On the Kottwitz conjecture for local shtuka spaces

Abstract

Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of $\operatorname {\mathrm {GL}}_n$ , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.

1 Introduction

Let F be a finite extension of the field $\mathbf {Q}_p$ of p-adic numbers, and let G be a connected reductive group defined over F. Scholze [SW20, §23] introduced a tower of moduli spaces of mixed-characteristic shtukas

$$\begin{align*}\operatorname{\mathrm{Sht}}_{G,b,\mu}=\varprojlim_K \operatorname{\mathrm{Sht}}_{G,b,\mu,K} \end{align*}$$

depending on a $\sigma $ -conjugacy class of $b\in G(\breve F)$ (where $\breve F$ is the completion of the maximal unramified extension of F) and on a conjugacy class of cocharacters $\mu \colon \mathbf {G}_{\mathrm {m}}\to G$ defined over $\overline {F}$ . Here, K ranges over open compact subgroups of $G(F)$ . Each $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ is a locally spatial diamond defined over $\operatorname {\mathrm {Spd}} \breve {E}$ , where E is the field of definition of the conjugacy class of $\mu $ .

When $\mu $ is minuscule, $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ is the diamond associated to a rigid-analytic variety $\mathcal {M}_{G,b,\mu ,K}$ [SW20, §24]. The latter is a local Shimura variety, whose general existence was conjectured in [RV14]. The theory of Rapoport–Zink spaces [RZ96] provides instances of $\mathcal {M}_{G,b,\mu ,K}$ admitting a moduli interpretation, as the generic fiber of a deformation space of p-divisible groups.

The Kottwitz conjecture [Rap95, Conjecture 5.1], [RV14, Conjecture 7.3] relates the cohomology of $\mathcal {M}_{G,b,\mu ,K}$ to the local Langlands correspondence in the case that b lies in the unique basic class in $B(G,\mu )$ . There is a natural generalization of this conjecture for $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ , as we now explain.

Let $G_b$ the inner form of G associated to b. The tower $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ admits commuting actions of $G_b(F)$ and $G(F)$ . The action of $G_b(F)$ preserves each $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ , whereas the action of $g\in G(F)$ sends $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ to $\operatorname {\mathrm {Sht}}_{G,b,\mu ,gKg^{-1}}$ . There is furthermore a (not necessarily effective) Weil descent datum on this tower from $\breve {E}$ down to E.

Let $\ell $ be a prime distinct from p. The geometric Satake equivalence produces an object $\mathcal {S}_{\mu }$ in the derived category of étale $\mathbf {Z}_{\ell }$ -sheaves on $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ ; this is compatible with the actions of $G(F)$ and $G_b(F)$ on the tower. Let C be the completion of an algebraic closure of $\breve {E}$ . For a smooth representation $\rho $ of $G_b(F)$ with coefficients in $\overline {\mathbf {Q}}_{\ell }$ , we define:

$$\begin{align*}R\Gamma(G,b,\mu)[\rho]=\varinjlim_K R\operatorname{\mathrm{Hom}}_{G_b(F)}(R\Gamma_c(\operatorname{\mathrm{Sht}}_{G,b,\mu,K,C},\mathcal{S}_{\mu}),\rho). \end{align*}$$

Then $R\Gamma (G,b,\mu )[\rho ]$ lies in the derived category of smooth representations of $G(F)\times W_E$ with coefficients in $\overline {\mathbf {Q}}_{\ell }$ , where $W_E$ is the Weil group. Informally, this is the $\rho $ -isotypic component of the cohomology of the tower $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ .

A recent result of Fargues–Scholze [FS21, Corollary I.7.3] states that, if $\rho $ is finite length and admissible, then $R\Gamma (G,b,\mu )[\rho ]$ is a complex of finite length admissible representations of $G(F)$ admitting a continuous action of $W_E$ .

Let $\operatorname {\mathrm {Groth}}(G_b(F))$ be the Grothendieck group of the category of finite length admissible representations of $G(F)$ with $\overline {\mathbf {Q}}_{\ell }$ coefficients. Also, let $\operatorname {\mathrm {Groth}}(G(F)\times W_E)$ be the Grothendieck group of the category of finite length admissible representations of $G(F)$ with $\overline {\mathbf {Q}}_{\ell }$ coefficients, which come equipped with a continuous action of $W_E$ commuting with the $G(F)$ -action. Following [Shi11], we define a map

$$\begin{align*}\operatorname{\mathrm{Mant}}_{b,\mu}\colon \operatorname{\mathrm{Groth}}(G_b(F))\to \operatorname{\mathrm{Groth}}(G(F)\times W_E) \end{align*}$$

(for ‘Mantovan’, referencing [Man04]) sending $\rho $ to the Euler characteristic of $R\Gamma (G,b,\mu )[\rho ]$ .

The Kottwitz conjecture (appropriately generalized) describes $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho )$ in terms of the local Langlands correspondence when $\rho $ lies in a supercuspidal L-packet. The complex dual groups of G and $G_b$ are canonically identified, and we write $\widehat {G}$ for either. Let $^LG=\widehat G \rtimes W_F$ be the L-group. The basic form of the local Langlands conjecture predicts that the set of isomorphism classes of essentially square-integrable representations of $G(F)$ (resp., $G_b(F)$ ) is partitioned into L-packets $\Pi _{\phi }(G)$ (resp., $\Pi _{\phi }(G_b)$ ) and that each such packet is indexed by a discrete Langlands parameter $\phi : W_F \times \mathrm {SL}_2(\mathbf {C}) \to {^LG}$ . When $\phi $ is discrete and trivial on $\mathrm {SL}_2(\mathbf {C})$ , we say $\phi $ is supercuspidal; in this case, it is expected that the packets $\Pi _{\phi }(G)$ and $\Pi _{\phi }(G_b)$ consist entirely of supercuspidal representations.

Our generalized Kottwitz conjecture is conditional on the refined local Langlands correspondence for supercuspidal L-parameters in the formulation of [Kal16a, Conjecture G]. In particular, it relies crucially on the endoscopic character identities satisfied by L-packets. These are reviewed in Appendix A. Note that we do not assume any compatibility between the validity of [Kal16a, Conjecture G] and the construction of [FS21], i.e., we do not require that the construction of [FS21] satisfy any portion of [Kal16a, Conjecture G].

We take this opportunity to give a brief summary of the status of [Kal16a, Conjecture G]. In short, the full conjecture is known for regular supercuspidal parameters [Kal19a, Definition 5.2.3] provided G splits over a tame extension of F, F has characteristic zero and p is sufficiently large (at least $(e+2)n$ , where e is the ramification index of $F/\mathbf {Q}_p$ and n is the smallest size of a faithful algebraic representation of G). The proof is contained in [Kal19a, §5.3] and [FKS19, §4.4]. However, various parts of that conjecture are known under less restrictive assumptions. To describe this, we remind the reader that [Kal16a, Conjecture G] consists of the following assertions:

1. 1. The existence of a finite set $\Pi _{\phi }$ of representations of rigid inner forms of G for each tempered L-parameter $\phi $ .

2. 2. The existence and uniqueness of a generic constituent of $\Pi _{\phi }$ for a fixed Whittaker datum.

3. 3. A bijection between $\Pi _{\phi }$ and the set $\mathrm {Irr}(\pi _0(S_{\phi }^+))$ of irreducible representations of the refined centralizer component group associated to $\phi $ .

4. 4. The character identities of ordinary endoscopy, as recalled in Appendix A.

At the moment, a set $\Pi _{\phi }$ has been constructed in [Kal19b, §§4.1,4.2] for every supercuspidal parameter $\phi $ provided G splits over a tame extension of F and p does not divide the order of the Weyl group of G (this assumption on p implies that any supercuspidal parameter maps wild inertia into a torus of $\widehat G$ ; under weaker assumptions on p, this is not automatically true, but for parameters $\phi $ that do have this property, the construction of [Kal19b] works under weaker assumptions on p). A bijection between $\Pi _{\phi }$ and $\mathrm {Irr}(\pi _0(S_{\phi }^+))$ has been constructed in [Kal19b, §§4.3-4.5] for any supercuspidal parameter $\phi $ . Assuming F has characteristic zero and $p \geq (e+2)n$ , the existence and uniqueness of a generic constituent in $\Pi _{\phi }(G)$ , as well as the character identities of ordinary endoscopy, are proved in [FKS19, §4.4] for all regular supercuspidal parameters $\phi $ . They are also proved for nonregular supercuspidal parameters $\phi $ but only for certain endoscopic elements.

Returning to the subject of this paper, let $S_{\phi }=\mathrm {Cent}(\phi ,\widehat {G})$ . For any $\pi \in \Pi _{\phi }(G)$ and $\rho \in \Pi _{\phi }(G_b)$ , the refined form of the local Langlands conjecture implies the existence of an algebraic representation $\delta _{\pi ,\rho }$ of $S_{\phi }$ , which can be thought of as measuring the relative position of $\pi $ and $\rho $ . (The representation $\delta _{\pi ,\rho }$ also depends on b, but we suppress this from the notation.) The conjugacy class of $\mu $ determines by duality a conjugacy class of weights of $\widehat G$ ; we denote by $r_{\mu }$ the irreducible representation of $\widehat G$ of highest weight $\mu $ . There is a natural extension of $r_{\mu }$ to $^LG_E$ , the L-group of the base change of G to E [Kot84a, Lemma 2.1.2]. Write $r_{\mu } \circ \phi _E$ for the representation of $S_{\phi } \times W_E$ , given by

$$\begin{align*}r_{\mu} \circ \phi_E(s,w) = r_{\mu}(s\cdot \phi(w)). \end{align*}$$

Conjecture 1.0.1. Let $\phi : W_F \to {^LG}$ be a supercuspidal Langlands parameter. Given $\rho \in \Pi _{\phi }(G_b)$ , we have the following equality in $\operatorname {\mathrm {Groth}}(G(F)\times W_E)$ :

(1.0.1) $$ \begin{align} \operatorname{\mathrm{Mant}}_{b,\mu}(\rho) = \sum_{\pi \in \Pi_{\phi}(G)} \pi \boxtimes \mathrm{Hom}_{S_{\phi}}(\delta_{\pi,\rho},r_{\mu} \circ \phi_E). \end{align} $$

This conjecture is more general than the formulation of Kottwitz’s conjecture in [Rap95] and [RV14], in that two conditions are removed. The first is that we are allowing the cocharacter $\mu $ to be nonminuscule—this is what requires passage from the local Shimura varieties $\mathcal {M}_{G,b,\mu }$ to the local shtuka spaces $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ . The second is that we do not require G to be a B-inner form of its quasi-split inner form $G^*$ . This condition, reviewed in §2.2, has the effect of making the definition of $\delta _{\pi ,\rho }$ straightforward. To remove it, we use the formulation of the refined local Langlands correspondence [Kal16a, Conjecture G] based on the cohomology sets $H^1(u \to W,Z \to G)$ of [Kal16b]. The definition of $\delta _{\pi ,\rho }$ in this setting is a bit more involved and is given in §2.3; see Definition 2.3.2.

We now present our main theorem.

Theorem 1.0.2. Assume the refined local Langlands correspondence [Kal16a, Conjecture G]. Let $\phi \colon W_F \times \mathrm {SL}_2 \to {^LG}$ be a discrete Langlands parameter with coefficients in $\overline {\mathbf {Q}}_{\ell }$ , and let $\rho \in \Pi _{\phi }(G_b)$ be a member of its L-packet. After ignoring the action of $W_E$ , we have an equality in $\operatorname {\mathrm {Groth}}(G(F))$ :

$$\begin{align*}\operatorname{\mathrm{Mant}}_{b,\mu}(\rho)=\sum_{\pi\in\Pi_{\phi}(G)} \left[\dim \operatorname{\mathrm{Hom}}_{S_{\phi}}(\delta_{\pi,\rho},r_{\mu})\right]\pi + \mathrm{err}, \end{align*}$$

where $\mathrm {err} \in \operatorname {\mathrm {Groth}}(G(F))$ is a virtual representation whose character vanishes on the locus of elliptic elements of $G(F)$ .

If the packet $\Pi _{\phi }(G)$ consists entirely of supercuspidal representations and the semisimple L-parameter $\varphi _{\rho }$ associated with $\rho $ as in [FS21, §I.9.6] is supercuspidal, then in fact $\mathrm {err}=0$ .

Of course we expect that $\varphi _{\rho }=\phi ^{\mathrm {ss}}$ so that, if $\phi $ is supercuspidal, the error term should vanish. In that case, we obtain Conjecture 1.0.1 modulo ignoring the action of $W_E$ . For a discrete but nonsupercuspidal parameter $\phi $ , the error term in Theorem 1.0.2 is often provably nonzero; cf. [Ima] for some examples. However, for applications to the local Langlands correspondence, it is crucial to have Theorem 1.0.2 in this extra generality.

The shtukas appearing in our work have only one ‘leg’. Scholze defines moduli spaces of mixed-characteristic shtukas $\operatorname {\mathrm {Sht}}_{G,b,\left \{ \mu _i \right \}}$ with arbitrarily many legs, fibered over a product $\prod _{i=1}^r \operatorname {\mathrm {Spd}} \breve {E}_i$ . It is straightforward to extend Conjecture 1.0.1 and Theorem 1.0.2 to this setting as well. In fact, Theorem 1.0.2 in this extended level of generality follows immediately from the results already proved in this paper, by allowing the legs to coalesce and using the fact that cohomology of shtuka spaces forms a local system over $(\mathrm {Div}^1)^I$ . We leave the details to the interested reader.

Theorem 1.0.2 has an application to the local Langlands correspondence.

Theorem 1.0.3. Let G be any inner form of $\mathrm {GL}_n/F$ , and let $\pi $ be an irreducible smooth representation of $G(F)$ . Then the L-parameter $\varphi _{\pi }$ associated with $\pi $ by the construction of Fargues–Scholze [FS21, §I.9] agrees with the usual semisimplified L-parameter attached to $\pi $ .

1.1 Remarks on the proof and relation with prior work

Ultimately, Theorem 1.0.2 is proved by an application of a Lefschetz–Verdier trace formula. Let us illustrate the idea in the Lubin–Tate case: Say $F=\mathbf {Q}_p$ , $G=\operatorname {\mathrm {GL}}_n$ , $\mu =(1,0,\dots ,0)$ , and b is basic of slope $1/n$ . Let $H_0$ be the p-divisible group over $\overline {\mathbf {F}}_p$ with isocrystal b so that $H_0$ has dimension $1$ and height n. In this case, $G_b(F)=\operatorname {\mathrm {Aut}}^0 H_0=D^{\times }$ , where $D/\mathbf {Q}_p$ is the division algebra of invariant $1/n$ . The spaces $\mathcal {M}_K=\mathcal {M}_{G,b,\mu ,K}$ are known as the Lubin–Tate tower; we consider these as rigid-analytic spaces over C, where $C/\mathbf {Q}_p$ is a complete algebraically closed field.

Atop the tower sits the infinite-level Lubin–Tate space $\mathcal {M}=\varprojlim _K\mathcal {M}_K$ as described in [SW13]. This is a perfectoid space admitting an action of $G(\mathbf {Q}_p)\times G_b(\mathbf {Q}_p)$ . The C-points of $\mathcal {M}$ classify equivalence classes of triples $(H,\alpha ,\iota )$ , where $H/\mathcal {O}_{C}$ is a p-divisible group, $\alpha \colon \mathbf {Q}_p^n\to VH$ is a trivialization of the rational Tate module and $\iota \colon H_0\otimes _{\overline {\mathbf {F}}_p} \mathcal {O}_C/p\to H\otimes _{\mathcal {O}_{C}} \mathcal {O}_C/p$ is an isomorphism in the isogeny category. (Equivalence between two such triples is a quasi-isogeny between p-divisible groups which makes both diagrams commute.) Then $\mathcal {M}$ admits an action of $G(\mathbf {Q}_p)\times G_b(\mathbf {Q}_p)$ via composition with $\alpha $ and $\iota $ , respectively.

The Hodge–Tate period map exhibits $\mathcal {M}$ as a pro-étale $D^{\times }$ -torsor over Drinfeld’s upper half-space $\Omega ^{n-1}$ (the complement in $\mathbf {P}^{n-1}$ of all $\mathbf {Q}_p$ -rational hyperplanes). This map $\mathcal {M}\to \Omega ^{n-1}$ is equivariant for the action of $G(\mathbf {Q}_p)$ .

Now suppose $g\in G(\mathbf {Q}_p)$ is a regular elliptic element (that is, an element with irreducible characteristic polynomial). Then g has exactly n fixed points on $\Omega ^{n-1}$ . For each such fixed point $x\in (\Omega ^{n-1})^g$ , the element g acts on the fiber $\mathcal {M}_x$ . Because $\mathcal {M}\to \Omega ^{n-1}$ is a $G_b(F)$ -torsor, there must exist $g'\in G_b(\mathbf {Q}_p)$ such that $(g,g')$ fixes a point in the fiber $\mathcal {M}_x$ .

Key observation. The elements $g\in G(\mathbf {Q}_p)$ and $g'\in G_b(\mathbf {Q}_p)$ are related, meaning they become conjugate over $\overline {\mathbf {Q}}_p$ .

We sketch the proof of this claim. Suppose y corresponds to the triple $(H,\alpha ,\iota )$ . This means there exists an automorphism $\gamma $ of H (in the isogeny category) which corresponds to g on the Tate module and $g'$ on the special fiber, respectively. We verify now that g and $g'$ are related. Let $B_{\operatorname {\mathrm {cris}}}=B_{\operatorname {\mathrm {cris}}}(C)$ be the crystalline period ring. There are isomorphisms

$$\begin{align*}B_{\operatorname{\mathrm{cris}}}^n {\to} VH \otimes_{\mathbf{Q}_p} B_{\operatorname{\mathrm{cris}}} {\to} M(H_0) \otimes B_{\operatorname{\mathrm{cris}}}, \end{align*}$$

where the first map is induced from $\alpha $ , and the second map comes from the comparison isomorphism between étale and crystalline cohomology of H (using $\iota $ to identify the latter with $M(H_0)$ ). The composite map carries the action of g onto that of $g'$ , which is to say that g and $g'$ become conjugate over $B_{\operatorname {\mathrm {cris}}}$ . This implies that g and $g'$ are related.

Suppose that $\rho $ is an admissible representation of $D^{\times }$ with coefficients in $\overline {\mathbf {Q}}_{\ell }$ . There is a corresponding $\overline {\mathbf {Q}}_{\ell }$ -local system $\mathcal {L}_{\rho }$ on $\Omega ^{n-1}_{C,\mathrm {\acute {e}t}}$ .

Let $g\in G(F)$ be elliptic. A naïve form of the Lefschetz trace formula would predict that:

$$\begin{align*}\operatorname{\mathrm{tr}}\left(g\vert R\Gamma_c(\Omega^{n-1},\mathcal{L}_{\rho})\right) = \sum_{x\in (\Omega^{n-1})^g} \operatorname{\mathrm{tr}}(g\vert \mathcal{L}_{\rho,x}). \end{align*}$$

For each fixed point x, the key observation above gives $\operatorname {\mathrm {tr}}(g\vert \mathcal {L}_{\rho ,x})=\operatorname {\mathrm {tr}} \rho (g')$ , where g and $g'$ are related. By the Jacquet–Langlands correspondence, there exists a discrete series representation $\pi $ of $G(\mathbf {Q}_p)$ satisfying $\operatorname {\mathrm {tr}} \pi (g)=(-1)^{n-1}\operatorname {\mathrm {tr}} \rho (g')$ (here $\operatorname {\mathrm {tr}} \pi (g)$ is interpreted as a Harish–Chandra character). Thus, the Euler characteristic of $R\Gamma _c(\Omega ^{n-1},\mathcal {L}_{\rho })$ equals $(-1)^{n-1}n\pi $ up to a virtual representation with trace zero on the elliptic locus.

In this situation, $\mathcal {S}_{\mu } = \mathbf {Z}_{\ell }[n-1]$ (up to a Tate twist), and we find that $R\Gamma (G,b,\mu )[\rho ]$ is the shift by $n-1$ of the dual of $R\Gamma _c(\Omega ^{n-1},\mathcal {L}_{\rho ^{\vee }})$ . Therefore, in $\operatorname {\mathrm {Groth}}(\operatorname {\mathrm {GL}}_n(\mathbf {Q}_p))$ , we have

$$\begin{align*}\operatorname{\mathrm{Mant}}_{b,\mu}(\rho) = n\pi+\mathrm{err}, \end{align*}$$

where the character of err vanishes on the locus of elliptic elements. This is in accord with Theorem 1.0.2.

