in (X - D)^I(S) \\ \mathcal{E}_0,\ldots, \mathcal{E}_r = \acute{\text{e}}\text{tale }G\text{-torsors over }X \times S \\ \varphi_j \colon \mathcal{E}_{j-1}|_{X \times S - \bigcup_{i \in I_j} \Gamma_{x_i}} \xrightarrow{\sim} \mathcal{E}_{j}|_{X \times S - \bigcup_{i \in I_j} \Gamma_{x_i}} \quad j = 1, \ldots, r \\ \varphi \colon \mathcal{E}_r \xrightarrow{\sim} {}^{\tau} \mathcal{E}_0 \\ \upsilon = \text{level structure over }D \times S \end{array} \right\}. \end{align*}$$

Here, by ‘level structure’ we mean a trivialization of the restriction of each $\mathcal {E}_i$ over $D \times S$ , compatible with the $\varphi _j$ and $\varphi $ . There is a map $\nu \colon \operatorname {\mathrm {Sht}}_{G,D,I}^{(I_1, \ldots , I_r)} \rightarrow \operatorname {\mathrm {Sht}}_{G,D,I}$ . A key property of this morphism is that it is stratified small (with respect to the Schubert stratification), which is a consequence of the same property of the convolution morphism for Beilinson-Drinfeld Grassmannians.

Define

$$\begin{align*}\stackrel{\circ}{\mathrm{Sht}}_{G,D,I} = \operatorname{\mathrm{Sht}}_{G,D,I} \times_{(X \setminus D)^I} (\stackrel{\circ}X \setminus D)^I \end{align*}$$

and

$$\begin{align*}\stackrel{\circ}{\mathrm{Sht}}_{G,D,I}^{(I_1, \ldots, I_r)} = \operatorname{\mathrm{Sht}}_{G,D,I}^{(I_1, \ldots, I_r)} \times \times_{(X \setminus D)^I} (\stackrel{\circ}X \setminus D)^I. \end{align*}$$

5.1.6 Partial Frobenius

There is a partial Frobenius $F_{I_1} \colon \operatorname {\mathrm {Sht}}_{G,D,I}^{(I_1, I_2, \ldots , I_r)} \rightarrow \operatorname {\mathrm {Sht}}_{G,D,I}^{(I_2, \ldots , I_r, I_1)} $ sending

$$ \begin{align*} x_i & \mapsto \begin{cases} {}^{\tau} x_i & i \in I_1\\ x_i &\text{otherwise}\end{cases} \\ (\mathcal{E}_0, \ldots, \mathcal{E}_r) & \mapsto (\mathcal{E}_1, \ldots, \mathcal{E}_r, {}^{\tau} \mathcal{E}_0) \\ (\varphi_1, \ldots, \varphi_r) &\mapsto (\varphi_2, \ldots, \varphi_r, {}^{\tau} \varphi_1). \end{align*} $$

It lies over the partial Frobenius $\operatorname {\mathrm {Frob}}_{I_1}$ on $X^I$ (applying $\operatorname {\mathrm {Frob}}_\ell $ to the coordinates indexed by $i \in I_1$ ), so that the diagram below is commutative (and cartesian up to radiciel maps):

(5.1)

5.1.7 Base change setup

We now consider the following ‘base change setup’. Let F be the function field of X and $H_F$ a reductive group over F. We choose a parahoric extension of $H_F$ to a smooth affine group scheme H over X.

Let $E/F$ be a cyclic extension of F having degree p, so E corresponds to the function field of a smooth projective curve $X'$ . Define $G := \operatorname {\mathrm {Res}}_{X'/X} (H_{X'})$ , which is an algebraic group scheme over X with generic fiber $G_F \cong \operatorname {\mathrm {Res}}_{E/F}(H_E)$ . The group scheme $G \rightarrow X$ comes with an induced action of $\langle \sigma \rangle = \operatorname {\mathrm {Aut}}(X'/X)$ .

5.2.1 Constructing actions of the excursion algebra

We will explain an abstract setup that gives rise to actions of the excursion algebra.

Construction 5.2. Let denote the trivial representation of ${}^L {G}$ . Given an admissible family of functors $H_I \colon \operatorname {\mathrm {Rep}}_k(({}^L G)^I) \rightarrow \operatorname {\mathrm {Mod}}_A(\Gamma ^I)$ , we get an A-linear action of $\operatorname {\mathrm {Exc}}(\Gamma , {}^L{G})$ on as follows.