This argument goes back at least to the 1990s, as discussed in [Har15, Chap. 9] and as far as we know first appears in [Fal94]. The present article is our attempt to push this argument as far as it will go. If a suitable Lefschetz formula is valid, then the equality in Theorem 1.0.2 can be reduced to an endoscopic character identity relating representations of $G(F)$ and $G_b(F)$ (Theorem 3.2.9), which we prove in §3.

Therefore, the difficulty in Theorem 1.0.2 lies in proving the validity of the Lefschetz formula. Prior work of Strauch and Mieda proved Theorem 1.0.2 in the case of the Lubin–Tate tower [Str05], [Str08], [Mie12], [Mie14a] and also in the case of a basic Rapoport–Zink space for GSp(4) [Mie].

In applying a Lefschetz formula to a nonproper rigid space, care must be taken to treat the boundary. For instance, if X is the affinoid unit disc $\left \{ \left \lvert T \right \rvert \leq 1 \right \}$ in the adic space $\mathbf {A}^1$ , then the automorphism $T\mapsto T+1$ has Euler characteristic 1 on X, despite having no fixed points. The culprit is that this automorphism fixes the single boundary point in $\overline {X}\backslash X$ . Mieda [Mie14b] proves a Lefschetz formula for an operator on a rigid space under an assumption that the operator has no topological fixed points on a compactification. Now, in all of the above cases, $\mathcal {M}_{G,b,\mu ,K}$ admits a cellular decomposition. This means (approximately) that $\mathcal {M}_{G,b,\mu ,K}$ contains a compact open subset, whose translates by Hecke operators cover all of $\mathcal {M}_{G,b,\mu ,K}$ . This is enough to establish the ‘topological fixed point’ hypothesis necessary to apply Mieda’s Lefschetz formula. Shen [She14] constructs a cellular decomposition for a basic Rapoport–Zink space attached to the group $U(1,n-1)$ , which paves the way for a similar proof of Theorem 1.0.2 in this case as well. For general $(G,b,\mu )$ , however, the $\mathcal {M}_{G,b,\mu ,K}$ do not admit a cellular decomposition, and so there is probably no hope of applying the methods of [Mie14b].

We had no idea how to proceed until we learned of the shift of perspective offered by Fargues’ program on the geometrization of local Langlands [Far], followed by the work [FS21]. At the center of that program is the stack $\operatorname {\mathrm {Bun}}_G$ of G-bundles on the Fargues–Fontaine curve. This is a geometrization of the Kottwitz set $B(G)$ : There is a bijection $b\mapsto \mathcal {E}^b$ between $B(G)$ and points of the underlying topological space of $\operatorname {\mathrm {Bun}}_G$ . For basic b, there is an open substack $\operatorname {\mathrm {Bun}}_G^b\subset \operatorname {\mathrm {Bun}}_G$ classifing G-bundles which are everywhere isomorphic to $\mathcal {E}^b$ ; in this situation, $\operatorname {\mathrm {Aut}} \mathcal {E}^b=G_b(F)$ , and so we have an isomorphism $\operatorname {\mathrm {Bun}}_G^b\cong [\ast /G_b(F)]$ .

Let $\mu $ be a cocharacter of G. As in geometric Langlands, there is a stack $\operatorname {\mathrm {Hecke}}_{G,\leq \mu }$ lying over the product $\operatorname {\mathrm {Bun}}_G\times \operatorname {\mathrm {Bun}}_G$ , which parametrizes $\mu $ -bounded modifications of G-bundles at one point of the curve. For each $\mu $ , one uses $\operatorname {\mathrm {Hecke}}_{G,\leq \mu }$ to define a Hecke operator $T_{\mu }$ on a suitable derived category $D(\operatorname {\mathrm {Bun}}_G,\mathbf {Z}_{\ell })$ of étale $\mathbf {Z}_{\ell }$ -sheaves on $\operatorname {\mathrm {Bun}}_G$ . If $b\in B(G,\mu )$ , then the moduli space of local shtukas $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ appears as the fiber of $\operatorname {\mathrm {Hecke}}_{G,\leq \mu }$ over the point $(\mathcal {E}^b,\mathcal {E}^1)$ of $\operatorname {\mathrm {Bun}}_G\times \operatorname {\mathrm {Bun}}_G$ . Consequently, there is an expression for $R\Gamma (G,b,\mu )[\rho ]$ in terms of the Hecke operators $T_{\mu }$ ; see Proposition 6.4.5.

Heavy use is made in [FS21] of the notion of universal local acyclity (ULA) as a property of objects $A\in D(X,\mathbf {Z}_{\ell })$ for Artin v-stacks X. When $X=[*/G_b(F)]$ , a ULA object is an admissible complex of representations of $G_b(F)$ . It is proved in [FS21] that the Hecke operators $T_{\mu }$ preserve ULA objects; the admissibility of $R\Gamma (G,b,\mu )[\rho ]$ is deduced from this.

We learned from [LZ22] that the ULA condition is precisely the right hypothesis necessary to prove a Lefschetz–Verdier trace formula applicable to the cohomology of A. This explains the counterexample above: $j_!\mathbf {Z}_{\ell }$ fails to be ULA, where j is the inclusion of the affinoid disc X into its compactification $\overline {X}$ . In fact, [LZ22] is written in the context of schemes, but their formalism applies equally well in the context of rigid-analytic spaces and diamonds. Indeed, some interesting new phenomena occur in the diamond context. For instance, if H is a locally profinite group acting continuously on a proper diamond X and $A\in D(X,\mathbf {Z}_{\ell })$ is a ULA object which is H-equivariant, then $R\Gamma (X,A)$ is an admissible H-module. One gets a formula for the trace distribution of H acting on $R\Gamma (X,A)$ in terms of local terms living on the fixed-point locus in $H\times X$ . We explain the Lefschetz–Verdier trace formula for diamonds in §4.

In §5, we study the Lefschetz–Verdier trace formula as it pertains to the mixed-characteristic affine Grassmannian $\operatorname {\mathrm {Gr}}_{G,\leq \mu }$ . The object $\mathcal {S}_{\mu }$ is ULA on $\operatorname {\mathrm {Gr}}_{G,\leq \mu }$ and $G(F)$ -equivariant, so it makes sense to ask for its local term $\operatorname {\mathrm {loc}}_g(x,A)$ at a fixed point x of a regular element $g\in G(F)$ . (Such fixed points are all isolated.) We found quickly that that result we needed for Theorem 1.0.2 would follow if we knew that $\operatorname {\mathrm {loc}}_g(x,A)$ agreed with the naïve local term $\operatorname {\mathrm {tr}}(g\vert A_x)$ . We asked Varshavsky, who devised a method for proving this agreement in the scheme setting. We show how to deduce the required statement for $\operatorname {\mathrm {Gr}}_{G,\leq \mu }$ , using the Witt vector affine Grassmannian as a bridge between diamonds and schemes. (We thank the referee for pointing out that an earlier argument we had here was incorrect.)

Finally, in §6, we prove Theorem 1.0.2 by applying our trace formula to the Hecke stack $\operatorname {\mathrm {Hecke}}_{G,b,\leq \mu }$ . An important step is to show that fixed points of elliptic elements $g\in G(F)$ acting on $\operatorname {\mathrm {Gr}}_{G,\leq \mu }$ are admissible, as we observed above in the Lubin–Tate case.

2 Review of the objects appearing in Kottwitz’s conjecture

2.1 Basic notions

Let $\breve {F}$ be the completion of the maximal unramified extension of F, and let $\sigma \in \operatorname {\mathrm {Aut}} \breve {F}$ be the Frobenius automorphism. Let G be a connected reductive group defined over F. Fix a quasi-split group $G^*$ and a $G^*(\overline {F})$ -conjugacy class $\Psi $ of inner twists $G^*\to G$ ; thus, elements $\psi \in \Psi $ are isomorphisms $G^*_{\overline {F}} \to G_{\overline {F}}$ such that for each $\tau \in \Gamma $ the automorphism $\psi ^{-1}\circ \tau (\psi )$ of $G^*_{\overline {F}}$ is inner. Given an element $b\in G(\breve {F})$ , there is an associated inner form $G_b$ of a Levi subgroup of $G^*$ as described in [Kot97, §3.3,§3.4]. Its group of F-points is given by

$$\begin{align*}G_b(F) \cong \left\{ g \in G(\breve F)\; \big\vert \; \mathrm{Ad}(b)\sigma(g)=g \right\}. \end{align*}$$

Up to isomorphism, the group $G_b$ depends only on the $\sigma $ -conjugacy class $[b]$ . It will be convenient to choose b to be decent [RZ96, Definition 1.8]. Then there exists a finite unramified extension $F'/F$ such that $b \in G(F')$ . This allows us to replace $\breve F$ by $F'$ in the above formula. The slope morphism $\nu : \mathbf {D} \to G_{\breve F}$ of b [Kot85, §4] is also defined over $F'$ . The centralizer $G_{F',\nu }$ of $\nu $ in $G_{F'}$ is a Levi subgroup of $G_{F'}$ . The $G(F')$ -conjugacy class of $\nu $ is defined over F and then so is the $G(F')$ -conjugacy class of $G_{F',\nu }$ . There is a Levi subgroup $M^*$ of $G^*$ defined over F and $\psi \in \Psi $ that restricts to an inner twist $\psi : M^* \to G_b$ ; see [Kot97, §4.3].

From now on, assume that b is basic. This is equivalent to $M^*=G^*$ so that $G_b$ is in fact an inner form of $G^*$ and of G. Furthermore, $\Psi $ is an equivalence class of inner twists $G^* \to G$ as well as $G^* \to G_b$ . This identifies the dual groups of $G^*$ , G and $G_b$ , and we write $\widehat G$ for either of them.

Let $\phi : W_F \times \mathrm {SL}_2(\mathbf {C}) \to {^LG}$ be a discrete Langlands parameter, and let $S_{\phi }=\mathrm {Cent}(\phi ,\widehat G)$ . For $\lambda \in X^*(Z(\widehat {G})^{\Gamma })$ , write $\mathrm {Rep}(S_{\phi },\lambda )$ for the set of isomorphism classes of algebraic representations of the algebraic group $S_{\phi }$ whose restriction to $Z(\widehat {G})^{\Gamma }$ is $\lambda $ -isotypic, and write $\mathrm {Irr}(S_{\phi },\lambda )$ for the subset of irreducible such representations. The class of b corresponds to a character $\lambda _b : Z(\widehat G)^{\Gamma } \to \mathbf {C}^{\times }$ via the isomorphism $B(G)_{\mathrm {bas}} \to X^*(Z(\widehat G)^{\Gamma })$ of [Kot85, Proposition 5.6]. Assuming the validity of the refined local Langlands conjecture [Kal16a, Conjecture G], we will construct in the following two subsections for any $\pi \in \Pi _{\phi }(G)$ and $\rho \in \Pi _{\phi }(G_b)$ an element $\delta _{\pi ,\rho } \in \mathrm {Rep}(S_{\phi },\lambda _b)$ that measures the relative position of $\pi $ and $\rho $ .

2.2 Construction of $\delta _{\pi ,\rho }$ in a special case

The statements of the Kottwitz conjecture given in [Rap95, Conjecture 5.1] and [RV14, Conjecture 7.3] make the assumption that G is a B-inner form of $G^*$ . In that case, the construction of $\delta _{\pi ,\rho }$ is straightforward, and we shall now recall it.

The assumption on G means that some $\psi \in \Psi $ can be equipped with a decent basic $b^* \in G^*(F^{\mathrm {nr}})$ such that $\psi $ is an isomorphism $G^*_{F^{\mathrm {nr}}} \to G_{F^{\mathrm {nr}}}$ satisfying $\psi ^{-1}\sigma (\psi )=\mathrm {Ad}(b^*)$ . In other words, $\psi $ becomes an isomorphism over F from the group $G^*_{b^*}$ to G. Under this assumption and after choosing a Whittaker datum $\mathfrak {w}$ for $G^*$ , the isocrystal formulation of the refined local Langlands correspondence [Kal16a, Conjecture F], which is implied by the rigid formulation [Kal16a, Conjecture G] according to [Kal18], predicts the existence of bijections

$$ \begin{align*} \Pi_{\phi}(G) &\cong \operatorname{\mathrm{Irr}}(S_{\phi},\lambda_{b*})\\ \Pi_{\phi}(G_b) &\cong \operatorname{\mathrm{Irr}}(S_{\phi},\lambda_{b^*}+\lambda_b), \end{align*} $$

where we have used the isomorphisms $B(G)_{\operatorname {\mathrm {bas}}}\cong X^*(Z(\widehat {G})^{\Gamma }) \cong B(G^*)_{\mathrm {bas}}$ of [Kot85, Proposition 5.6] to obtain from $[b] \in B(G)_{\mathrm {bas}}$ and $[b^*] \in B(G^*)_{\mathrm {bas}}$ characters $\lambda _{b}$ and $\lambda _{b^*}$ of $Z(\widehat G)^{\Gamma }$ .

These bijections are uniquely characterized by the endoscopic character identities which are part of [Kal16a, Conjecture F]. Write $\pi \mapsto \tau _{b^*,\mathfrak {w},\pi }$ , $\rho \mapsto \tau _{b^*,\mathfrak {w},\rho }$ for these bijections, and define

(2.2.1) $$ \begin{align} \delta_{\pi,\rho} := \check\tau_{b^*,\mathfrak{w},\pi} \otimes\tau_{b^*,\mathfrak{w},\rho}. \end{align} $$

While these bijections depend on the choice of Whittaker datum $\mathfrak {w}$ and the choice of $b^*$ , we will argue in Subsection 2.3 that for any pair $\pi $ and $\rho $ the representation $\delta _{\pi ,\rho }$ is independent of these choices. Of course, it does depend on b, but this we take as part of the given data.

2.3 Construction of $\delta _{\pi ,\rho }$ in the general case

We now drop the assumption that G is a B-inner form of $G^*$ . Because of this, we no longer have the isocrystal formulation of the refined local Langlands correspondence. However, we do have the formulation based on rigid inner twists [Kal16a, Conjecture G]. What this means with regards to the Kottwitz conjecture is that neither $\pi $ nor $\rho $ correspond to representations of $S_{\phi }$ . Rather, they correspond to representations $\tau _{\pi }$ and $\tau _{\rho }$ of a different group $\pi _0(S_{\phi }^+)$ . Nonetheless, it will turn out that $\check \tau _{\pi } \otimes \tau _{\rho }$ provides in a natural way a representation $\delta _{\pi ,\rho }$ of $S_{\phi }$ .

In order to make this precise, we will need the material of [Kal16b] and [Kal18], some of which is summarized in [Kal16a]. First, we will need the cohomology set $H^1(u \to W,Z \to G^*)$ defined in [Kal16b, §3] for any finite central subgroup $Z \subset G^*$ defined over F. As in [Kal18, §3.2], it will be convenient to package these sets for varying Z into the single set

$$\begin{align*}H^1(u \to W,Z(G^*) \to G^*) := \varinjlim H^1(u \to W,Z \to G^*). \end{align*}$$

The transition maps on the right are injective, so the colimit can be seen as an increasing union.

Next, we will need the reinterpretation, given in [Kot], of $B(G)$ as the set of cohomology classes of algebraic 1-cocycles of a certain Galois gerbe $1 \to \mathbf {D}(\bar F) \to \mathcal {E} \to \Gamma \to 1$ . This reinterpretation is also reviewed in [Kal18, §3.1]. For this, we recall that inflation along $W_F \to \mathbf {Z}$ induces an isomorphism between $B(G)=H^1(\langle \sigma \rangle ,G(L))$ and $H^1(W_F,G(\bar L))$ , where we have written $L=\breve F$ to ease typesetting. In [Kot97, App B], Kottwitz constructs a continuous homomorphism $W_F \to \mathcal {E}$ whose composition with the natural projection $\mathcal {E} \to \Gamma $ is the natural map $W_F \to \Gamma $ . He proves in [Kot97, §8 and App B] that pulling back along this homomorphism and pushing along the inclusion $G(\bar F) \to G(\bar L)$ gives an isomorphism $H^1_{\mathrm {alg}}(\mathcal {E},G(\bar F)) \to B(G)$ and, in particular, $H^1_{\mathrm {bas}}(\mathcal {E},G(\bar F)) \to B_{\mathrm {bas}}(G)$ . While the section $W_F \to \mathcal {E}$ is not completely canonical, the induced isomorphism on cohomology is independent of the choice of section. Strictly speaking, Kottwitz gives the proof only in the case of tori, but the general case is immediate from that.

Finally, we will need the comparison map

$$\begin{align*}H^1_{\mathrm{bas}}(\mathcal{E},G(\bar F)) \to H^1(u \to W,Z(G) \to G) \end{align*}$$

of [Kal18, §3.3].

After this short review, we turn to the construction of $\delta _{\pi ,\rho } \in \operatorname {\mathrm {Rep}}(S_{\phi },\lambda _b)$ . For this, it is not enough to work with the cohomology class of b, because $\delta _{\pi ,\rho }$ is an invariant of the equivalence class of the triple $(b,\pi ,\rho )$ , and changing b within its cohomology class must be accompanied with a corresponding change in $\rho $ . Therefore, we must work with cocycles.

To that end, fix the section $W_F \to \mathcal {E}$ . If $z_b \in Z^1_{\mathrm {bas}}(\mathcal {E},G(\bar F))$ denotes a representative of the element of $H^1_{\mathrm {bas}}(\mathcal {E},G(\bar F))$ corresponding to the class of b, there exists $g \in G(\bar L)$ , unique up to right multiplication by elements of $G_b(F)$ such that

(2.3.1) $$ \begin{align} g^{-1} z_b(w)w(g) = b \cdot \sigma(b) \cdots \sigma^{|w|-1}(b) \qquad \forall w \in W_F \to \mathcal{E}, \end{align} $$

where $|w|$ is the image of w under $W_F \to \mathbf {Z}$ . Note that the image of g in $G_{\mathrm {ad}}(\bar L)$ lies in $G_{\mathrm {ad}}(\bar F)$ and that $\mathrm {Ad}(g)$ induces an F-isomorphism $G_{z_b} \to G_b$ . Therefore, $\rho \circ \mathrm {Ad}(g)$ is an irreducible representation of $G_{z_b}(F)$ whose isomorphism class does not depend on the choice of g.

Choose any inner twist $\psi \in \Psi $ and let $\bar z_{\sigma } := \psi ^{-1}\sigma (\psi ) \in G^*_{\mathrm {ad}}(\overline {F})$ . Then $\bar z \in Z^1(F,G^*_{\mathrm {ad}})$ and the surjectivity of the natural map $H^1(u \to W,Z(G^*) \to G^*) \to H^1(F,G^*_{\mathrm {ad}})$ asserted in [Kal16b, Corollary 3.8] allows us to choose $z \in Z^1(u \to W,Z(G^*) \to G^*)$ lifting $\bar z$ . Then $(\psi ,z) : G^* \to G$ is a rigid inner twist, and $(\psi ,\psi ^{-1}(z)\cdot z_b) : G^* \to G_{z_b}$ is also a rigid inner twist.

The L-packets $\Pi _{\phi }(G)$ and $\Pi _{\phi }(G_{z_b})$ are now parameterized by representations of a certain cover $S_{\phi }^+$ of $S_{\phi }$ . While [Kal16a, Conjecture G] is formulated in terms of a finite cover depending on an auxiliary choice of a finite central subgroup $Z \subset G^*$ , we will adopt here the point of view of [Kal18] and work with a canonical infinite cover, namely the preimage of $S_{\phi }$ in the universal cover of $\widehat G$ . Following [Kal18, §3.3], we can present this universal cover as follows. Let $Z_n \subset Z(G)$ be the subgroup of those elements whose image in $Z(G)/Z(G_{\mathrm {der}})$ is n-torsion, and let $G_n=G/Z_n$ . Then $G_n$ has adjoint derived subgroup and connected center. More precisely, $G_n=G_{\mathrm {ad}} \times C_n$ , where $C_n=C_1/C_1[n]$ and $C_1=Z(G)/Z(G_{\mathrm {der}})$ . It is convenient to identify $C_n=C_1$ as algebraic tori and take the $m/n$ -power map $C_1 \to C_1$ as the transition map $C_n \to C_m$ for $n|m$ . The isogeny $G \to G_n$ dualizes to $\widehat G_n \to \widehat G$ , and we have $\widehat G_n = \widehat G_{\mathrm {sc}} \times \widehat C_1$ . Note that $\widehat C_1=Z(\widehat G)^{\circ }$ . The transition map $\widehat G_m \to \widehat G_n$ is then the identity on $\widehat G_{\mathrm {sc}}$ , and the $m/n$ -power map on $\widehat C_1$ . Set $\widehat {\bar G} = \varprojlim \widehat G_n = \widehat G_{\mathrm {sc}} \times \widehat C_{\infty }$ , where $\widehat C_{\infty } = \varprojlim \widehat C_n$ . Then $\widehat {\bar G}$ is the universal cover of $\widehat G$ . Elements of $\widehat {\bar G}$ can be written as $(a,(b_n)_n)$ , where $a \in \widehat G_{\mathrm {sc}}$ and $(b_n)_n$ is a sequence of elements $b_n \in \widehat C_1$ satisfying $b_n=(b_m)^{\frac {m}{n}}$ for $n|m$ . In this presentation, the natural map $\widehat {\bar G} \to \widehat G$ sends $(a,(b_n))$ to $a_{\mathrm {der}} \cdot b_1$ , where $a_{\mathrm {der}} \in \widehat G_{\mathrm {der}}$ is the image of $a \in \widehat G_{\mathrm {sc}}$ under the natural map $\widehat G_{\mathrm {sc}} \to \widehat G_{\mathrm {der}}$ .

Definition 2.3.1. Let $Z(\widehat {\bar G})^+ \subset S_{\phi }^+ \subset \widehat {\bar G}$ be the preimages of $Z(\widehat G)^{\Gamma } \subset S_{\phi } \subset \widehat G$ under $\widehat {\bar G}\to \widehat G$ .

Given a character $\lambda : \pi _0(Z(\widehat {\bar G})^+) \to \mathbf {C}^{\times }$ (which we will always assume trivial on the kernel of $Z(\widehat {\bar G})^+ \to \widehat G_n$ for some n), let $\operatorname {\mathrm {Rep}}(\pi _0(S_{\phi }^+),\lambda )$ denote the set of isomorphism classes of representations of $\pi _0(S_{\phi }^+)$ whose pullback to $\pi _0(Z(\widehat {\bar G})^+)$ is $\lambda $ -isotypic, and let $\operatorname {\mathrm {Irr}}(\pi _0(S_{\phi }^+),\lambda )$ be the (finite) subset of irreducible representations. Let $\lambda _z$ be the character corresponding to the class of z under the Tate–Nakayama isomorphism

$$\begin{align*}H^1(u \to W,Z(G^*) \to G^*) \to \pi_0(Z(\widehat{\bar G})^+)^* \end{align*}$$

of [Kal16b, Corollary 5.4], and let $\lambda _{z_b}$ be the character corresponding to the class of $z_b$ in $H^1(u \to W,Z(G) \to G)$ . Then according to [Kal16a, Conjecture G], upon fixing a Whittaker datum $\mathfrak {w}$ for $G^*$ , there are bijections

$$ \begin{align*} \Pi_{\phi}(G) &\cong \operatorname{\mathrm{Irr}}(\pi_0(S_{\phi}^+),\lambda_{z})\\ \Pi_{\phi}(G_{z_b}) &\cong \operatorname{\mathrm{Irr}}(\pi_0(S_{\phi}^+),\lambda_{z}+\lambda_{z_b}) \end{align*} $$

again uniquely determined by the endoscopic character identities. We write $\pi \mapsto \tau _{z,\mathfrak {w},\pi }$ , $\rho \mapsto \tau _{z,\mathfrak {w},\rho }$ for these bijections and $\tau \mapsto \pi _{z,\mathfrak {w},\tau }$ , $\tau \mapsto \rho _{z,\mathfrak {w},\tau }$ for their inverses. We form the representation $\check \tau _{z,\mathfrak {w},\pi } \otimes \tau _{z,\mathfrak {w},\rho } \in \operatorname {\mathrm {Rep}}(\pi _0(S_{\phi }^+),\lambda _{z_b})$ , where we are identifying $\rho $ with the representation $\rho \circ \mathrm {Ad}(g)$ of $G_{z_b}(F)$ .

Recall the map [Kal18, (4.7)]

(2.3.2) $$ \begin{align} S_{\phi}^+ \to S_{\phi},\qquad (a,(b_n)) \mapsto \frac{a_{\mathrm{der}} \cdot b_1}{N_{E/F}(b_{[E:F]})}. \end{align} $$

Here, $a_{\mathrm {der}} \in \widehat G_{\mathrm {der}}$ is the image of $a \in \widehat G_{\mathrm {sc}}$ under the natural map $\widehat G_{\mathrm {sc}} \to \widehat G_{\mathrm {der}}$ and $E/F$ is a sufficiently large finite Galois extension. This map is independent of the choice of $E/F$ . According to [Kal18, Lemma 4.1], pulling back along this map defines a natural bijection $\operatorname {\mathrm {Irr}}(\pi _0(S_{\phi }^+),\lambda _{z_b}) \cong \operatorname {\mathrm {Irr}}(S_{\phi },\lambda _b)$ . Note that since $\phi $ is discrete the group $S_{\phi }^{\natural }$ defined in loc. cit. is equal to $S_{\phi }$ . The lemma remains valid, with the same proof, if we remove the requirement of the representations being irreducible, and we obtain the bijection $\operatorname {\mathrm {Rep}}(\pi _0(S_{\phi }^+),\lambda _{z_b}) \to \operatorname {\mathrm {Rep}}(S_{\phi },\lambda _b)$ .

Definition 2.3.2. Let $\delta _{\pi ,\rho }$ be the image of $\check \tau _{z,\mathfrak {w},\pi } \otimes \tau _{z,\mathfrak {w},\rho }$ under the bijection $\operatorname {\mathrm {Rep}}(\pi _0(S_{\phi }^+),\lambda _{z_b}) \to \operatorname {\mathrm {Rep}}(S_{\phi },\lambda _b)$ .

In the situation when G is a B-inner form of $G^*$ , this definition of $\delta _{\pi ,\rho }$ agrees with the one of Subsection 2.2, because then we can obtain z from $b^*$ just like we obtained $z_b$ from b, and then $\tau _{z,\mathfrak {w},\pi }$ and $\tau _{b^*,\mathfrak {w},\pi }$ are related via equation (2.3.2) and so are $\tau _{z,\mathfrak {w},\rho }$ and $\tau _{b^*,\mathfrak {w},\rho }$ ; see [Kal18, §4.2].

Lemma 2.3.3. Assume [Kal16a, Conjecture G]. The representation $\delta _{\pi ,\rho }$ is independent of the choices of Whittaker datum $\mathfrak {w}$ and of a rigidifying 1-cocycle $z \in Z^1(u \to W,Z(G^*) \to G^*)$ .

Proof. Both of these statements follow from [Kal16a, Conjecture G]. For the independence of Whittaker datum, one can prove that the validity of this conjecture implies that if $\mathfrak {w}$ is replaced by another choice $\mathfrak {w'}$ , then there is an explicitly constructed character $(\mathfrak {w},\mathfrak {w'})$ of $\pi _0(S_{\phi }/Z(\widehat G)^{\Gamma })$ whose inflation to $\pi _0(S_{\phi }^+)$ satisfies $\tau _{z,\mathfrak {w},\sigma }=\tau _{z,\mathfrak {w'},\sigma } \otimes (\mathfrak {w},\mathfrak {w'})$ for any $\sigma \in \Pi _{\phi }(G) \cup \Pi _{\phi }(G_b)$ . See §4 and in particular Theorem 4.3 of [Kal13], the proof of which is valid for a general G that satisfies [Kal16a, Conjecture G], bearing in mind that the transfer factor we use here is related to the one used there by $s \mapsto s^{-1}$ . The independence of z follows from the same type of argument but now using [Kal18, Lemma 6.2].

2.4 Spaces of local shtukas and their cohomology

We recall here some material from [SW20] and [Far] regarding the Fargues–Fontaine curve and moduli spaces of local shtukas.

Let k be the residue field of F. For a perfectoid space S over k, we have the Fargues–Fontaine curve $X_S$ [FF18], an adic space over F. For $S=\operatorname {\mathrm {Spa}}(R,R^+)$ affinoid with pseudouniformiser $\varpi $ , the adic space $X_S$ is defined as follows:

$$ \begin{align*} Y_S &= (\operatorname{\mathrm{Spa}} W_{\mathcal{O}_F}(R^+))\backslash\left\{ p[\varpi]=0 \right\} \\ X_S &= Y_S/\operatorname{\mathrm{Frob}}^{\mathbf{Z}}. \end{align*} $$

Here, $\operatorname {\mathrm {Frob}}$ is the qth power Frobenius on S.

For an affinoid perfectoid space S lying over the residue field of F, the following sets are in bijection:

1. 1. S-points of $\operatorname {\mathrm {Spd}} F$ ,

2. 2. Untilts $S^{\sharp }$ of S over F,

3. 3. Cartier divisors of $Y_S$ of degree 1.

Given an untilt $S^{\sharp }$ , we let $D_{S^{\sharp }}\subset Y_S$ be the corresponding divisor. If $S^{\sharp }=\operatorname {\mathrm {Spa}}(R^{\sharp },R^{\sharp +})$ is affinoid, then the completion of $Y_S$ along $D_{S^{\sharp }}$ is $\operatorname {\mathrm {Spf}} B_{\operatorname {\mathrm {dR}}}^+(R^{\sharp })$ , where $B_{\operatorname {\mathrm {dR}}}^+(R^{\sharp })$ is the de Rham period ring attached to the perfectoid algebra $R^{\sharp }$ . The untilt $S^{\sharp }$ determines a Cartier divisor on $X_S$ , which we still refer to as $D_{S^{\sharp }}$ .

There is a functor $b \mapsto \mathcal {E}^b$ from the category of isocrystals with G-structure to the category of G-bundles on $X_S$ (for any S). When S is a geometric point this functor induces a bijection between the sets of isomorphism classes [Far20].

We now recall Scholze’s definition of the local shtuka space. It is a set-valued functor on the pro-étale site of perfectoid spaces over $\mathbf {F}_p$ and is equipped with a morphism to $\operatorname {\mathrm {Spd}} C$ . Thus, it can be described equivalently as a set-valued functor on the pro-étale site of perfectoid spaces over C.

Definition 2.4.1. The local shtuka space $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ inputs a perfectoid C-algebra $(R,R^+)$ and outputs the set of isomorphisms

$$\begin{align*}\gamma\colon \mathcal{E}^1\vert_{X_{R^{\flat}}\backslash D_R} \cong \mathcal{E}^b\vert_{X_{R^{\flat}}\backslash D_R} \end{align*}$$

of G-torsors that are meromorphic along $D_R$ and bounded by $\mu $ pointwise on $\operatorname {\mathrm {Spa}} R$ .

Let us briefly recall the condition of being pointwise bounded by $\mu $ . If $\operatorname {\mathrm {Spa}}(C,O_C) \to \operatorname {\mathrm {Spa}} R$ is a geometric point, we obtain via pullback $\gamma : \mathcal {E}^1|_{X_{C^{\flat }} \smallsetminus \{x_C\}} \to \mathcal {E}^b|_{X_{C^{\flat }} \smallsetminus \{x_C\}}$ , where we have written $x_C$ in place of $D_C$ to emphasize that this a point on $X_{C^{\flat }}$ . The completed local ring of $X_{C^{\flat }}$ at $x_C$ is Fointaine’s ring $B_{\operatorname {\mathrm {dR}}}^+(C)$ . A trivialization of both bundles $\mathcal {E}^1$ and $\mathcal {E}^b$ on a formal neighborhood of $x_C$ , together with $\gamma $ , leads to an element of $G(B_{\operatorname {\mathrm {dR}}}(C))$ , well-defined up to left and right multiplication by elements of $G(B_{\operatorname {\mathrm {dR}}}^+(C))$ . The corresponding element of the double coset space $G(B_{\operatorname {\mathrm {dR}}}^+(C)) \setminus G(B_{\operatorname {\mathrm {dR}}}(C)) / G(B_{\operatorname {\mathrm {dR}}}^+(C))$ is indexed by a conjugacy class of cocharacters of $G/C$ according to the Cartan decomposition, and we demand that this conjugacy class is dominated by $\mu $ in the usual order (given by the simple roots of the universal Borel pair).

The space $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ is a locally spatial diamond [SW20, §23]. Since the automorphism groups of $\mathcal {E}^1$ and $\mathcal {E}^b$ are the constant group diamonds $\underline {G(\mathbf {Q}_p)}$ and $\underline {G_b(\mathbf {Q}_p)}$ , respectively, the space $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ is equipped with commuting actions of $G(\mathbf {Q}_p)$ and $G_b(\mathbf {Q}_p)$ , acting by pre- and postcomposition on $\gamma $ .

Remark 2.4.2. According to [SW20, Corollary 23.2.2], the above definition recovers the moduli space of local shtukas with one leg and infinite level structure. We have dropped the subscript $\infty $ used in [SW20] to denote the infinite level structure.

We will use the cohomology theory developed in [Sch17]. For any compact open subgroup $K \subset G(F)$ , the quotient $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}=\operatorname {\mathrm {Sht}}_{G,b,\mu }/K$ is again a locally spatial diamond [SW20, §23]. For each $n=1,2,\dots $ , let $V_{\mu ,n}\in \operatorname {\mathrm {Rep}}(\widehat {G},\mathbf {Z}/\ell ^n\mathbf {Z})$ be the Weyl module associated to $\mu $ . By the geometric Satake equivalence (Theorem 5.1.1), there is a corresponding object $\mathcal {S}_{\mu ,n}$ of $D_{\mathrm {\acute {e}t}}(\operatorname {\mathrm {Gr}}_{G,b,\leq \mu },\mathbf {Z}/\ell ^n\mathbf {Z}[\sqrt {q}])$ . Define

$$\begin{align*}R\Gamma_c(\operatorname{\mathrm{Sht}}_{G,b,\mu}/K,\mathcal{S}_{\mu}) = \varinjlim_U R\Gamma_c(U,\mathcal{S}_{\mu}), \end{align*}$$

where $U\subset \operatorname {\mathrm {Sht}}_{G,b,\mu }/K$ runs over quasicompact open subsets and where we have put

$$\begin{align*}R\Gamma_c(U,\mathcal{S}_{\mu})=\varprojlim_n R\Gamma_c(U,\mathcal{S}_{\mu,n}). \end{align*}$$

Then $R\Gamma _c(\operatorname {\mathrm {Sht}}_{G,b,\mu }/K,\mathcal {S}_{\mu })$ is a complex of $\mathbf {Z}_{\ell }[\sqrt {q}]$ -modules carrying an action of $G_b(F)\times W_E$ .

Definition 2.4.3. Let $\rho $ be a finite-length admissible representation of $G_b(F)$ with coefficients in $\overline {\mathbf {Q}_{\ell }}$ . Then we define

$$\begin{align*}R\Gamma(G,b,\mu)[\rho]= \varinjlim_{K\subset G(F)} R\operatorname{\mathrm{Hom}}_{G_b(F)}(R\Gamma_c(\operatorname{\mathrm{Sht}}_{G,b,\mu}/K,\mathcal{S}_{\mu}) \otimes \overline{\mathbf{Q}_{\ell}}, \rho),\end{align*}$$

where K runs over the set of open compact subgroups of $G(F)$ .

By Proposition 6.4.5 below, this defines a finite-length $W_E$ -equivariant object in the derived category of smooth representations of $G(F)$ with coefficients in $\overline {\mathbf {Q}_{\ell }}$ , and we write $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho )$ for the image of $R\Gamma (G,b,\mu )[\rho ]$ in $\operatorname {\mathrm {Groth}}(G(F) \times W_E)$ .

Remark 2.4.4. We now discuss the relationship between our definition of $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho )$ and the virtual representation $H^*(G,b,\mu )[\rho ]$ defined in [RV14].

When $\mu $ is minuscule, $\operatorname {\mathrm {Sht}}_{G,b,\mu ,K}$ is the diamond $\mathcal {M}_{G,b, \mu ,K}^{\diamond }$ associated to the local Shimura variety $\mathcal {M}_{G,b,\mu ,K}$ [SW20, §24.1]. The latter is a rigid-analytic variety of dimension $d=\left < \mu ,2\rho _G \right>$ , where $2\rho _G$ is the sum of the positive roots. Moreover, in that case, $\mathcal {S}_{\mu }=\mathbf {Z}_{\ell }[\sqrt {q}][d](\tfrac {d}{2})$ is a shift and twist of the constant sheaf. In [RV14], $H^*(G,b,\mu )[\rho ]$ is defined as the alternating sum

$$\begin{align*}\sum_{i,j\in\mathbf{Z}} (-1)^{i+j} H^{i,j}(G,b,\mu)[\rho](-d),\end{align*}$$

where

$$\begin{align*}H^{i,j}(G,b,\mu)[\rho]=\varinjlim_K \mathrm{Ext}^i_{G_b(F)}(H^{j}_{c}(\mathcal{M}_{G,b,\mu,K},\mathbf{Z}_{\ell}) \otimes \overline{\mathbf{Q}_{\ell}},\rho) \end{align*}$$

Note that $H^{i,j}(G,b,\mu )[\rho ]$ vanishes for all but finitely many $(i,j)$ , and each $H^{i,j}(G,b,\mu )[\rho ]$ is an admissible representation of $G_b(F)$ by the analysis in [FS21]. On the other hand, unwinding definitions, we see that there is a spectral sequence $H^{i,j-d}(G,b,\mu )[\rho ](-\tfrac {d}{2}) \implies H^{i+j}\left ( R\Gamma (G,b,\mu )[\rho ] \right )$ .

Putting these observations together, we get the equality

$$\begin{align*}\operatorname{\mathrm{Mant}}_{b,\mu}(\rho) = (-1)^d H^*(G,b,\mu)[\rho](\tfrac{d}{2}).\end{align*}$$

Note that in our formulation, the Tate twist appearing in [RV14, Conjecture 7.3] has been absorbed into the normalization of $\operatorname {\mathrm {Mant}}_{b,\mu }$ .

3 Transfer of conjugation-invariant functions from $G(F)$ to $G_b(F)$

Throughout, $F/\mathbf {Q}_p$ is a finite extension, and $G/F$ is a connected reductive group.

3.1 The space of strongly regular conjugacy classes in $G(F)$

The following definitions are important for our work.

• • $G_{\operatorname {\mathrm {rs}}}\subset G$ is the open subvariety of regular semisimple elements, meaning those whose connected centralizer is a maximal torus.

• • $G_{\operatorname {\mathrm {sr}}}\subset G$ is the open subvariety of strongly regular semisimple elements, meaning those regular semisimple elements whose centralizer is connected, i.e., a maximal torus.

• • $G(F)_{\operatorname {\mathrm {ell}}}\subset G(F)$ is the open subset of strongly regular elliptic elements, meaning those strongly regular semisimple elements in $G(F)$ whose centralizer is an elliptic maximal torus.

We put $G(F)_{\operatorname {\mathrm {sr}}}=G_{\operatorname {\mathrm {sr}}}(F)$ and $G(F)_{\operatorname {\mathrm {rs}}}=G_{\operatorname {\mathrm {rs}}}(F)$ . Note that $G(F)_{\operatorname {\mathrm {ell}}}\subset G(F)_{\operatorname {\mathrm {sr}}}\subset G(F)_{\operatorname {\mathrm {rs}}}$ . The inclusion $G(F)_{\operatorname {\mathrm {sr}}}\subset G(F)_{\operatorname {\mathrm {rs}}}$ is dense.

If g is regular semisimple, then it is necessarily contained in a unique maximal torus T, namely the neutral component $\operatorname {\mathrm {Cent}}(g,G)^{\circ }$ , but this is not necessarily all of $\operatorname {\mathrm {Cent}}(g,G)$ . If $G_{\mathrm {der}}$ is simply connected, then $\operatorname {\mathrm {Cent}}(g,G)$ is connected; thus, in such a group, regular semisimple and strongly regular semisimple mean the same thing.

Observe that if g is regular semisimple, then $\alpha (g)\neq 1$ for all roots $\alpha $ relative to the action of T. Indeed, if $\alpha (g)=1$ , then the root subgroup of $\alpha $ would commute with g, and then it would have dimension strictly greater than $\dim T$ .

All of the sets $G(F)_{\operatorname {\mathrm {sr}}}$ , $G(F)_{\operatorname {\mathrm {rs}}}$ , $G(F)_{\operatorname {\mathrm {ell}}}$ are conjugacy-invariant, so we may for instance consider the quotient , considered as a topological space.

Lemma 3.1.1. is locally profinite, in fact equal to the disjoint union of the locally profinite sets $T(F)_{\operatorname {\mathrm {rs}}}/N(T,G)(F)$ , where T runs over the set of $G(F)$ -conjugacy classes of F-rational maximal tori in G, and $N(T,G)$ is the normalizer of T in G. The same is true with ‘rs’ replaced by ‘sr’.