For a tuple $(I, W, x, \xi , (\gamma _i)_{i \in I})$ , we define an endomorphism, which gives the image of $S_{I, W, x, \xi , (\gamma _i)_{i \in I}}$ in , by the following composition:

(Here, we again conflate the vector $x \in W^{\Delta (\widehat {G})}$ with a $\widehat {G}$ -equivariant map .) From the assumptions of admissibility, it is straightforward to check the relations in §2.4.2.

5.2.2 Excursion action on the cohomology of shtukas

Let $\mathcal {H}_G$ be the Hecke algebra acting on $\operatorname {\mathrm {Sht}}_{G, D,I}$ ; it is the tensor product of local Hecke algebras with the level structure dictated by D. For any finite set I, we have a map

$$\begin{align*}\pi_I \colon \operatorname{\mathrm{Sht}}_{G,D, I} \rightarrow (X-D)^I \end{align*}$$

remembering the points of the curve indexed by I (which avoid D by definition). Let $\eta ^I$ denote the generic point of $X^I$ and $\overline {\eta ^I}$ the spectrum of an algebraic closure, viewed as a geometric generic point of $X^I$ . When I is a singleton, we will just abbreviate these by $\eta $ and $\overline {\eta }$ .

We will define a family of functors indexed by finite sets I:

(5.3) $$ \begin{align} H_I^j \colon \operatorname{\mathrm{Rep}}_k(({}^L {G}^{\operatorname{\mathrm{alg}}})^I) \rightarrow \operatorname{\mathrm{Mod}}_{\mathcal{H}_G}(\Gamma^I) \end{align} $$

sending $V \in \operatorname {\mathrm {Rep}}_k(({}^L G^{\operatorname {\mathrm {alg}}})^I)$ to

(5.4) $$ \begin{align} H_c^j(\operatorname{\mathrm{Sht}}_{G,D, I}|_{\overline{\eta^{I}}}; \operatorname{\mathrm{Sat}}(V)). \end{align} $$

Note that a priori $H_I^j(V)$ has an action of $\pi _1(\eta ^{I}, \overline {\eta ^{I}})$ , which mapsFootnote ¹² to $\Gamma ^{I}$ but neither injectively nor surjectively.

5.2.3

We explain why the action of $\pi _1(\eta ^I, \overline {\eta ^I})$ extends canonically to an action of $\Gamma ^I$ . Assume I is nonempty, since otherwise there is nothing to prove. The Satake functor of §5.1.3 generalizes to a functor

$$\begin{align*}\operatorname{\mathrm{Sat}}^{(I_1, \ldots, I_r)} \colon \operatorname{\mathrm{Rep}}_k(({}^L G)^I) \rightarrow D^b(\stackrel{\circ}{\mathrm{Sht}}_{G,D, I}^{(I_1, \ldots, I_r)};k), \end{align*}$$

such that the map

$$\begin{align*}\nu \colon \stackrel{\circ}{\mathrm{Sht}}_{G,D, I}^{(I_1, \ldots, I_r)} \rightarrow \stackrel{\circ}{\mathrm{Sht}}_{G,D,I} \end{align*}$$

has the property that $R\nu _! \operatorname {\mathrm {Sat}}^{(I_1, \ldots , I_r)}(V) \cong \operatorname {\mathrm {Sat}}(V)$ . Furthermore, there are natural isomorphisms

$$\begin{align*}F_{I_1}^* \operatorname{\mathrm{Sat}}^{(I_1, I_2, \ldots, I_r)}(V) \cong \operatorname{\mathrm{Sat}}^{(I_2, \ldots, I_r, I_1)}(V), \end{align*}$$

where $F_{I_1}$ is the partial Frobenius from §5.1.6.

Write $I = \{1, \ldots , n\}$ . Thanks to the above properties and (5.1), the partial Frobenius maps on $\operatorname {\mathrm {Sht}}_{G,D, I}^{(\{1\}, \ldots , \{n\})}$ then induce maps

$$\begin{align*}\operatorname{\mathrm{Frob}}_{\{1\}}^* H_I^j(V) \xrightarrow{\sim} H_I^j(V). \end{align*}$$