Proof. Let $T\subset G$ be a F-rational maximal torus. The set $H^1(F,N(T,G))$ classifies conjugacy classes of F-rational tori, as follows: Given a F-rational torus $T'$ , we must have $T'=xTx^{-1}$ for some $x\in G(\overline {F})$ . Then for all $\sigma \in \operatorname {\mathrm {Gal}}(\overline {F}/F)$ , $x^{-1} x^{\sigma }$ normalizes T. We associate to $T'$ the class of $\sigma \mapsto x^{-1}x^{\sigma }$ in $H^1(F,N(T,G))$ , and it is a simple matter to see that this defines a bijection as claimed. (In fact $H^1(F,N(T,G))$ is finite.)

There is a map , sending the conjugacy class of $g\in G(F)_{\operatorname {\mathrm {rs}}}$ to the conjugacy class of the unique F-rational torus containing it, namely $\operatorname {\mathrm {Cent}}(g,G)^{\circ }$ . We claim that this map is locally constant.

To prove the claim, we consider

$$\begin{align*}\varphi : G(F) \times T_{\operatorname{\mathrm{rs}}}(F) \to G_{\operatorname{\mathrm{rs}}}(F), \;\;\; (g,t)\mapsto gtg^{-1}, \end{align*}$$

a morphism of p-adic analytic varieties. We would like to show that $\varphi $ is open. To do this, we will compute its differential at the point $(g,t)$ by means of a change of variable. Consider the map

$$\begin{align*}\psi = L_{gtg^{-1}}^{-1}\circ\varphi \circ (L_g \times L_t).\end{align*}$$

Explicitly, for $(z,w) \in G(F) \times T(F)$ , we have $\psi (z,w)=gt^{-1}ztwz^{-1}g^{-1}$ .

Let $\mathfrak {g}=\operatorname {\mathrm {Lie}} G$ , $\mathfrak {t}=\operatorname {\mathrm {Lie}} T$ . The derivative $d\psi (1,1) : \mathfrak {g} \times \mathfrak {t} \to \mathfrak {g}$ is given by the formula

$$\begin{align*}d\psi(1,1)(Z,W)=\mathrm{Ad}(g)[ (\mathrm{Ad}(t^{-1})-\mathrm{id})Z+W].\end{align*}$$

We would like to check that $d\psi (1,1)$ is surjective. We may decompose $\mathfrak {g}=\mathfrak {t} \oplus \mathfrak {t}^{\perp }$ , where $\mathfrak {t}^{\perp }$ is the descent to F of the direct sum of all root subspaces of $\mathfrak {g}_{\overline {F}}$ for the action of T.

The element t is regular, hence $\alpha (t)\neq 1$ for all roots of $\mathfrak {g}$ for the action of T. Therefore, $\mathrm {Ad}(t^{-1})-\mathrm {id} : \mathfrak {g}/\mathfrak {t} \to \mathfrak {t}^{\perp }$ is an isomorphism. It follows that $d\psi $ is surjective. The derivative of $\varphi $ at $(g,t)$ is

$$\begin{align*}d\varphi(g,t)=dL_{gtg^{-1}}(gtg^{-1})\circ d\psi(1,1)\circ (dL_g(1) \times dL_t(1)).\end{align*}$$

All terms $dL$ are isomorphisms, so $d\varphi (g,t)$ is also surjective. Thus, $\varphi $ is a submersion in the sense of Bourbaki VAR §5.9.1; hence, it is open by loc. cit. §5.9.4.

Therefore, if $g\in T(F)_{\operatorname {\mathrm {rs}}}$ and $g'$ is sufficiently close to g in $G(F)$ , then $g'$ is conjugate in $G(F)$ to an element of $T(F)$ , which proves the claim about the local constancy of .

The fiber of this map over $T'$ is $T'(F)_{\operatorname {\mathrm {rs}}}$ modulo the action of the finite group $N(T',G)(F)/T'(F)$ . Since $T'(F)_{\operatorname {\mathrm {rs}}}$ is locally profinite so is its quotient by the action of a finite group.

3.2 Hecke transfer maps

Suppose that $b\in G(\breve {F})$ is basic. The goal of this section is to define a family of explicit maps, which input a conjugation-invariant function on $G(F)_{\operatorname {\mathrm {sr}}}$ and output a conjugation-invariant function on $G_b(F)_{\operatorname {\mathrm {sr}}}$ . We shall call them Hecke transfer maps as a way of foreshadowing their relation to the Hecke operators defined on the stack $\operatorname {\mathrm {Bun}}_G$ .

Given a sufficiently strong version of the local Langlands conjectures, we will show that the Hecke transfer maps act predictably on the trace characters attached to irreducible admissible representations.

We begin by recalling the concept of related elements and the definition of their invariant in the isocrystal setting from [Kal14].

Lemma 3.2.1. Suppose $g\in G(F)$ and $g'\in G_b(F)$ are strongly regular elements which are conjugate over an algebraic closure of $\breve {F}$ . Then they are conjugate over $\breve F$ .

Proof. Let K be an algebraic closure of $\breve {F}$ . Say $g'=zgz^{-1}$ with $z\in G(K)$ . Let $T=\operatorname {\mathrm {Cent}}(g,G)$ ; then for all $\tau $ in the inertia group $ \operatorname {\mathrm {Gal}}(\overline {F}/F^{\operatorname {\mathrm {nr}}})$ , $z^{-\tau }z$ commutes with g and therefore lies in $T(K)$ . Then $\tau \mapsto z^{-\tau } z$ is a cocycle in $H^1(\breve {F},T)$ . Since T is a connected algebraic group, $H^1(\breve {F},T)=0$ [Ste65, Theorem 1.9]. If $x\in T(K)$ splits the cocycle, then $y=zx^{-1}\in G(\breve {F})$ , and $g'=ygy^{-1}$ so that g and $g'$ are related.

It is customary to call elements $g,g'$ as in the above lemma stably conjugate, or related. Suppose we have strongly regular elements $g\in G(F)_{\operatorname {\mathrm {sr}}}$ and $g'\in G_b(F)_{\operatorname {\mathrm {sr}}}$ which are related. Let $T=\operatorname {\mathrm {Cent}}(g,G)$ , and suppose $y\in G(\breve {F})$ with $g'=ygy^{-1}$ . The rationality of g means that $g^{\sigma }=g$ , whereas the rationality of $g'$ in $G_b$ means that $(g')^{\sigma } = b^{-1}g' b$ . Combining these statements shows that $b_0:=y^{-1}by^{\sigma }$ commutes with g and therefore lies in $T(\breve {F})$ .

Definition 3.2.2. For strongly regular related elements $g\in G(F)_{\operatorname {\mathrm {sr}}}$ and $g'\in G_b(F)_{\operatorname {\mathrm {sr}}}$ , the invariant $\operatorname {\mathrm {inv}}[b](g,g')$ is the class of $y^{-1}by^{\sigma }$ in $B(T)$ , where $y\in G(\breve {F})$ satisfies $g'=ygy^{-1}$ .

Fact 3.2.3. The invariant $\operatorname {\mathrm {inv}}[b](g,g')\in B(T)$ only depends on b, g and $g'$ and not on the element y which conjugates g into $g'$ . It depends on the rational conjugacy classes of g and $g'$ as follows:

• • For $z\in G(F)$ , we have $\operatorname {\mathrm {inv}}[b]((\operatorname {\mathrm {ad}} z)(g),g') = (\operatorname {\mathrm {ad}} z)(\operatorname {\mathrm {inv}}[b](g,g'))$ , a class in $B((\operatorname {\mathrm {ad}} z)(T))$ .

• • For $z\in G_b(F)$ , we have $\operatorname {\mathrm {inv}}[b](g,(\operatorname {\mathrm {ad}} z)(g')) = \operatorname {\mathrm {inv}}[b](g,g')$ .

The image of $\mathrm {inv}[b](g,g')$ under the composition of $B(T) \to B(G)$ and $\kappa \colon B(G) \to \pi _1(G)_{\Gamma }$ equals $\kappa (b)$ .

Definition 3.2.4. We define a diagram of topological spaces

(3.2.1)

as follows. The space $\operatorname {\mathrm {Rel}}_b$ is the set of conjugacy classes of triples $(g,g',\lambda )$ , where $g\in G(F)_{\operatorname {\mathrm {sr}}}$ and $g'\in G_b(F)_{\operatorname {\mathrm {sr}}}$ are related, and $\lambda \in X_*(T)$ , where $T=\operatorname {\mathrm {Cent}}(g,G)$ . It is required that $\kappa (\operatorname {\mathrm {inv}}[b](g,g'))$ agrees with the image of $\lambda $ in $X_*(T)_{\Gamma }$ . We consider $(g,g',\lambda )$ conjugate to $((\operatorname {\mathrm {ad}} z)(g),(\operatorname {\mathrm {ad}} z')(g'),(\operatorname {\mathrm {ad}} z)(\lambda ))$ whenever $z\in G(F)$ and $z'\in G_b(F)$ . We give $\operatorname {\mathrm {Rel}}_b\subset (G(F) \times G_b(F)\times X_*(G))/(G(F)\times G_b(F))$ the subspace topology, where $X_*(G)$ is taken to be discrete.

Remark 3.2.5. Given $g\in G(F)_{\operatorname {\mathrm {sr}}}$ and $\lambda $ a cocharacter of its torus, there is at most one conjugacy class of $g'\in G_b(F)$ with $(g,g',\lambda )\in \operatorname {\mathrm {Rel}}_b$ . In other words, g and $\operatorname {\mathrm {inv}}[b](g,g')$ determine the conjugacy class of $g'$ . Indeed, suppose $(g,g',\lambda )$ and $(g,g'',\lambda )$ are both in $\operatorname {\mathrm {Rel}}_b$ . Then $g'=ygy^{-1}$ and $g''=zgz^{-1}$ for some $y,z\in G(F^{\operatorname {\mathrm {nr}}})$ , and $y^{-1}by^{\sigma }$ and $z^{-1}bz^{\sigma }$ are $\sigma $ -conjugate in $T(\breve {F})$ . This means there exists $t\in T(\breve {F})$ such that $y^{-1}by^{\sigma }=(zt)^{-1}b (zt)^{\sigma }$ . We see that $x=zty^{-1}\in G_b(F)$ , and that x conjugates $g'$ onto $g''$ .

Lemma 3.2.6. The map is a homeomorphism locally on the source. Its image consists of those classes that transfer to $G_b$ . In particular, the image is open and closed.

The analogous statement is true for .

Proof. The proof of Lemma 3.1.1 shows that is the disjoint union of spaces $T(F)_{\operatorname {\mathrm {sr}}}/W_T$ , as $T\subset G$ runs through the finitely many conjugacy classes of F-rational maximal tori, and $W_T=N(T,G)(F)/T(F)$ is a finite group. By the above remark, $\operatorname {\mathrm {Rel}}_b$ injects into the disjoint union of the spaces $T(F)_{\operatorname {\mathrm {sr}}}/W_T\times X_*(T)$ , with the map to corresponding to the projection $T(F)_{\operatorname {\mathrm {sr}}}/W_T\times X_*(T)\to T(F)_{\operatorname {\mathrm {sr}}}/W_T$ . Since $X_*(T)$ is discrete, this map is a homeomorphism locally on the source. The other statements are evident from the definitions.

The definition of $\operatorname {\mathrm {Rel}}_b$ already suggests a means for transferring functions from to , namely, by pulling back from to $\operatorname {\mathrm {Rel}}_b$ , multiplying by a compactly supported kernel function and then pushing forward to . We will define one such kernel function for each geometric conjugacy class of cocharacters $\mu \colon \mathbf {G}_{\mathrm {m}}\to G_{\overline {F}}$ .

Let $\widehat {G}$ be the Langlands dual group. It comes equipped with a splitting, in particular with a torus and Borel $\widehat T \subset \widehat B \subset \widehat G$ . Given a conjugacy class of cocharacters $\mu $ for G as above, we obtain a character $\widehat \mu : \widehat T \to \mathbf {G}_{\mathrm {m}}$ which is $\widehat B$ -dominant. Let $r_{\mu }$ be the Weyl module of the dual group $\widehat G$ whose highest weight with respect to $(\widehat T,\widehat B)$ is $\widehat \mu $ .

A cocharacter $\lambda \in X_*(T)$ corresponds to a character $\widehat {\lambda }\in X^*(\widehat {T})$ . Let $r_{\mu }[\lambda ]$ be the $\widehat {\lambda }$ -weight space of $r_{\mu }$ . The quantity $\dim r_{\mu }[\lambda ]$ will give us our kernel function. While we will not need it here, we note that there is an explicit formula for $\dim r_{\mu }[\lambda ]$ coming from the Weyl character formula.

We now fix a commutative ring $\Lambda $ in which p is invertible. For a topological space X, we let $C(X,\Lambda )$ be the space of continuous $\Lambda $ -valued functions on X, where $\Lambda $ is given the discrete topology.

Definition 3.2.7. Let $d=\left < \mu ,2\rho _G \right>$ , where $2\rho _G$ is the sum of the positive roots of G. We define the Hecke transfer map

by

$$\begin{align*}[T_{b,\mu}^{G\to G_b} f](g') = (-1)^d\sum_{(g,g',\lambda) \in \operatorname{\mathrm{Rel}}_{b}} f(g) \dim r_{\mu}[\lambda]. \end{align*}$$

Analogously, we define

by

$$\begin{align*}[T_{b,\mu}^{G_b\to G} f'](g) =(-1)^d \sum_{(g,g',\lambda) \in \operatorname{\mathrm{Rel}}_{b}} f'(g') \dim r_{\mu}[\lambda]. \end{align*}$$

Since $r_{\mu }$ is finite-dimensional, the sum is finite. If $f'$ has compact support, then so does its image.

Lemma 3.2.8. The Hecke transfer map $T_{b,\mu }^{G\to G_b}$ is zero unless $[b]$ is the unique basic class in $B(G,\mu )$ .

Proof. Suppose there exists an F-rational maximal torus $T\subset G$ and a cocharacter $\lambda \in X_*(T)$ such that $r_{\mu }[\lambda ]\neq 0$ . Then $\widehat {\mu }$ and $\widehat {\lambda }$ must agree when restricted to the center $Z(\widehat {G})$ , which is to say that $\widehat {\mu }$ and $\widehat {\lambda }$ have the same image in $X^*(Z(\widehat {G}))$ . Equivalently, if we conjugate $\mu $ so as to assume it is a cocharacter of T, then $\mu $ and $\lambda $ have the same image under $X_*(T)\cong \pi _1(T)\to \pi _1(G)$ . By Fact 3.2.3 and the functoriality of $\kappa $ the image of $\lambda $ in $\pi _1(G)_{\Gamma }$ equals $\kappa (b)$ . We conclude that $\kappa ([b])$ equals the image of $\mu $ in $\pi _1(G)_{\Gamma }$ . This means that $[b]$ is the unique basic class in $B(G,\mu )$ .

Assume therefore that $[b]$ is the unique basic class in $B(G,\mu )$ . We may define a ‘truncation’ $\operatorname {\mathrm {Rel}}_{b,\mu }\subset \operatorname {\mathrm {Rel}}_b$ , consisting of conjugacy classes of triples $(g,g',\lambda )$ for which $\lambda \leq \mu $ . Then the kernel function $(g,g',\lambda )\mapsto \dim r_{\mu }[\lambda ]$ is supported on $\operatorname {\mathrm {Rel}}_{b,\mu }$ . In the diagram

(3.2.2)

both maps are finite étale over their respective images.

The following theorem is proved in §3.3. It relates the Hecke transfer map $T_{b,\mu }^{G\to G_b}$ to the local Jacquet–Langlands correspondence for G.

Theorem 3.2.9. Assume that $b\in B(G,\mu )$ is basic and that $\Lambda $ is an algebraically closed field of characteristic 0 abstractly isomorphic to $\mathbf {C}$ . Let $\phi \colon W_F \times \mathrm {SL}_2 \to \;^LG$ be a discrete L-parameter with coefficients in $\Lambda $ , and let $\rho \in \Pi _{\phi }(G_b)$ . Let be its Harish–Chandra character. Then for any $g\in G(F)_{\operatorname {\mathrm {sr}}}$ that transfers to $G_b(F)$ , we have

(3.2.3) $$ \begin{align} \left[T_{b,\mu}^{G_b\to G}\Theta_{\rho}\right](g)=\sum_{\pi\in \Pi_{\phi}(G)} \dim \operatorname{\mathrm{Hom}}_{S_{\phi}}(\delta_{\pi,\rho},r_{\mu})\Theta_{\pi}(g), \end{align} $$

assuming the validity of the refined local Langlands conjecture, i.e., [Kal16a, Conjecture G].

Example 3.2.10. Let $G=\mathrm {GL}_2$ , and let $\mu \colon \bf {G}_m \to G$ the cocharacter sending x to the diagonal matrix with entries $(x,1)$ . We have $\pi _1(G)=\mathbf {Z}$ as a trivial $\Gamma $ -module. Let $b \in B(G,\mu )$ be the basic class. Then b corresponds to the isocrystal of slope $1/2$ , and $G_b(F)$ is the multiplicative group of the nonsplit quaternion algebra over F. Let $\phi $ be a discrete Langlands parameter. The L-packets $\Pi _{\phi }(G)=\left \{ \pi \right \}$ and $\Pi _{\phi }(G_b)=\left \{ \rho \right \}$ are singletons. We have $S_{\phi }=Z(\widehat G)=\mathbf {C}^{\times }$ , and $\delta _{\pi ,\rho }$ is the identity character of $S_{\phi }$ . The representation $r_{\mu }$ is the standard representation of $\widehat G=\mathrm {GL}_2(\mathbf {C})$ , and $\dim \operatorname {\mathrm {Hom}}_{S_{\phi }}(\delta _{\pi ,\rho },r_{\mu })=2$ . Therefore, the right-hand side of equation (3.2.3) equals $2\Theta _{\pi }(g)$ .

Let $w\mu $ be the cocharacter sending x to the diagonal matrix with entries $(1,x)$ . The map $\lambda \mapsto r_{\mu }[\lambda ]$ sends $\mu $ and $w\mu $ to $1$ and all other cocharacters to $0$ . For any strongly regular $g' \in G_b(F)$ , there is a unique $G(F)$ -conjugacy class of strongly regular $g \in G(F)$ related to $g'$ . Let $S \subset G$ be the centralizer of one such g. Then $X_*(S) \cong \mathbf {Z}[\Gamma _{E/F}]$ for a quadratic extension $E/F$ , and the map $\pi _1(S)_{\Gamma } \to \pi _1(G)_{\Gamma }$ is an isomorphism. There are exactly two elements $\lambda ,w\lambda \in X_*(S)$ that map to $\mathrm {inv}[b](g,g')$ . Finally, $d=1$ . Therefore, $T_{b,\mu }^{G \to G_b}f(g')=-2f(g)$ . Setting $f=\Theta _{\rho }$ , we find that Theorem 3.2.9 reduces to the Jacquet–Langlands character identity

$$\begin{align*}\Theta_{\rho}(g')=-\Theta_{\pi}(g). \end{align*}$$

3.3 Proof of Theorem 3.2.9

We now give the proof of Theorem 3.2.9. We will use the notation and results of §A.1.

We are given a discrete L-parameter $\phi $ , a representation $\rho \in \Pi _{\phi }(G_b)$ in its L-packet, and an element $g\in G(F)_{\operatorname {\mathrm {sr}}}$ . We assume that g is related to an element of $G_b(F)$ . This means there exists a triple in $\operatorname {\mathrm {Rel}}_b$ of the form $(g,g',\lambda )$ . For the moment, we fix such a triple $(g,g',\lambda )$ .

Let $s\in S_{\phi }$ be a semisimple element, and let $\dot s \in S_{\phi }^+$ be a lift of it. Then we have the refined endoscopic datum $\mathfrak {\dot e}=(H,\mathcal {H},\dot s,\eta )$ defined in equation (A.1.1); we choose as in that section a z-pair $\mathfrak {z}=(H_1,\eta _1)$ . Then

$$ \begin{align*} \begin{array}{cll} &&\; e(G_b)\displaystyle\sum_{\rho'\in \Pi_{\phi}(G_b)} \operatorname{\mathrm{tr}}\tau_{z,\mathfrak{w},\rho'}(\dot s)\Theta_{\rho'}(g')\\[2pt] &\stackrel{(A.1.2)}{=}& \!\! \displaystyle\sum_{h_1\in H_1(F)/\mathrm{st}} \Delta(h_1,g')S\Theta_{\phi^s}(h_1) \\[2pt] &\stackrel{(A.1.4)}{=}&\!\! \displaystyle\sum_{h_1\in H_1(F)/\mathrm{st}} \Delta(h_1,g)\left< \operatorname{\mathrm{inv}}[b](g,g'),s^{\natural}_{h,g} \right>S\Theta_{\phi^s}(h_1) \\[2pt] & \quad = &\!\! \displaystyle\sum_{h_1\in H_1(F)/\mathrm{st}} \Delta(h_1,g)\lambda(s^{\natural}_{h,g})S\Theta_{\phi^s}(h_1). \end{array} \end{align*} $$

We now multiply this expression by the kernel function $\dim r_{\mu }[\lambda ]$ and then sum over all $G_b(F)$ -conjugacy classes of elements $g' \in G_b(F)$ and all $\lambda \in X_*(T_g)$ such that $(g,g',\lambda )$ lies in $\operatorname {\mathrm {Rel}}_b$ . We obtain

$$ \begin{align*} \begin{array}{cll} & &\; e(G_b) \displaystyle\sum_{(g',\lambda)}\; \sum_{\rho'\in \Pi_{\phi}(G_b)} \operatorname{\mathrm{tr}}\tau_{z,\mathfrak{w},\rho'}(\dot s)\Theta_{\rho'}(g')\dim r_{\mu}[\lambda]\\[2pt] &\ \ = &\!\displaystyle\sum_{h_1\in H_1(F)/\mathrm{st}}\Delta(h_1,g) S\Theta_{\phi^s}(h_1)\sum_{(g',\lambda)}\lambda(s^{\natural}_{h,g})\dim r_{\mu}[\lambda]\\[2pt] &\ \stackrel{(*)}{=}&\!\!\displaystyle\sum_{h_1\in H_1(F)/\mathrm{st}}\Delta(h_1,g)S\Theta_{\phi^s}(h_1)\operatorname{\mathrm{tr}} r_{\mu}(s^{\natural}_{h,g}) \\[2pt] &\ \stackrel{(**)}{=}&\!\!\operatorname{\mathrm{tr}} r_{\mu}(s^{\natural})\displaystyle\sum_{h_1\in H_1(F)/\mathrm{st}}\Delta(h_1,g)S\Theta_{\phi^s}(h_1)\\[2pt] &\stackrel{(A.1.2)}{=}&\!\! \operatorname{\mathrm{tr}} r_{\mu}(s^{\natural})e(G)\displaystyle\sum_{\pi\in \Pi_{\phi}(G)} \operatorname{\mathrm{tr}}\tau_{z,\mathfrak{w},\pi}(\dot s)\Theta_{\pi}(g).\\[2pt] \end{array} \end{align*} $$

We justify $(**)$ : Let $T \subset G$ be the centralizer of g. The image of $s^{\natural }_{h,g}$ under any admissible embedding $\widehat T \to \widehat G$ is conjugate to $s^{\natural }$ in $\widehat {G}$ and $\operatorname {\mathrm {tr}} r_{\mu }$ is conjugation-invariant. Recall here that $s^{\natural } \in S_{\phi }$ is the image of $\dot s$ under equation (2.3.2).

We justify $(*)$ : $\lambda \in X_*(T)$ determines the $G_b(F)$ -conjugacy class of $g'$ since $\mathrm {inv}[b](g,g') \in B(T)$ determines it. Therefore, the sum over $(g',\lambda )$ is in reality a sum only over $\lambda $ . There exists $g' \in G_b(F)$ with $\kappa (\mathrm {inv}[b](g,g'))$ being the image of $\lambda $ in $X_*(T)_{\Gamma }$ if and only if the image of $\lambda $ under $X_*(T) \to X_*(T)_{\Gamma } \to \pi _1(G)_{\Gamma }$ equals $\kappa (b)$ . Since the image of $\mu $ in $\pi _1(G)_{\Gamma }$ also equals $\kappa (b)$ , the sum over $(g',\lambda )$ is in fact the sum over $\lambda \in X_*(T)$ having the same image as $\mu $ in $\pi _1(G)_{\Gamma }$ . In terms of the dual torus $\widehat T$ , this is the sum over $\lambda \in X^*(\widehat T)$ whose restriction to $Z(\widehat G)^{\Gamma }$ equals that of $\mu $ . Since, for $\lambda $ not satisfying this condition the number $\mathrm {dim}r_{\mu }[\lambda ]$ is zero, we may extend the sum to be over all $\lambda \in X_*(T)=X^*(\widehat T)$ .

We now continue with the equation. Multiply both sides of the above equation by $\operatorname {\mathrm {tr}}\check \tau _{z,\mathfrak {w},\rho }(\dot s)$ . As functions of $\dot s \in S_{\phi }^+$ , both sides then become invariant under $Z(\widehat {\bar G})^+$ and thus become functions of the finite quotient $\bar S_{\phi } = S_{\phi }^+/Z(\widehat {\bar G})^+=S_{\phi }/Z(\widehat G)^{\Gamma }$ . Now apply $\left \lvert \bar S_{\phi } \right \rvert ^{-1}\sum _{\bar s \in \bar S_{\phi }}$ to both sides to obtain an equality between

(3.3.1) $$ \begin{align} \left\lvert \bar S_{\phi} \right\rvert^{-1} e(G_b)\sum_{\bar s \in \bar S_{\phi}} \sum_{(g',\lambda)} \sum_{\rho'\in \Pi_{\phi}(G_b)} \operatorname{\mathrm{tr}}\check\tau_{z,\mathfrak{w},\rho}(\dot s)\operatorname{\mathrm{tr}}\tau_{z,\mathfrak{w},\rho'}(\dot s)\Theta_{\rho'}(g')\dim r_{\mu}[\lambda] \end{align} $$

and

(3.3.2) $$ \begin{align} \left\lvert \bar S_{\phi} \right\rvert^{-1}e(G) \sum_{\bar s \in \bar S_{\phi}}\operatorname{\mathrm{tr}} r_{\mu}(s^{\natural})\sum_{\pi\in \Pi_{\phi}(G)} \operatorname{\mathrm{tr}}\check\tau_{z,\mathfrak{w},\rho}(\dot s) \operatorname{\mathrm{tr}}\tau_{z,\mathfrak{w},\pi}(\dot s)\Theta_{\pi}(g), \end{align} $$

where in both formulas $\dot s$ is an arbitrary lift of $\bar s$ and $s^{\natural } \in S_{\phi }$ is the image of $\dot s$ under equation (2.3.2). Executing the sum over $\bar s$ in equation (3.3.1) gives

$$\begin{align*}e(G_b)\sum_{(g',\lambda)} \Theta_{\rho}(g')\dim r_{\mu}[\lambda] = e(G_b)[T^{G_b\to G}_{b,\mu}\Theta_{\rho}](g). \end{align*}$$

To treat equation (3.3.2) note that $\check \tau _{z,\mathfrak {w},\rho }\otimes \tau _{z,\mathfrak {w},\pi }(\dot s) = \check {\delta }_{\pi ,\rho }(s^{\natural })$ . Furthermore, the composition of the map (2.3.2) with the natural projection $S_{\phi } \to S_{\phi }/Z(\widehat G)^{\Gamma }$ is equal to the natural projection $S_{\phi }^+ \to S_{\phi }^+/Z(\widehat {\bar G})^+ = S_{\phi }/Z(\widehat G)^{\Gamma } = \bar S_{\phi }$ . Thus, $s^{\natural }$ is simply a lift of $\bar s$ to $S_{\phi }$ . We find that equation (3.3.2) equals

$$\begin{align*}e(G)\left\lvert \bar S_{\phi} \right\rvert^{-1} \sum_{\bar s \in \bar S_{\phi}}\operatorname{\mathrm{tr}} r_{\mu}(s^{\natural}) \operatorname{\mathrm{tr}}\check{\delta}_{\pi,\rho}(s^{\natural}) = e(G)\dim \operatorname{\mathrm{Hom}}_{S_{\phi}}(\delta_{\pi,\rho}, r_{\mu}).\end{align*}$$

We have now reduced Theorem 3.2.9 to the identity

(3.3.3) $$ \begin{align} e(G)e(G_b)=(-1)^{\left< 2\rho_G,\mu \right>}, \end{align} $$

where $\rho _G$ is the sum of the positive roots. Recall that $G^*$ is a quasi-split inner form of G. Let $\mu _1,\mu _2 \in X^*(Z(\widehat G_{\mathrm {sc}})^{\Gamma })$ be the elements corresponding to the inner twists $G^* \to G$ and $G^* \to G_b$ by Kottwitz’s homomorphism [Kot86, Theorem 1.2]. By Lemma A.2.1, we have $e(G_b)e(G)=(-1)^{\langle 2\rho ,\mu _2-\mu _1\rangle }$ . But since $G_b$ is obtained from G by twisting by b, the difference $\mu _2-\mu _1$ is equal to the image of $\kappa (b) \in X^*(Z(\widehat G)^{\Gamma })$ under the map $X^*(Z(\widehat G)^{\Gamma }) \to X^*(Z(\widehat G_{\mathrm {sc}})^{\Gamma })$ dual to the natural map $Z(\widehat G_{\mathrm {sc}}) \to Z(\widehat G)$ . Since $b \in B(G,\mu )$ , we see that $\mu _2-\mu _1=\mu $ , and equation (3.3.3) follows. The proof of Theorem 3.2.9 is complete.

3.4 An adjointness property

In this section, we will discuss an adjointness property of the Hecke transfer maps $T_{b,\mu }^{G \to G_b}$ . This will be used in §6.3.

Let $\Lambda $ be an algebraically closed field of characteristic zero. For a topological space X, we let $C_c(X,\Lambda )$ be the space of compactly supported locally constant $\Lambda $ -valued functions. The space of distributions $\operatorname {\mathrm {Dist}}(G(F),\Lambda )$ is the $\Lambda $ -linear dual of $C_c(G(F),\Lambda )$ . The subspace of invariant distributions $\operatorname {\mathrm {Dist}}(G(F),\Lambda )^{G(F)}$ is the linear dual of the space of coinvariants $C_c(G(F),\Lambda )_{G(F)}$ .

Given a $\Lambda $ -valued Haar measure $dx$ on $G(F)$ , integration against a function is a $G(F)$ -invariant distribution on $G(F)$ . Due to the functions in $C_c(G(F),\Lambda )$ being locally constant and having compact support, the ‘integral’ is in reality a finite sum. For our purposes, we will work with functions and integrate them against test functions in $C_c(G(F)_{\mathrm {sr}})$ .

The Weyl integration formula can be used to compute this distribution in terms of orbital integrals. In fact, we will need a ‘stable’ variant of this formula. Before we can explain this, we need to discuss choices of measures.

Choose a $\Lambda $ -valued Haar measure on F. Then a choice of an element $\eta \in \bigwedge ^{\dim (G)}(\mathrm {Lie}(G)(F)^*)=\bigwedge ^{\dim (G)}(\mathrm {Lie}(G)^*)(F)$ leads to a $\Lambda $ -valued Haar measure $dx_{\eta }$ on $G(F)$ ; note that multiplying $\eta $ by an element of $\mathcal {O}_F^{\times }$ doesn’t affect the measure $dx_{\eta }$ . More generally, any element of $\bigwedge ^{\dim (G)}(\mathrm {Lie}(G)^*)(\breve F)$ leads to a $\Lambda $ -valued Haar measure $dx_{\eta }$ on $G(F)$ by choosing $a \in \mathcal {O}_{\breve F}^{\times }$ with the property that $a\eta \in \bigwedge ^{\dim (G)}(\mathrm {Lie}(G)^*)(F)$ and defining $dx_{\eta } := dx_{a\eta }$ , noting that this does not depend on the choice of a. In fact, this procedure allows us to even attach a measure to an element $\eta \in \bigwedge ^{\dim (G)}(\mathrm {Lie}(G)^*)(\overline {F})$ by taking $a \in \overline {F}^{\times }$ such that $a\eta \in \bigwedge ^{\dim (G)}(\mathrm {Lie}(G)^*)(F)$ and letting $dx_{\eta } := |a|_{\Lambda }^{-1}dx_{a\eta }$ . But for this we need to make sense of $|a|_{\Lambda }$ , which requires choosing a compatible system of roots of p in $\Lambda $ . For us, elements of $\bigwedge ^{\dim (G)}(\mathrm {Lie}(G)^*)(\breve F)$ will suffice, so we will not make such a choice.

This procedure allows us to choose Haar measures compatibly in the following two situations. First, consider the inner forms G and $G_b$ . They are canonically identified over $\breve F$ , which gives an identification $\bigwedge ^{\dim (G)}(\mathrm {Lie}(G)^*)(\breve F)= \bigwedge ^{\dim (G)}(\mathrm {Lie}(G_b)^*)(\breve F)$ . Haar measures on $G(F)$ and $G_b(F)$ corresponding to the same $\eta $ will be called compatible. Second, consider two maximal F-rational tori $T_1$ and $T_2$ , each either in G or $G_b$ . They are called related if there exists $g \in G(\overline {F})$ or, equivalently (cf. Lemma 3.2.1) $g \in G(\breve F)$ such that $gT_1g^{-1}=T_2$ and the isomorphism $\mathrm {Ad}(g) :T_1 \to T_2$ is F-rational; we are using here the identification $G_{\breve F} = (G_b)_{\breve F}$ . We obtain an isomorphism $\mathrm {Ad}(g) : \bigwedge ^{\dim (G)}(\mathrm {Lie}(T_1)^*)(\breve F)= \bigwedge ^{\dim (G)}(\mathrm {Lie}(T_2)^*)(\breve F)$ , which leads again to the notion of compatible measures on $T_1(F)$ and $T_2(F)$ . The choice of $g \in G(\breve F)$ is unique up to multiplication by $N(T_1,G)(\breve F)$ , and since this group acts on $\bigwedge ^{\dim (G)}(\mathrm {Lie}(T_1)^*)(\breve F)$ via a $\mathcal {O}_{\breve F}^{\times }$ -valued character of the Weyl group, the notion of compatible measures does not depend on the choice of g.

From now on, we assume that the Haar measures on $G(F)$ and $G_b(F)$ have been chosen compatibly, and the Haar measures on all tori of G and $G_b$ that are related to each other have been chosen compatibly.

We now return to the discussion of distributions. For $\phi \in C_c(G(F)_{\mathrm {sr}},\Lambda )$ , let be the orbital integral function,

$$\begin{align*}\phi_G(y) = \int_{x\in G(F)/G(F)_y} \phi(xyx^{-1})\; dx.\end{align*}$$

As remarked by the referee, the map $\phi \to \phi _G$ induces an isomorphism

(3.4.1)

cf. Lemma 3.1.1. The stable Weyl integration formula states

(3.4.2) $$ \begin{align} \int_{G(F)} f(x)\phi(x)\; dx = \left< f,\phi_G \right>_G. \end{align} $$

We explain now the notation $\left < f,\phi _G \right>_G$ . For a function , we define

$$\begin{align*}\left< f,h \right>_G = \sum_{T} \left\lvert W(T,G)(F) \right\rvert^{-1}\int_{t \in T(F)_{\operatorname{\mathrm{sr}}}} \left\lvert D(t) \right\rvert \sum_{t_0 \sim t} f(t_0)h(t_0)dt, \end{align*}$$

where

• • T runs over a set of representatives for the stable classes of maximal tori,

• • $W(T,G)=N(T,G)/T$ is the absolute Weyl group,

• • $D(t)=\det \left (\operatorname {\mathrm {Ad}}(t)-1\biggm \vert \operatorname {\mathrm {Lie}} G/\operatorname {\mathrm {Lie}} T\right )$ is the usual Weyl discriminant and

• • $t_0$ runs over the $G(F)$ -conjugacy classes inside of the stable class of t.

Note that the integral does not depend on the chosen representative since any two are isomorphic over F by definition of stable conjugacy, and the isomorphism is canonical up to the action of the Weyl group $W(T,G)(F)$ , which is irrelevant given the sum $t_0 \sim t$ .

The following lemma shows that the Hecke transfer maps $T_{b,\mu }^{G\to G_b}$ and $T_{b,\mu }^{G_b\to G}$ are adjoint with respect to the pairing $\left < \cdot ,\cdot \right>_G$ and its analogue $\left < \cdot ,\cdot \right>_{G_b}$ , defined similarly.

Lemma 3.4.1. Given and , one of which has compact support, we have

$$\begin{align*}\langle T_{b,\mu}^{G_b \to G}f',f\rangle_G = \langle f',T_{b,\mu}^{G \to G_b}f\rangle_{G_b}. \end{align*}$$

Proof. By definition $\langle T_{b,\mu }^{G_b \to G}f',f\rangle _G$ equals

(3.4.3) $$ \begin{align} (-1)^d\sum_T |W(T,G)(F)|^{-1}\int_{t \in T(F)_{\operatorname{\mathrm{sr}}}} |D(t)| \sum_{t_0 \sim t} \sum_{(t_0,t_0',\lambda)}r_{\mu,\lambda}f'(t_0')f(t_0)dt. \end{align} $$

The first sum runs over a set of representatives for the stable classes of maximal tori in G. The second sum runs over the set $t_0$ of $G(F)$ -conjugacy classes of elements that are stably conjugate to t. Let $T_{t_0}$ denote the centralizer of $t_0$ . The third sum runs over triples $(t_0,t_0',\lambda )$ , where $t_0'$ is a $G_b(F)$ -conjugacy class that is stably conjugate to $t_0$ , and $\lambda \in X_*(T_{t_0})$ maps to $\mathrm {inv}(t_0,t_0') \in X_*(T_{t_0})_{\Gamma }$ . Note that if T does not transfer to $G_b$ , then it does not contribute to the sum because the sum over $(t_0,t_0',\lambda )$ is empty. Let $\mathcal {X}$ be a set of representatives for those stable classes of maximal tori in G that transfer to $G_b$ . The above expression becomes

$$\begin{align*}(-1)^d\sum_{T \in \mathcal{X}} |W(T,G)(F)|^{-1}\int_{t \in T(F)_{\operatorname{\mathrm{sr}}}} |D(t)| \sum_{(t_0,t_0',\lambda)}r_{\mu,\lambda}f'(t_0')f(t_0)dt, \end{align*}$$

where now the second sum runs over triples $(t_0,t_0',\lambda )$ with $t_0$ a $G(F)$ -conjugacy class and $t_0'$ a $G_b(F)$ -conjugacy class, both stably conjugate to t, and $\lambda \in X_*(T_{t_0})$ mapping to $\mathrm {inv}(t_0,t_0') \in X_*(T_{t_0})_{\Gamma }$ .

Let $\mathcal {X}'$ be a set of representatives for those stable classes of maximal tori of $G_b$ that transfer to G. We have a bijection $\mathcal {X} \leftrightarrow \mathcal {X}'$ . Fix arbitrarily an admissible isomorphism $T \to T'$ for any $T \in \mathcal {X}$ and $T' \in \mathcal {X}'$ that correspond under this bijection. It induces an isomorphism $W(T,G) \to W(T',G_b)$ of finite algebraic groups, as well as an isomorphism $T(F) \to T'(F)$ of toplogical groups that preserves the chosen measures (since we have arranged the measures to be compatible). Given $t \in T(F)_{\operatorname {\mathrm {sr}}}$ , let $t' \in T'(F)_{\operatorname {\mathrm {sr}}}$ be its image under the admissible isomorphism. Then $|D(t)|=|D(t')|$ , and equation (3.4.3) becomes

(3.4.4) $$ \begin{align} (-1)^d\sum_{T' \in \mathcal{X}'} |W(T',G_b)(F)|^{-1}\int_{t' \in T^{\prime}_{\operatorname{\mathrm{sr}}}(F)} |D(t')| \sum_{(t_0,t_0',\lambda)}r_{\mu,\lambda}f'(t_0')f(t_0)dt, \end{align} $$

where now the second sum runs over triples $(t_0,t_0',\lambda )$ , where $t_0$ is a $G(F)$ -conjugacy class, $t_0'$ is a $G_b(F)$ -conjugacy class, both are stably conjugate to $t'$ and $\lambda \in X_*(T_{t_0})$ maps to $\mathrm {inv}(t_0,t_0') \in X_*(T_{t_0})_{\Gamma }$ . Reversing the arguments from the beginning of this proof, we see that this expression equals $\langle f',T_{b,\mu }^{G \to G_b}f\rangle _{G_b}$ .

In §6.3, we will define by geometric means an operator

$$\begin{align*}\widetilde T_{b,\mu}^{G \to G_b} : C_c(G(F)_{\operatorname{\mathrm{ell}}},\Lambda)_{G(F)} \to C_c(G_b(F)_{\operatorname{\mathrm{ell}}},\Lambda)_{G(F)} \end{align*}$$

and show (Proposition 6.3.3) that $T_{b,\mu }^{G \to G_b}$ corresponds to $\widetilde T_{b,\mu }^{G \to G_b}$ under equation (3.4.1).