That equips $H_I^j(V)$ with the action of the larger group $\operatorname {\mathrm {FWeil}}(\eta ^I, \overline {\eta ^I}) $ that we now recall, summarizing [Reference LafforgueLaf18, Remarque 8.18]. Let $F^I$ denote the function field of $X^I$ , so $\eta ^I = \operatorname {\mbox {Spec }} F^I$ , and $\overline {F^I}$ an algebraic closure, so we may take $\overline {\eta ^I} = \operatorname {\mbox {Spec }} \overline {F^I}$ . Write $(F^I)^{\operatorname {\mathrm {perf}}}$ for the perfect closure of $F^I$ and $\operatorname {\mathrm {Frob}}_{\{i\}}$ for the ‘partial Frobenius’ automorphism of $(F^I)^{\operatorname {\mathrm {perf}}}$ induced by $\operatorname {\mathrm {Frob}}_\ell $ on the ith factor. We define

$$\begin{align*}\operatorname{\mathrm{FWeil}} (\eta^I, \overline{\eta^I}) := \{ \gamma \in \operatorname{\mathrm{Aut}}_{\overline{\mathbf{F}}_q} (\overline{F^I}) \colon \exists (n_i)_{i \in I} \in \mathbf{Z}^I \text{ such that } \gamma|_{(F^I)^{\operatorname{\mathrm{perf}}}} = \prod_{i \in I} (\operatorname{\mathrm{Frob}}_{\{i\}})^{n_i}\}. \end{align*}$$

Writing $\pi _1^{\operatorname {\mathrm {geom}}}(\eta ^I, \overline {\eta ^I}) := \ker (\pi _1(\eta ^I, \overline {\eta ^I}) \xrightarrow {\deg } \widehat {\mathbf {Z}})$ , this fits into an extension

$$\begin{align*}0 \rightarrow \pi_1^{\operatorname{\mathrm{geom}}}(\eta^I, \overline{\eta^I}) \rightarrow \operatorname{\mathrm{FWeil}} (\eta^I, \overline{\eta^I}) \rightarrow \mathbf{Z}^I \rightarrow 0. \end{align*}$$

Fixing a specialization morphism $\overline {\eta ^I} \rightsquigarrow \Delta (\eta ^{\{1\}})$ induces a surjection

$$\begin{align*}\operatorname{\mathrm{FWeil}} (\eta^I, \overline{\eta^I}) \twoheadrightarrow \operatorname{\mathrm{Weil}}(\eta, \overline{\eta})^I. \end{align*}$$

A form of Drinfeld’s Lemma [Reference XueXuea, Lemma 7.4.2] is used to show that the action of $\operatorname {\mathrm {FWeil}} (\eta ^I, \overline {\eta ^I})$ on $H_I^j(V)$ factors through $\operatorname {\mathrm {Weil}}(F^s/F)^I$ .

Example 5.4. Let us unravel

(5.5)

By Remark 5.3, the underlying Hecke module of is isomorphic to . According to [Reference LafforgueLaf18, Remarque 12.2], this is the space of compactly supported k-valued functions on the discrete groupoid

(5.6) $$ \begin{align} \operatorname{\mathrm{Bun}}_{G,D}(\mathbf{F}_\ell) = \coprod_{\alpha \in \ker^1(F,G)} \left( G_{\alpha}(F) \backslash G_{\alpha}(\mathbf{A}_F) / \prod_v K_v \right), \end{align} $$

where $G_{\alpha }$ is the pure inner form of G corresponding to $\alpha $ , $K_v = G(\mathcal {O}_v)$ for $v \notin D$ , and $K_v = \ker (G(\mathcal {O}_v) \rightarrow G_D)$ .

The excursion action preserves the decomposition (5.6) and so gives an action of $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G)$ on each .

The family of functors $H_I^j$ is admissible; this is an immediate consequence of the fact that $\operatorname {\mathrm {Sat}}$ is already compatible with composition and fusion. Hence, Construction 5.2 applies to define an action of $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G)$ on $C_c^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k)$ . Elements of the image of $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G)$ in $\operatorname {\mathrm {End}}(C_c^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k))$ are called ‘excursion operators’.

5.2.4 Xue’s generalization

Lafforgue defined an $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G)$ -action on the finite-dimensional subspace of cuspidal functions $C_{\mathrm {cusp}}^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k) \subset C_c^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k)$ . This decomposes $C_{\mathrm {cusp}}^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k)$ into a direct sum of generalized eigenspaces under the action of $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G)$ . Using Proposition 2.4, this decomposition corresponds to a parametrization by Langlands parameters.

Thanks to Xue’s extension of the action to $\operatorname {\mathrm {Exc}}(\Gamma , {}^L G) \curvearrowright C_c^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k)$ , it is meaningful to speak of Langlands parameters arising from $C_c^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k)$ . However, since the excursion action does not stabilize any finite-dimensional subspaces of $C_c^{\infty }(\operatorname {\mathrm {Bun}}_{G,D}(\mathbf {F}_\ell ); k)$ unless they are contained in the space of cusp forms, we must broaden what it means to have an L-parameter ‘come from’ an automorphic function.