4 The Lefschetz–Verdier trace formula for v-stacks

The goal of this section is to build up some machinery related to the Lefschetz–Verdier trace formula.

We briefly review the setup in the context of a separated finite-type morphism of schemes $p\colon X\to \operatorname {\mathrm {Spec}} k$ , where k is an algebraically closed field. Let $\ell $ be a prime unequal to the characteristic of k, and let A be an object of $D(X_{\mathrm {\acute {e}t}},\overline {\mathbf {Q}}_{\ell })$ , the derived category of étale $\overline {\mathbf {Q}}_{\ell }$ -sheaves on X. Suppose $f\colon X\to X$ is a k-linear endomorphism. If we are given the additional datum of a morphism¹ $Rf_!A\to A$ , we obtain an operator $Rp_!A\cong Rp_!Rf_!A\to Rp_!A$ on the compactly supported cohomology $Rp_!A=R\Gamma _c(X,A)$ .

In the special case that X is proper over k so that $R\Gamma _c(X,A)=R\Gamma (X,A)$ , the Lefschetz–Verdier trace formula [SGA77], [Var07] expresses $\operatorname {\mathrm {tr}} (f\vert R\Gamma (X,A))$ in terms of data living on the fixed point locus $\operatorname {\mathrm {Fix}}(f)$ of f. In particular, at isolated fixed points $x\in \operatorname {\mathrm {Fix}}(f)$ , there are local terms $\operatorname {\mathrm {loc}}_x(f,A) \in \overline {\mathbf {Q}}_{\ell }$ , and if all fixed points are isolated, then $\operatorname {\mathrm {tr}} (f\vert R\Gamma (X,A))$ is the sum of the $\operatorname {\mathrm {loc}}_x(f,A)$ .

In order to apply the Lefschetz–Verdier trace formula, we need to assume that A satisfies a suitable finiteness hypothesis (constructible of bounded amplitude). Under this hypothesis, one establishes an isomorphism [SGA77, Exposé III, (3.1.1)]

(4.0.1) $$ \begin{align} \mathbf{D} A \boxtimes_k^{\mathbf{L}} A \cong \underline{\mathrm{RHom}}(\operatorname{\mathrm{pr}}_1^*A, \operatorname{\mathrm{pr}}_2^!A) \end{align} $$

in $D((X\times _k X)_{\mathrm {\acute {e}t}},\overline {\mathbf {Q}}_{\ell })$ , where $\operatorname {\mathrm {pr}}_1,\operatorname {\mathrm {pr}}_2\colon X\times _k X\to X$ are the projection maps, and $\mathbf {D}$ is Verdier duality relative to k. Once equation (4.0.1) is established, the definition of local terms and the validity of the Lefschetz–Verdier trace formula can be derived by applying Grothendieck’s six functor formalism. The special case where $X=\operatorname {\mathrm {Spec}} k$ is instructive; the finiteness condition on A is that it be a perfect complex of $\overline {\mathbf {Q}}_{\ell }$ -vector spaces, and then equation (4.0.1) reduces to the fact that $A^{\vee }\otimes ^{\mathbf {L}} A\to \operatorname {RHom}(A,A)$ is an isomorphism. This allows us to express the trace of an endomorphism $f\in \operatorname {\mathrm {End}} A$ as the image of f under the evaluation map $A^{\vee }\otimes ^{\mathbf {L}} A\to \overline {\mathbf {Q}}_{\ell }$ .

In this section, we extend the formalism of the Lefschetz–Verdier trace formula to the setting of perfectoid spaces, diamonds and v-stacks. The main result is Theorem 4.3.8 and its Corollary 4.3.9. We very closely follow the approach of [LZ22], putting a suitable symmetric monoidal 2-category of cohomological correspondences at center stage. In both the schematic and perfectoid settings, the finiteness condition required of the object A can be stated in terms of the property of universal local acyclicity (ULA); as noted in [FS21, Theorem IV.2.23], this is precisely the hypothesis necessary to obtain the isomorphism in equation (4.0.1).

The statement of Lefschetz–Verdier is formally identical in the schematic and perfectoid settings. However, in the perfectoid setting, there arises the possibility that the fixed point locus $\operatorname {\mathrm {Fix}}(f)$ has the structure of a locally profinite set, in which case the local terms appearing in Lefschetz–Verdier are not a function on $\operatorname {\mathrm {Fix}}(f)$ , but rather a distribution on $\operatorname {\mathrm {Fix}}(f)$ . This observation is critical to our applications.

For our applications, we have included two additional theorems concerning local terms in the perfectoid setting, which could also have been stated in the schematic setting and may be of independent interest. Theorem 4.5.3 is a sort of Künneth isomorphism for local terms on a fiber product of stacks. Theorem 4.6.1 states that, in the situation of a smooth group G acting on a diamond X, the local terms corresponding to individual elements $g\in G$ agree with local terms computed on the quotient stack $[X/G]$ .

4.1 Decent v-stacks and the six-functor formalism

We recall here some material from [Sch17] and [GHW22] on the main classes of geometric objects we deal with—perfectoid spaces, diamonds and v-stacks—and their associated étale cohomology formalism.

Let $\operatorname {\mathrm {Perf}}$ be the category of perfectoid spaces in characteristic p. There are four topologies we consider on $\operatorname {\mathrm {Perf}}$ , which we list from coarsest to finest: the analytic topology, the étale topology, the pro-étale topology and the v-topology. The v-topology is a rough analogue of the fpqc topology on schemes. All representable presheaves on $\operatorname {\mathrm {Perf}}$ are sheaves for the v-topology [Sch17, Theorem 1.2].

A diamond is a pro-étale sheaf on $\operatorname {\mathrm {Perf}}$ of the form $X/R$ , where X is a perfectoid space and $R\subset X \times X$ is a pro-étale equivalence relation. Diamonds are automatically v-sheaves [Sch17, Proposition 11.9]. A particularly well-behaved class of diamonds is locally spatial diamonds [Sch17, Definition 1.4]. There is a natural functor $X\mapsto X^{\diamond }$ from analytic adic spaces over $\mathbf {Z}_p$ to locally spatial diamonds.

A v-sheaf Y on $\operatorname {\mathrm {Perf}}$ is small if there exists a surjective map $X\to Y$ from a perfectoid space X. A v-stack is a stack over $\operatorname {\mathrm {Perf}}$ with its v-topology. A small v-stack [Sch17, Definition 12.4] is a v-stack Y on $\operatorname {\mathrm {Perf}}$ such that there exists a surjective map $X\to Y$ from a perfectoid space X such that $X\times _Y X$ is a small v-sheaf.

As with any category of stacks, v-stacks form a strict $(2,1)$ -category. The objects of this category are v-stacks X, which are themselves categories fibered in groupoids over $\operatorname {\mathrm {Perf}}$ . The morphisms between v-stacks $X\to Y$ are functors between fibered categories. Given two morphisms $f_1,f_2\colon X\to Y$ , a 2-morphism $\alpha \colon f_1\Rightarrow f_2$ is an invertible natural transformation between functors.

Example 4.1.1. Let S be a diamond, and let $G\to S$ be a group diamond. The stack $BG=[S/G]$ classifying G-torsors is a small v-stack, as $S\times _{[S/G]} S\cong G$ is already a diamond. Let $H\to S$ be another group diamond. The morphisms $[S/G]\to [S/H]$ correspond to S-homomorphisms $G\to H$ . Suppose we are given two homomorphisms $f_1,f_2\colon G\to H$ , inducing morphisms $\phi _1,\phi _2\colon [S/G]\to [S/H]$ . The set of 2-morphisms $\phi _1\Rightarrow \phi _2$ may be identified with the set of $h\in H(S)$ satisfying $f_1 = (\operatorname {\mathrm {ad}} h)\circ f_2$ .

We use the notation $\ast $ to indicate ‘horizontal’ composition between 2-morphisms. Thus, if $X,Y,Z$ are v-stacks, $f_1,f_2\colon X\to Y$ and $g_1,g_2\colon Y\to Z$ are morphisms, and $\alpha \colon f_1\Rightarrow f_2$ and $\beta \colon g_1\Rightarrow g_2$ are 2-morphisms, then $\beta \ast \alpha \colon g_1\circ f_1\Rightarrow g_2\circ f_2$ is another 2-morphism.

Let $\Lambda $ be a ring which is n-torsion for some n prime to p. For every small v-stack X, there is a triangulated category $D_{\mathrm {\acute {e}t}}(X,\Lambda )$ [Sch17, Definition 1.7]. If X is a locally spatial diamond, then $D_{\mathrm {\acute {e}t}}(X,\Lambda )$ is equivalent to the left-completion of the derived category of sheaves of $\Lambda $ -modules on the étale topology of X [Sch17, Proposition 14.15].

The familiar six functors of Grothendieck have analogues in the world of small v-stacks [Sch17, Definition 1.7]. There is a derived tensor product $\otimes _{\Lambda }^{\mathbf {L}}$ and a derived internal hom $\underline {\mathrm {RHom}}_{\Lambda }$ . For any morphism $f\colon Y\to X$ of small v-stacks, there is a pair of adjoint functors $f^*$ and $Rf_*$ .

Remark 4.1.2. The adjointness between $f^*$ and $Rf_*$ is compatible with 2-morphisms, in the following sense. Suppose $\alpha \colon f\Rightarrow g$ is a 2-morphism between $f,g\colon Y\to X$ . Then there are natural isomorphisms $\alpha _*\colon f_*\to g_*$ and $\alpha ^*\colon f^*\to g^*$ such that the following diagrams commute:

We propose for convenience the following definition.

Definition 4.1.3. A morphism $f\colon Y\to X$ is representable in nice diamonds or simply nice if is compactifiable [Sch17, Definition 22.2], representable in locally spatial diamonds [Sch17, Definition 13.3] and locally of finite geometric transcendence degree [Sch17, Definition 21.7].

If $f\colon Y\to X$ is representable in nice diamonds, then there is an adjoint pair of functors $Rf_!$ and $Rf^!$ [Sch17, Sections 22 and 23].

Theorem 4.1.4 [Sch17, Theorem 1.8]

The six operations $\otimes _{\Lambda }$ , $\underline {\mathrm {RHom}}_{\Lambda }$ , $f^*$ , $Rf_*$ , $Rf_!$ and $Rf^!$ obey the rules:

• (P1.) $f^*A \otimes _{\Lambda }^{\mathbf {L}} f^*B \cong f^*(A\otimes _{\Lambda }^{\mathbf {L}} B)$ ,

• (P2.) $Rf_*\underline {\mathrm {RHom}}_{\Lambda }(f^*A,B)\cong \underline {\mathrm {RHom}}_{\Lambda }(A,Rf_*B)$ ,

• (P3.) $Rf_!(A\otimes _{\Lambda }^{\mathbf {L}} f^*B)\cong Rf_!A\otimes _{\Lambda }^{\mathbf {L}} B$ (the projection formula),

• (P4.) $\underline {\mathrm {RHom}}_{\Lambda }(Rf_!A,B)\cong Rf_*\underline {\mathrm {RHom}}_{\Lambda }(A,Rf^!B)$ (local Verdier duality),

• (P5.) $Rf^!\underline {\mathrm {RHom}}_{\Lambda }(A,B)\cong \underline {\mathrm {RHom}}_{\Lambda }(f^*A,Rf^!B)$ .

We also need the following base change results.

Theorem 4.1.5 [Sch17, Theorem 1.9]

Let

(4.1.1)

be a Cartesian diagram of small v-stacks.

• (BC1.) If f is representable in nice diamonds, then $g^*Rf_! \cong Rf^{\prime }_!\widetilde {g}^*$ .

• (BC2.) If g is representable in nice diamonds, then $Rg^!Rf_*\cong Rf_*'R\widetilde {g}^!$ .

There is a notion of cohomological smoothness [Sch17, Definition 23.8] for morphisms between small v-stacks which are representable in nice diamonds. Let $\Lambda $ be an n-torsion ring for some n not divisible by p. For a morphism $f\colon Y\to X$ of small v-stacks which is representable in nice diamonds, there is a natural map of functors

(4.1.2) $$ \begin{align} Rf^!\Lambda_X\otimes_{\Lambda}^{\mathbf{L}} f^*\to Rf^!, \end{align} $$

adjoint to

$$\begin{align*}Rf_!(Rf^!\Lambda_X \otimes^{\mathbf{L}}_{\Lambda} f^*A) \stackrel{(P3)}{\stackrel{\cong}{\longrightarrow}} Rf_!Rf^!\Lambda_X \otimes^{\mathbf{L}}_{\Lambda} A \stackrel{\text{counit}}{\to} A.\end{align*}$$

If f is cohomologically smooth, then equation (4.1.2) is an equivalence [Sch17, Theorem 1.10]. Furthermore, the object $Rf^!\Lambda _X$ is invertible in the monoidal category $D_{\mathrm {\acute {e}t}}(Y,\Lambda )$ . (For X a small v-stack, an object A of $D_{\mathrm {\acute {e}t}}(X,\Lambda )$ is invertible if and only if étale locally on X there is an isomorphism $A\cong L[n]$ for some invertible $\Lambda $ -module L.)

There is also the following base change theorem for cohomologically smooth morphisms.

Theorem 4.1.6 [Sch17, Theorem 1.10]

In the Cartesian diagram (4.1.1), assume that f is cohomologically smooth. Then $\widetilde {g}^*Rf^!\cong R(f')^!g^*$ and $(f')^*Rg^!\cong R\widetilde {g}^! f^*$ .

In our applications, we will crucially need to deal with stacky morphisms $f\colon Y\to X$ between v-stacks. These morphisms are never representable in nice diamonds, and the hoped-for functors $Rf_!$ and $Rf^!$ were not constructed in [Sch17]. In the companion paper [GHW22], we have extended the $!$ -functor formalism to certain stacky maps between certain small v-stacks, using the $\infty $ -categorical machinery of [LZ]. Here, we briefly recall the main results from [GHW22], referring the reader to that paper for a more detailed discussion.

Definition 4.1.7 [GHW22, Definition 1.1]

A decent v-stack is a small v-stack X such that the diagonal $\Delta _X \colon X\to X \times X$ is representable in locally separated locally spatial diamonds [GHW22, Definition 4.3] and such that there is a locally separated locally spatial diamond U with a morphism $ U\to X$ which is strictly surjective [GHW22, Definition 4.1], representable in locally spatial diamonds and which locally on U is compactifiable of finite dim.trg and cohomologically smooth. Any such morphism $U \to X$ is called a chart for X.

A morphism $f:X\to Y$ between decent v-stacks is fine if there exists a commutative diagram

where the vertical maps are charts and g is locally on W compactifiable of finite dim.trg.

Note that these definitions rely on the notion of cohomological smoothness for morphisms representable in nice diamonds.

In our applications, we will often need to deal with decent v-stacks equipped with a structure map to a fixed v-stack S. We refer to such objects as decent S-v-stacks.

In [GHW22], we showed that decent v-stacks and fine morphisms between them are very reasonable notions:

• • Any locally separated locally spatial diamond is a decent v-stack. In particular, if X is any analytic adic space over $\operatorname {\mathrm {Spa}} \mathbf {Z}_p$ , the associated diamond $X^{\lozenge }$ is a decent v-stack.

• • Decent v-stacks are Artin v-stacks in the sense of [FS21].

• • Any absolute product or fiber product of decent v-stacks is decent.

• • Fine morphisms are stable under composition and (decent) base change.

• • Any morphism of decent v-stacks which is representable in nice diamonds is fine.

The key motivation for singling out fine morphisms of decent v-stacks is the following result.

Theorem 4.1.8 [GHW22, Theorem 1.4]

If $f\colon Y\to X$ is any fine map of decent v-stacks, there exist functors $Rf_!$ and $Rf^!$ satisfying the properties listed in Theorems 4.1.4 and 4.1.5 and agreeing with the constructions in [Sch17] when f is representable in nice diamonds. Moreover, the associations $f \rightsquigarrow Rf_!$ and $f \rightsquigarrow Rf^!$ naturally have the structure of pseudo-functors, and on the class of proper morphisms, there is a pseudo-natural isomorphism $Rf_! \to Rf_*$ .

Finally, there is a notion of cohomological smoothness for fine maps between decent v-stacks, which can be defined extrinsically in terms of charts or intrinsically in terms of the $!$ -functors [GHW22, Proposition 4.17], agreeing with the notion discussed above for morphisms representable in nice diamonds and with the same formal properties as in the representable case. In particular, the map (4.1.2) is an isomorphism for cohomologically smooth morphisms, and the evident analogue of Theorem 4.1.6 holds.

Again, we refer the reader to [GHW22] for a complete discussion.

Finally, we need the notion of the relative dualising complex for v-stacks.

Definition 4.1.9 (The dualising complex)

Let $f\colon X\to S$ be a fine morphism of decent v-stacks. We define $K_{X/S}=Rf^!\Lambda $ , an object in $D_{\mathrm {\acute {e}t}}(X,\Lambda )$ .

Suppose that S is connected, and that f is proper. Then $Rf_*=Rf_!$ , and so there is a morphism $Rf_*K_{X/S}=Rf_!Rf^!\Lambda \overset {\mathrm {counit}}{\longrightarrow } \Lambda $ , which induces a morphism on the level of global sections²

$$\begin{align*}H^0(X,K_{X/S})=H^0(S,Rf_*K_{X/S})\to H^0(S,\Lambda)=\Lambda, \end{align*}$$

which we notate as $\omega \mapsto \int _X \omega $ .

We record the following lemmas for convenience.

Lemma 4.1.10. Let $f\colon Y\to X$ be a fine morphism of decent v-stacks, and let $A,I\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ be any objects with I invertible. The natural map $Rf^!A\otimes ^{\mathbf {L}}_{\Lambda } f^*I \to Rf^!(A\otimes ^{\mathbf {L}}_{\Lambda } I)$ of equation (4.1.2) is an isomorphism.

Proof. Let $w_{A,I}$ be this morphism. We also get a map $w_{A\otimes ^{\mathbf {L}}_{\Lambda } I,I^{-1}}\colon Rf^!(A \otimes ^{\mathbf {L}}_{\Lambda } I) \otimes ^{\mathbf {L}}_{\Lambda } f^*I^{-1} \to Rf^!A$ , which induces $Rf^!(A\otimes ^{\mathbf {L}}_{\Lambda } I)\to Rf^!A \otimes ^{\mathbf {L}}_{\Lambda } f^*I$ . This is the inverse to $w_{A,I}$ .

Lemma 4.1.11. Let $f\colon Y\to X$ be a fine morphism of decent v-stacks which is cohomologically smooth. Then there is a canonical isomorphism

(4.1.3) $$ \begin{align} R\Delta_{Y/X}^!\Lambda_{Y\times_X Y} \otimes^{\mathbf{L}}_{\Lambda} Rf^!\Lambda_X \cong \Lambda_Y \end{align} $$

so that $K_{Y/Y\times _X Y} \cong K_{Y/X}^{-1}$ is invertible.

Proof. Let $\operatorname {\mathrm {pr}}_1,\operatorname {\mathrm {pr}}_2\colon Y\times _X Y\to Y$ be the projection morphisms; each is cohomologically smooth. We have

$$\begin{align*}\Lambda_Y\cong R\operatorname{\mathrm{id}}_Y^!\Lambda_Y\cong R\Delta_{Y/X}^!R\operatorname{\mathrm{pr}}_1^!\Lambda_Y \cong R\Delta_{Y/X}^!\Lambda_{Y\times_X Y} \otimes^{\mathbf{L}}_{\Lambda} \Delta_{Y/X}^*R\operatorname{\mathrm{pr}}_1^!\Lambda_Y,\end{align*}$$

where in the last isomorphism we used Lemma 4.1.10 combined with the cohomological smoothness of $\operatorname {\mathrm {pr}}_1$ . Now use Theorem 4.1.6 to obtain an isomorphism $\Delta _{Y/X}^*R\operatorname {\mathrm {pr}}_1^!f^*\Lambda _X \cong \Delta _{Y/X}^*\operatorname {\mathrm {pr}}_2^*Rf^!\Lambda _X\cong Rf^!\Lambda _X$ .

For the remainder of the section, we fix $\Lambda $ , an n-torsion ring for some n not divisible by p. We will now start writing $f_!$ for $Rf_!$ and $\otimes $ for $\otimes ^{\mathbf {L}}_{\Lambda }$ , etc.

4.2 Examples

We wish to illustrate the behavior of the functors $f_!$ and $f^!$ through a long list of examples. In the following, assume $S=\operatorname {\mathrm {Spd}} C$ for an algebraically closed perfectoid field C or else $S=\operatorname {\mathrm {Spd}} k$ for an algebraically closed discrete field of characteristic p. In both cases, we freely identify $D_{\mathrm {\acute {e}t}}(S,\Lambda )$ with the derived category of $\Lambda $ -modules. Observe that, in both cases, S is decent: This is trivial for $S=\operatorname {\mathrm {Spd}} C$ , while for $S=\operatorname {\mathrm {Spd}} k$ the condition on the diagonal is easy to check, and one can show that $U = S \times \operatorname {\mathrm {Spd}} \mathbf {F}_p((t^{1/p^{\infty }}))\to S$ is a chart.

Example 4.2.1. Let T be a locally profinite set, and let $T_S=\underline {T} \times S$ be the associated constant diamond over S. Then $f\colon T_S\to S$ is representable in nice diamonds. Let $C(T,\Lambda )$ be the ring of continuous functions $T\to \Lambda $ , for the discrete topology on $\Lambda $ . We may naturally identify $D_{\mathrm {\acute {e}t}}(T_S,\Lambda )$ with the derived category of the abelian category of smooth $C(T,\Lambda )$ -modules in the sense of §B.2. Indeed, $D_{\mathrm {\acute {e}t}}(T_S,\Lambda ) \cong D(T_{S,\mathrm {\acute {e}t}},\Lambda )$ since $T_S$ locally has cohomological dimension zero, and then the site $T_{S,\mathrm {\acute {e}t}}$ agrees with the site associated with the topological space T. Finally, Lemma B.2.5 identifies $\operatorname {\mathrm {Sh}}(T,\Lambda )$ with the category of smooth $C(T,\Lambda )$ -modules. This identification matches the constant sheaf $\Lambda $ with the smooth $C(T,\Lambda )$ -module $C_c(T,\Lambda )$ . Under this identification, we have concrete descriptions of the four operations associated with $f\colon T_S\to S$ . Here, we freely use some language and notation from §B.2.

• • $f^*$ sends a $\Lambda $ -module M to the smooth $C(T,\Lambda )$ -module $C_c(T,\Lambda )\otimes _{\Lambda } M$ .

• • $f_*$ sends a smooth $C(T,\Lambda )$ -module M to $M^c$ regarded as a $\Lambda $ -module.

• • $f_!$ sends a smooth $C(T,\Lambda )$ -module M to its underlying $\Lambda $ -module.

• • $f^!$ sends a $\Lambda $ -module M to $\mathrm {RHom}_{\Lambda }(C_c(T,\Lambda ),M)^s$ . In particular, $H^0(T_S, f^!\Lambda _S) \cong \operatorname {\mathrm {Dist}}(T,\Lambda )$ , the module of $\Lambda $ -valued distributions on T.

Example 4.2.2. Suppose G is a locally pro-p group. Let $[S/G_S]$ be the classifying v-stack of $G_S$ -torsors. Assume that there is a separated locally spatial diamond X together with a strictly surjective cohomologically smooth map $X\to S$ and admitting a free $G_S$ -action.³ Then $[S/G_S]$ is a decent v-stack. Indeed, the condition on the diagonal is easy to check, and one can also check that the natural map $X/G_S \to [S/G_S]$ is a chart.

Let $q\colon S\to [S/G_S]$ be the quotient map. Then q is representable in nice diamonds. The functor $q^*$ is an equivalence of monoidal categories between $D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ and the derived category of $\Lambda $ -modules with a smooth G-action [FS21, Theorem V.1.1]. (Strictly speaking, if M is an object of $D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ , then $q^*M$ is a bare $\Lambda $ -module, but then for each $g\in G$ there is a 2-morphism $\alpha _g\colon q\implies q$ as in Example 4.1.1, inducing an automorphism of $q^*M$ .)

With respect to this equivalence of categories, the functors $q^*,q^!$ (resp., $q_*,q_!$ ) take the following values on a $\Lambda $ -module M (resp., a $\Lambda $ -module M with smooth G-action).

• • $q^*M=M$ with its G-action forgotten.

• • $q_*M=C(G,M)^{G\text {-sm}}$ , the module of continuous M-valued functions on G which are smooth with respect to the action of G by right translation.

• • $q_!M=C_c(G,M)=C_c(G,\Lambda )\otimes M$ .

• • $q^!M=\operatorname {\mathrm {Hom}}_G(C_c(G,\Lambda ),M)$ . In particular, $q^!\Lambda =\mathrm {Haar}(G,\Lambda )$ is the module of left-invariant Haar measures on G.

We give justifications for these expressions in the next example, which is more general.

Example 4.2.3. This example generalizes the previous one. Let G be a locally pro-p group as in Example 4.2.2. Suppose $H\subset G$ is a closed subgroup. Then $[S/H_S]$ is also a decent v-stack. Let

$$\begin{align*}q\colon [S/H_S]\to [S/G_S] \end{align*}$$

be the quotient map, so q is representable in nice diamonds. The functors $q^*,q^!$ (resp., $q_*,q_!$ ) take the following values on a $\Lambda $ -module M with smooth G-action (resp., smooth H-action):

• • $q^*M$ is the restriction of M from G to H.

• • $q_*M=\mathrm {Ind}_H^G M$ is the smooth induction of M from H to G.

• • $q_!M=\mathrm {cInd}_H^G M$ is the compact induction of M from H to G.

• • $q^!M$ seems difficult to describe explicitly in general, but there are two special cases:

  1. 1. If $H\subset G$ is open, then $q^!M\cong q^*M$ is the restriction of M from G to H.

  2. 2. If H is a direct factor of G so that $G=H\times H'$ , then

     $$\begin{align*}q^!M= \operatorname{\mathrm{Hom}}_{H'}(C_c(H',\Lambda),M)^{H\mathrm{-sm}}. \end{align*}$$

For the claims regarding $q^*$ and $q_*$ : Let $q_H\colon S\to [S/H_S]$ be the quotient map for H and similarly for $q_G\colon S\to [S/G_S]$ . Then the underlying module of $q^*M$ is $q_H^*q^*M=q_G^*M$ , which we have identified with M itself. For $h\in H$ , we have a 2-morphism $\beta _h\colon q_H\implies q_H$ , which induces an action of $h\in H$ on $q_G^*M$ , as well as the 2-morphism $\alpha _h\colon q_G\implies q_G$ inducing the action of $h\in G$ on $q_G^*M$ as discussed in Example 4.2.2; these actions agree because $\alpha _h = \operatorname {\mathrm {id}}_q\ast \beta _h$ . Since $q^*$ is restriction, $q_*$ must be its right adjoint, which is smooth induction.

For the claim about $q_!$ , consider the Cartesian diagram:

The underlying module of $q_!M$ is $q_G^*q_!M$ . By base change property (BC1), we have $q_G^*q_!M \cong \widetilde {q}_!\widetilde {q}_G^*M$ , which by Example 4.2.1 is identified with the underlying $\Lambda $ -module of $\widetilde {q}_G^*M$ . The latter is the descent of M along the H-torsor in topological spaces $G\to G/H$ . In our dictionary between sheaves on $G/H$ and smooth $C(G/H,\Lambda )$ -modules, $\widetilde {q}_G^*M$ is the module of smooth H-equivariant functions $G\to M$ which are compactly supported modulo H. This is none other than the compact induction of M from H to G. (To show that the action of G is by right translation on such functions, one has to appeal to the compatibility of base change with the 2-isomorphisms $\alpha _g\colon q_G\implies q_G$ .)

We now turn to the claims for $q^!$ . In the case that $H\subset G$ is open, q is étale, and so $q^!\cong q^*$ . In the case that $G=H\times H'$ , suppose M is a smooth H-module; we have an isomorphism of smooth G-modules

$$\begin{align*}q_!M = \mathrm{cInd}_H^G M \cong M\boxtimes C_c(H',\Lambda),\end{align*}$$

from which it is easy to see that the right adjoint to $q_!M$ is as claimed.

Example 4.2.4. Suppose G is a locally pro-p-group as in Example 4.2.2. Let $f\colon [S/G_S]\to S$ be the structure morphism. Then f is fine and, in fact, is cohomologically smooth. The functors associated with f have the following descriptions:

• • $f^*M=M$ with trivial G-action.

• • $f_*M=M^G$ is the (derived) G-invariants of M, that is, the group cohomology.

• • $f_!M = (M\otimes \mathrm {Haar}(G,\Lambda ))_G$ is the group homology of M twisted by the module of Haar measures.

• • $f^!M = \mathrm {Haar}(G,\Lambda )^* \otimes M$ is M (with trivial G-action) twisted by the dual of the module of Haar measures. In particular, $K_{[S/G_S]/S}=\mathrm {Haar}(G,\Lambda )^*$ .

The claim about $f^*$ is clear from the definitions, and $f_*$ is the right adjoint to $f^*$ . Next, we consider $f^!$ . Since f is cohomologically smooth, we have $f^!M\cong f^*M\otimes f^!\Lambda $ . Let $q\colon S\to [S/G_S]$ be as in Example 4.2.2 so that $f\circ q = \operatorname {\mathrm {id}}_S$ . We have $M=q^!f^!M=\mathrm {Haar}(G,\Lambda )\otimes f^!M$ so that $f^!M=\mathrm {Haar}(G,\Lambda )^* \otimes M$ as claimed. From here, it is easy to compute $f_!$ as the left adjoint of $f^!$ .

Example 4.2.5. This example combines Examples 4.2.1 with 4.2.4. Let G be a locally pro-p group as in Example 4.2.2. Let T be a locally profinite set equipped with a continuous action of G, and let $T_S$ be the constant diamond over S. Then the stacky quotient $[T_S/G_S]$ is a decent v-stack, and the structure map $[T_S/G_S] \to S$ is fine. Indeed, we have already seen that $[S/G_S]$ is a decent v-stack fine over S, and the evident morphism $[T_S/G_S] \to [S/G_S]$ is representable in nice diamonds, so the claim immediately follows from [GHW22, Proposition 4.11]. The stack $[T_S/G_S]$ is not in general cohomologically smooth over S. The category $D_{\mathrm {\acute {e}t}}([T_S/G_S],\Lambda )$ may be identified with the derived category of G-equivariant smooth $C(T,\Lambda )$ -modules. Using this identification, we get a natural isomorphism

(4.2.1) $$ \begin{align} H^0([T_S / G_S],K_{[T_S/G_S]/S}) \cong \operatorname{\mathrm{Hom}}_G(C_c(T,\Lambda)\otimes \mathrm{Haar}(G,\Lambda), \Lambda) \end{align} $$

which we can think of as the space of G-invariant distributions on T with values in $\mathrm {Haar}(G,\Lambda )^*$ . If G is unimodular and if we choose a Haar measure on G, then $H^0([T_S / G_S],K_{[T_S/G_S]/S})$ becomes isomorphic to $\operatorname {\mathrm {Dist}}(T,\Lambda )^G$ , the module of G-invariant distributions on T.

4.3 The category of cohomological correspondences

The Lefschetz–Verdier trace formula was expressed elegantly by Lu and Zheng [LZ22] in the language of symmetric monoidal 2-categories. In brief, [LZ22] constructs such a category of cohomological correspondences, where the objects are pairs $(X,A)$ , where X is a scheme over a fixed base scheme S and A is an object of $D(X_{\mathrm {\acute {e}t}},\Lambda )$ , and a morphism $(X,A)\to (X',A')$ is a correspondence $c=(c_1,c_2)\colon C\to X\times _S X'$ together with a morphism $c_1^*A\to c_2^!A'$ . An endomorphism of a dualizable object $(X,A)$ has a categorical trace, which lives over the fixed point locus of c. In the special case that $X=X'=C=S$ and A is a perfect complex of $\Lambda $ -modules, the categorical trace is just the Euler characteristic of an endomorphism of A. The trace formula is interpreted as the statement that the categorical trace is compatible with proper pushforwards.

We adapt here [LZ22] to the setting of v-stacks, but the same language could be used in the world of stacks in the scheme setting. The main point of departure from [LZ22] is that stacks form a 2-category, and so one must keep track of the 2-morphisms witnessing commutativity of diagrams of stacks. This means that the definition of cohomological correspondences we give (Definition 4.3.4) is a little more delicate than its analogue in [LZ22].

First, we recall some definitions and constructions concerning the categorical trace.

Definition 4.3.1 [LZ22, Definition 1.1, Construction 1.6]

An object X of a symmetric monoidal 2-category $(\mathcal {C},\otimes ,1_{\mathcal {C}})$ is dualizable if there exists an object $X^{\vee }$ together with morphisms $\text {ev}_X\colon X^{\vee }\otimes X \to 1_{\mathcal {C}}$ and $\text {coev}_X\colon 1_{\mathcal {C}} \to X\otimes X^{\vee }$ such that the compositions

and

are isomorphic to the identities on X and $X^{\vee }$ , respectively. Consequently, the functor $Y\mapsto X\otimes Y$ has right adjoint $Y\mapsto X^{\vee }\otimes Y$ . If X is dualizable, then $X^{\vee }\otimes Y$ serves as an internal mapping object $\underline {\mathrm {Hom}}(X,Y)$ .

Let $\Omega \mathcal {C}=\operatorname {\mathrm {End}}(1_{\mathcal {C}})$ be the (1-)category of endomorphisms of the unit object of $\mathcal {C}$ . Let $f\in \operatorname {\mathrm {End}} X$ be an endomorphism of a dualizable object X. Define the categorical trace $\operatorname {\mathrm {tr}}(f)$ as the composite:

so that $\operatorname {\mathrm {tr}}(f)$ is an object of $\Omega \mathcal {C}$ .

Example 4.3.2. Let $\Lambda $ be an arbitrary ring, and let $D(\Lambda )$ be the derived category of $\Lambda $ -modules. An object A of $D(\Lambda )$ is dualizable if and only if it is a perfect complex, in which case $\mathbf {D} A=\underline {\mathrm {RHom}}(A,\Lambda [0])$ is a dual object. (See Lemma B.1.2 in the Appendix for a proof of this claim and related conditions.) If f is an endomorphism of the perfect complex of A, then the categorical trace $\operatorname {\mathrm {tr}}(f)$ agrees with the Euler characteristic $\operatorname {\mathrm {tr}}(f\vert A)$ of f.

Next, we define the symmetric monoidal 2-category $\operatorname {\mathrm {Corr}}_S$ of correspondences of v-stacks and its cohomological enhancement $\operatorname {\mathrm {CoCorr}}_S\to \operatorname {\mathrm {Corr}}_S$ . In the following discussion, we fix a decent v-stack S.

Definition 4.3.3 (The category of correspondences)

We define a symmetric monoidal 2-category $\operatorname {\mathrm {Corr}}_S$ as follows:

• • The objects of $\operatorname {\mathrm {Corr}}_S$ are decent S-v-stacks X whose structure map $X \to S$ is fine.

• • Given objects X and $X'$ , the category $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {Corr}}_S}(X,X')$ has for its objects the correspondences:

  (4.3.1)

  
  where each $c_i$ a morphism of decent v-stacks, and $c_2$ is assumed to be fine.⁴ The composition of $(c_1,c_2)\colon C\to X\times _S X'$ with $(d_1,d_2)\colon D\to X'\times _S X''$ is the correspondence $(c_1d_1',d_2c_2')$ defined by the diagram:
  (4.3.2)

  

• • If $c=(c_1,c_2)\colon C\to X\times _S X'$ and $d=(d_1,d_2)\colon D\to X\times _S X'$ represent two objects in $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {Corr}}_S}(X,X')$ , a 2-morphism $c\Rightarrow d$ is an equivalence class of 2-commutative diagrams

  (4.3.3)

  
  where p is proper and $\alpha _i\colon c_i\Rightarrow d_i\circ p$ (for $i=1,2$ ) is a 2-isomorphism witnessing the 2-commutativity of the appropriate triangle. We write $(p,\alpha _1,\alpha _2)$ as shorthand for the datum of such a diagram or just p if the 2-isomorphisms are clear from context.

  We declare two such diagrams $(p,\alpha _1,\alpha _2)$ and $(q,\beta _1,\beta _2)$ equivalent if there is a 2-isomorphism $\gamma \colon p\Rightarrow q$ such that $\beta _i=(\operatorname {\mathrm {id}}_{d_i}\ast \gamma )\circ \alpha _i$ for $i=1,2$ .

• • The monoidal structure is defined by $X\otimes Y = X\times _S Y$ , with unit object S. Given $c=(c_1,c_2)\colon C\to X\times _S X'$ and $d=(d_1,d_2)\colon D\to Y\times _S Y'$ representing objects in $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {Corr}}_S}(X,X')$ and $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {Corr}}_S}(Y,Y')$ , respectively, we define the object $c\otimes d$ of $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {Corr}}_S}(X\times _S Y,X'\times _S Y')$ as the correspondence:

  (4.3.4)

  

The next thing to do is to construct a symmetric monoidal 2-category $\operatorname {\mathrm {CoCorr}}_S$ of cohomological correspondences, which lies over $\operatorname {\mathrm {Corr}}_S$ .

Definition 4.3.4 (The category of cohomological correspondences)

We define a symmetric monoidal 2-category $\operatorname {\mathrm {CoCorr}}_S$ , which comes equipped with a functor to $\operatorname {\mathrm {Corr}}_S$ .

• • An object of $\operatorname {\mathrm {CoCorr}}_S$ is a pair $\mathfrak {X}=(X,A)$ , where X is a decent S-v-stack whose structure map $X \to S$ is fine, and $A\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ is arbitrary.

• • Given objects $\mathfrak {X}=(X,A)$ and $\mathfrak {X}'=(X',A')$ of $\operatorname {\mathrm {CoCorr}}_S$ , the category $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {CoCorr}}_S}(\mathfrak {X},\mathfrak {X}')$ consists of pairs $\mathfrak {c}=(c,u)$ , where $c=(c_1,c_2)$ is a correspondence as in equation (4.3.1), and $u\colon c_1^*A\to c_2^!A'$ is a morphism in $D_{\mathrm {\acute {e}t}}(C,\Lambda )$ . The composition of $\mathfrak {c}=(c,u)\colon (X,A)\to (X',A')$ with $\mathfrak {d}=(d,v)\colon (X',A')\to (X'',A'')$ is $\mathfrak {d}\circ \mathfrak {c}=(e,w)$ , where $e\colon C\times _{X'} D\to X\times _S X''$ is the correspondence in equation (4.3.2), and w is the composition

  $$\begin{align*}(d_1')^*c_1^*A\stackrel{u}{\to} (d_1')^*c_2^! A' \stackrel{a}{\to} (c_2')^!d_1^*A' \stackrel{v}{\to} (c_2')^! d_2^! A'', \end{align*}$$
  where the map labeled a is adjoint to the base change isomorphism $(c_2')_!(d_1')^* \cong d_1^* (c_2)_!$ .

• • Let $\mathfrak {X}=(X,A)$ and $\mathfrak {X}'=(X',A')$ be two objects of $\operatorname {\mathrm {CoCorr}}_S$ . Let $\mathfrak {c}=(c,u)$ and $\mathfrak {d}=(d,v)$ be two objects in $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {CoCorr}}_S}(\mathfrak {X},\mathfrak {X}')$ , where $c=(c_1,c_2)\colon C\to X\times _S X'$ and $d=(d_1,d_2)\colon D\to X\times _S X'$ are correspondences, and $u\colon c_1^*A\to c_2^!A'$ and $v\colon d_1^*A\to d_2^!A'$ are morphisms. A 2-morphism $\mathfrak {p}\colon \mathfrak {c}\implies \mathfrak {d}$ is an equivalence class of triples $(p,\alpha _1,\alpha _2)$ as in the diagram (4.3.3) such that the composition

  $$ \begin{align*} \begin{array}{lcl} d_1^* A &\! \stackrel{\text{unit}}{\longrightarrow}&\!\! p_*p^*d_1^* A \\ &\!\stackrel{(\alpha_1^*)^{-1}}{\longrightarrow}&\!\! p_*c_1^* A \\ &\!\stackrel{u}{\to}&\!\! p_* c_2^! A' \\ &\!\stackrel{\alpha_2^!}{\longrightarrow} &\!\!p_*p^! d_2^! A' \cong p_!p^!d_2^! A'\\ &\!\stackrel{\text{counit}}{\longrightarrow}&\!\! d_2^!A'\\ \end{array} \end{align*} $$

  agrees with $v\colon d_1^*A\to d_2^!A'$ . Here, $\alpha _1^*$ and $\alpha _2^!$ are the natural isomorphisms $c_1^*\stackrel {\cong }{\longrightarrow } p^*d_1^*$ and $c_2^!\stackrel {\cong }{\longrightarrow } p^!d_2^!$ , respectively.

  We need to check that the condition on the aforementioned composition depends only on the equivalence class of the triple $(p,\alpha _1,\alpha _2)$ . To check this, let $(q,\beta _1,\beta _2)$ be another triple which is equivalent to $(p,\alpha _1,\alpha _2)$ by a 2-isomorphism $\gamma \colon p\Rightarrow q$ in the sense of Definition 4.3.3, so assume that $\beta _i=(\operatorname {\mathrm {id}}_{d_i}\ast \gamma )\circ \alpha _i$ for $i=1,2$ . Consider the diagram in $D_{\mathrm {\acute {e}t}}(D,\Lambda )$ :

  
  The first and fifth squares commute by the compatibility described in Remark 4.1.2, the second and fourth squares commute because of the condition $\beta _i=(\operatorname {\mathrm {id}}_{d_i}\ast \gamma )\circ \alpha _i$ , taking into account the pseudo-functor structures on the four nonbinary operations and the third square commutes because of the equalities $p_*=p_!$ , $q_*=q_!$ , and $\gamma _*=\gamma _!$ . The commutativity of the outside rectangle says that if the composition along the left vertical arrow is v, then so is the composition along the right vertical arrow. This shows that our notion of 2-morphsm in $\operatorname {\mathrm {CoCorr}}_S$ is well-defined.

• • The symmetric monoidal structure on $\operatorname {\mathrm {CoCorr}}_S$ is given by $(X,A)\otimes (X',A')= (X\times _S X', A\boxtimes _S A')$ . The unit object is $1_{\operatorname {\mathrm {CoCorr}}_S}=(S,\Lambda _S)$ . Finally, given $(c,u)\colon (X_1,A_1)\to (X^{\prime }_1,A^{\prime }_1)$ and $(d,v)\colon (X_2,A_2)\to (X^{\prime }_2,A^{\prime }_2)$ , the tensor product $(c,u)\otimes (d,v)$ is $(c\otimes d, w)$ , where $c\otimes d$ is the correspondence in equation (4.3.4), and w is the composition

  $$\begin{align*}(c_1\times_S d_1)^*(A_1\boxtimes_S A_2)\cong c_1^*A_1 \boxtimes d_1^*A_2 \stackrel{u\boxtimes_S v}{\to} c_2^!A_1' \boxtimes_S d_2^!A_2' \stackrel{\kappa}{\to} (c_2\times_S d_2)^! (A_2\boxtimes A_2'), \end{align*}$$
  where $\kappa $ is adjoint to the Künneth isomorphism $(c_2\times d_2)_!(B_1\boxtimes _S B_2)\cong (c_2)_!B_1 \boxtimes _S (d_2)_!B_2$ .

The category $\operatorname {\mathrm {CoCorr}}_S$ has internal mapping objects: if $\mathfrak {X}_1=(X_1,A_1)$ and $\mathfrak {X}_2=(X_2,A_2)$ , then $\underline {\mathrm {Hom}}(\mathfrak {X}_1,\mathfrak {X}_2)=(X_1\times _S X_2,\underline {\mathrm {RHom}}(\operatorname {\mathrm {pr}}_1^*A_1,\operatorname {\mathrm {pr}}_2^!A_2))$ , where $\operatorname {\mathrm {pr}}_i\colon X_1\times _S X_2\to X_i$ is the projection.

We have the following characterization of dualizable objects in $\operatorname {\mathrm {CoCorr}}_S$ .

Proposition 4.3.5. Let X be a decent S-v-stack whose structure map $\pi \colon X\to S$ is fine, and let $A\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ be any object. The following are equivalent.

1. 1. The object $(X,A)$ is dualizable in $\operatorname {\mathrm {CoCorr}}_S$ .

2. 2. The natural map $m\colon \mathbf {D}_{X/S} A\boxtimes _S A \to \underline {\mathrm {RHom}}(\operatorname {\mathrm {pr}}_1^*A,\operatorname {\mathrm {pr}}_2^!A)$ (see proof for construction) is an isomorphism.

In this situation, the dual of $(X,A)$ is $(X,\mathbf {D}_{X/S}A)$ .

Proof. Let $\mathfrak {X}=(X,A)$ , and let $\mathfrak {X}'=(X,\mathbf {D}_{X/S}A)$ . There is a morphism $\mathfrak {e}\colon \mathfrak {X}'\otimes \mathfrak {X}\to 1_{\operatorname {\mathrm {CoCorr}}_S}$ , defined as the pair $(c,u)$ , where $c=(\Delta _{X/S},\pi )$ and u is the composition

$$\begin{align*}\Delta_{X/S}^*(\mathbf{D}_{X/S} A \boxtimes_S A) \stackrel{\cong}{\longrightarrow} \mathbf{D}_{X/S} A \otimes A \to K_{X/S}=\pi^!\Lambda_S.\end{align*}$$

Then $\mathfrak {e}$ induces a morphism $\mathfrak {X}'\otimes \mathfrak {Y}\to \underline {\mathrm {Hom}}(\mathfrak {X},\mathfrak {Y})$ for any object $\mathfrak {Y}$ of $\operatorname {\mathrm {CoCorr}}_S$ . For $\mathfrak {Y}=\mathfrak {X}$ , the map u becomes the map m in (2).

Suppose $\mathfrak {X}$ is dualizable, with witnesses $\mathfrak {X}^{\vee }$ , $\operatorname {\mathrm {ev}}_{\mathfrak {X}}$ , and $\operatorname {\mathrm {coev}}_{\mathfrak {X}}$ . Then $\mathfrak {X}^{\vee }\otimes \mathfrak {Y}\to \underline {\mathrm {Hom}}(\mathfrak {X},\mathfrak {Y})$ is an isomorphism for all objects $\mathfrak {Y}$ . Setting $\mathfrak {Y}=1_{\operatorname {\mathrm {CoCorr}}_S}$ , we find an isomorphism $\mathfrak {X}^{\vee }\cong \mathfrak {X}'$ which identifies $\operatorname {\mathrm {ev}}_{\mathfrak {X}}$ with $\mathfrak {e}$ . Setting $\mathfrak {Y}=\mathfrak {X}$ , we find that m is an isomorphism.

Conversely, if m is an isomorphism, let $\operatorname {\mathrm {coev}}_{\mathfrak {X}}=(d,w)$ , where $d=(\pi ,\Delta _{X/S})$ and w is the composition

$$\begin{align*}\pi^*\Lambda_S \cong \Lambda_X \stackrel{\epsilon}{\to} \underline{\mathrm{RHom}}_{\Lambda}(A,A) \stackrel{(P5)}{\stackrel{\cong}{\longrightarrow}} \Delta_{X/S}^!\underline{\mathrm{RHom}}_{\Lambda}(\operatorname{\mathrm{pr}}_1^*A, \operatorname{\mathrm{pr}}_2^!A) \end{align*}$$

followed by $m^{-1}\colon \Delta _{X/S}^!\underline {\mathrm {RHom}}_{\Lambda }(\operatorname {\mathrm {pr}}_1^*A, \operatorname {\mathrm {pr}}_2^!A)\to \Delta _{X/S}^!(A\boxtimes _S \mathbf {D}_{X/S}A)$ . Here, $\epsilon $ is adjoint to $\operatorname {\mathrm {id}}_A\colon A\to A$ . A diagram chase now shows that $\operatorname {\mathrm {coev}}_{\mathfrak {X}}$ and $\operatorname {\mathrm {ev}}_{\mathfrak {X}}$ witness the dualizability of $\mathfrak {X}$ .

In the scheme setting, a pair $(X,A)$ is dualizable if and only if A is locally acyclic over S [LZ22, Theorem 2.16], under some mild assumptions. Similarly, if $f:X\to S$ is a morphism of v-stacks which is representable in nice diamonds, then $(X,A)$ is dualizable in $\operatorname {\mathrm {CoCorr}}_S$ if and only if A is f-universally locally acyclic [FS21, Theorem IV.2.24]. This result extends immediately to the situation of fine morphisms between decent v-stacks, using that universal local acyclicity is cohomologically smooth-local on the source.

If $\mathfrak {X}$ is a dualizable object of $\operatorname {\mathrm {CoCorr}}_S$ , and $\mathfrak {f}\colon \mathfrak {X}\to \mathfrak {X}$ is an endomorphism, we may define the categorical trace $\operatorname {\mathrm {tr}}(\mathfrak {f})$ , an object of $\Omega \operatorname {\mathrm {CoCorr}}_S=\operatorname {\mathrm {End}} 1_{\operatorname {\mathrm {CoCorr}}_S}$ . Let us make this explicit. The category $\Omega \operatorname {\mathrm {CoCorr}}_S$ has objects $(X,\omega )$ , where X is a decent S-v-stack with fine structure map $X\to S$ , and $\omega \in H^0(X,K_{X/S})$ is arbitrary. A morphism $(X,\omega )\to (X',\omega ')$ is a diagram

with p proper such that $\omega '=p_*\omega $ . Here,

$$\begin{align*}p_*\colon H^0(X,K_{X/S})\to H^0(X',K_{X'/S}) \end{align*}$$

is induced from $\pi _*K_{X/S}\cong \pi ^{\prime }_*p_*p^!K_{X'/S} \cong \pi ^{\prime }_*p_!p^!K_{X'/S} \stackrel {\text {counit}}{\to } \pi ^{\prime }_*K_{X'/S}$ .

Now let $\mathfrak {X}=(X,A)$ be a dualizable object in $\operatorname {\mathrm {CoCorr}}_S$ , and let $\mathfrak {f}=(c,u)\colon \mathfrak {X}\to \mathfrak {X}$ be an endomorphism, with $c=(c_1,c_2)\colon C\to X\times _S X$ and $u\colon c_1^*A\to c_2^!A$ . By definition, $\operatorname {\mathrm {tr}}(\mathfrak {f}) = \operatorname {\mathrm {ev}}_{\mathfrak {X}}\circ (f\otimes \operatorname {\mathrm {id}}_{\mathfrak {X}^{\vee }}) \circ \operatorname {\mathrm {coev}}_{\mathfrak {X}}$ . The object $\operatorname {\mathrm {tr}}(\mathfrak {f})$ is represented by a pair $(\operatorname {\mathrm {tr}}(c),\operatorname {\mathrm {tr}}(u))$ . Here, $\operatorname {\mathrm {tr}}(c)\in \Omega 1_{\operatorname {\mathrm {Corr}}_S}$ is the correspondence $\operatorname {\mathrm {Fix}}(c)\to S\times _S S = S$ , where $\operatorname {\mathrm {Fix}}(c)$ is the fixed-point locus of the correspondence c, as in the Cartesian diagram:

For its part, the element $\operatorname {\mathrm {tr}}(u)$ is an element of $H^0(\operatorname {\mathrm {Fix}}(c),K_{\operatorname {\mathrm {Fix}}(c)/S})$ . ⁵It is the image of $u\in \operatorname {\mathrm {Hom}}(c_1^*A,c_2^!A)$ under

$$ \begin{align*} \begin{array}{lcl} H^0(C,\underline{\mathrm{RHom}}(c_1^*A, c_2^!A)) &\stackrel{(P5)}{\stackrel{\cong}{\longrightarrow}} &\!\!H^0(C,c^!\underline{\mathrm{RHom}}(\operatorname{\mathrm{pr}}_1^*A, \operatorname{\mathrm{pr}}_2^!A)) \\[2pt] &\stackrel{(4.0.1)}{\stackrel{\cong}{\longrightarrow}}&\!\! H^0(C,c^!(\mathbf{D}_{X/S}A \boxtimes_S A)) \\[2pt] &\stackrel{\alpha}{\to}&\!\! H^0(C,c^!(\Delta_{X/S})_* (\mathbf{D}_{X/S}A \otimes A) ) \\[2pt] &\stackrel{\operatorname{\mathrm{ev}}_A}{\to}&\!\! H^0(C,c^!(\Delta_{X/S})_* K_{X/S} )\\[2pt] &\stackrel{{\text{(BC2)}}}{\stackrel{\cong}{\longrightarrow}} &\!\! H^0(C,(\Delta^{\prime}_{X/S})_* (c')^! K_{X/S}) \\[2pt] &\cong &\!\! H^0(\operatorname{\mathrm{Fix}}(c),K_{\operatorname{\mathrm{Fix}}(c)/S}). \\ \end{array} \end{align*} $$

Here, the map labeled $\alpha $ is adjoint to $(\Delta _{X/S})^*(\mathbf {D}_{X/S}A \boxtimes _S A)\stackrel {\cong }{\longrightarrow } \mathbf {D}_{X/S} A \otimes A$ .

Definition 4.3.6 (Inertia stack, characteristic class)

In the special case that $\mathfrak {f}=\operatorname {\mathrm {id}}_{\mathfrak {X}}=(\Delta _{X/S},\operatorname {\mathrm {id}}_A)$ , the object $\operatorname {\mathrm {tr}}(\Delta _{X/S})=X\times _{X\times _S X} X$ is the inertia stack of X, which we notate as $\operatorname {\mathrm {In}}_S(X)$ . Its objects are pairs $(x,g)$ , where $x\in X$ and $g\in \operatorname {\mathrm {Aut}} x$ . Then $\operatorname {\mathrm {tr}}(\operatorname {\mathrm {id}}_A)$ is an element of $H^0(\operatorname {\mathrm {In}}_S(X),K_{\operatorname {\mathrm {In}}_S(X)})$ , which we call the characteristic class of A. We notate this element as $\operatorname {\mathrm {cc}}_{X/S}(A)$ .

We record one lemma here for later reference.

Lemma 4.3.7. Let $i\colon U\to X$ be an open immersion of decent S-v-stacks fine over S. Then $\operatorname {\mathrm {In}}_S(i)\colon \operatorname {\mathrm {In}}_S(U)\to \operatorname {\mathrm {In}}_S(X)$ is also an open immersion. If $A\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ is ULA over S, then so is $i^*A$ , and then

$$\begin{align*}\operatorname{\mathrm{cc}}_{U/S}(i^*A)=\operatorname{\mathrm{In}}_S(i)^*\operatorname{\mathrm{cc}}_{X/S}(A). \end{align*}$$

Proof. All constructions are local on X.

Theorem 4.3.8 (Relative Lefschetz–Verdier trace formula)

Let $\mathfrak {X}=(X,A)$ be a dualizable object in $\operatorname {\mathrm {CoCorr}}_S$ , and let $\mathfrak {f}\in \operatorname {\mathrm {End}} \mathfrak {X}$ lie over the correspondence $c\colon C\to X\times _S X$ . Suppose we are given a diagram

with $p,q$ proper. Then $\mathfrak {X}'=(X',q_*A)$ is also dualizable. Let $\mathfrak {q}\colon \mathfrak {X}\to \mathfrak {X}'$ be the evident 1-morphism lying over q. There is a unique morphism $\mathfrak {f}'\in \operatorname {\mathrm {End}}\mathfrak {X}'$ lying over $C'\to X'\times _S X'$ such that p defines a 2-morphism $\widetilde {p}$ filling in the square

Finally, there exists a (necessarily unique) 2-morphism $\operatorname {\mathrm {tr}}(\widetilde {p})\colon \operatorname {\mathrm {tr}}(\mathfrak {f})\implies \operatorname {\mathrm {tr}}(\mathfrak {f}')$ lying over $\operatorname {\mathrm {tr}}(p)\colon \operatorname {\mathrm {tr}}(c)\implies \operatorname {\mathrm {tr}}(c')$ .

Proof. This is formally the same as the proof of (a special case of) [LZ22, Theorem 2.21], so we give a brief sketch. (In [LZ22] one gets a statement about the more general Lefschetz–Verdier pairing, which we also could have established.) The existence and uniqueness of $\mathfrak {f}'\in \operatorname {\mathrm {End}}\mathfrak {X}'$ follows from the definition of 2-morphisms in $\operatorname {\mathrm {CoCorr}}_S$ . The dualizability of $\mathfrak {X}'$ is [LZ22, Proposition 2.23]. The dual of $\mathfrak {X}'$ is $(\mathfrak {X}')^{\vee }=(X',q_*\mathbf {D}_{X/S}A)$ ; there is another natural map $\mathfrak {q}^{\vee }\colon \mathfrak {X}^{\vee }\to (\mathfrak {X}')^{\vee }$ defined similarly to $\mathfrak {q}$ .

Consider the diagram:

The two outer triangles can be filled in with a 2-morphism, as can the inner square (via $p\otimes \operatorname {\mathrm {id}}_{\mathfrak {q}^{\vee }}$ ). See [LZ22, Construction 1.7] for details. Composing, we find the required 2-morphism $\operatorname {\mathrm {tr}}(\widetilde {p})\colon \operatorname {\mathrm {tr}}(\mathfrak {f})\implies \operatorname {\mathrm {tr}}(\mathfrak {f}')$ .

Corollary 4.3.9. Let $(X,A)$ be a dualizable object, and let $p\colon X\to X'$ be proper. Then $(X',p_!A)$ is dualizable. The morphism $\operatorname {\mathrm {In}}_S(p)\colon \operatorname {\mathrm {In}}_S(X)\to \operatorname {\mathrm {In}}_S(X')$ is proper, and

$$\begin{align*}\operatorname{\mathrm{In}}_S(p)_*\operatorname{\mathrm{cc}}_{X/S}(A) = \operatorname{\mathrm{cc}}_{X'/S}(p_!A).\end{align*}$$

Example 4.3.10. We can immediately deduce a familiar-looking trace formula from Theorem 4.3.8 in the case of proper diamonds. Suppose $S=\operatorname {\mathrm {Spd}} C$ for an algebraically closed perfectoid field C, and suppose $q\colon X\to S$ is a nice diamond. Let $A\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ be ULA over S. Then $\mathfrak {X}=(X,A)$ is a dualizable object of $\operatorname {\mathrm {CoCorr}}_S$ . Let $\mathfrak {f}=(c,u)$ be an endomorphism of $\mathfrak {X}$ lying over a correspondence $c\colon Y\to X\times _S X$ . The categorical trace $\operatorname {\mathrm {tr}}(\mathfrak {f})$ is an endomorphism of $1_{\operatorname {\mathrm {CoCorr}}_S}$ consisting of the pair $(\operatorname {\mathrm {Fix}}(c),\omega )$ , where $\operatorname {\mathrm {Fix}}(c)=Y\times _{c,X\times _S X,\Delta _{X/S}} X$ is the fixed point locus of the correspondence, and $\omega $ is a global section of $K_{\operatorname {\mathrm {Fix}}(c)/S}$ .

Now suppose that X is proper over S. In the setting of Theorem 4.3.8, we put $\mathfrak {X}'=(S,q_*A)$ and $C'=S$ . The dualizability of $\mathfrak {X}'$ means that $q_*A=R\Gamma (X,A)$ is a perfect complex. The morphism $\mathfrak {f}'\colon \mathfrak {X}'\to \mathfrak {X}'$ supplied by Theorem 4.3.8 is the endomorphism $q_*(\mathfrak {f})\colon q_*A \to q_*A$ . Finally, the existence of $\operatorname {\mathrm {tr}}(\widetilde {p})\colon \operatorname {\mathrm {tr}}(\mathfrak {f})\implies \operatorname {\mathrm {tr}}(\mathfrak {f}')$ lying over $\operatorname {\mathrm {tr}}(p)\colon \operatorname {\mathrm {tr}}(c)\implies \operatorname {\mathrm {tr}}(\operatorname {\mathrm {id}}_S)$ implies that

(4.3.5) $$ \begin{align} \operatorname{\mathrm{tr}}\left(q_*(\mathfrak{f})\biggm\vert R\Gamma(X,A)\right) = \int_{\operatorname{\mathrm{Fix}}(c)} \omega. \end{align} $$

Definition 4.3.11. Let $x\in \operatorname {\mathrm {Fix}}(c)$ be an isolated point such that $x\to S$ an isomorphism. The local term $\operatorname {\mathrm {loc}}_x(\mathfrak {f})$ is the restriction of $\omega $ to x, considered as an element of $\Lambda $ .

If $\mathfrak {f}$ arises from an automorphism $g\colon X\to X$ along with a morphism $u\colon g^*A\to A$ , we write the local term as $\operatorname {\mathrm {loc}}_x(g,A)$ (the dependence on u being implicit).

In the latter situation, if it so happens that $\operatorname {\mathrm {Fix}}(g)$ consists of finitely many isolated S-points $x_1,\dots ,x_n$ , then equation (4.3.5) reduces to

$$\begin{align*}\operatorname{\mathrm{tr}}\left(q_*(g)\biggm\vert R\Gamma(X,A)\right) = \sum_{i=1}^n \operatorname{\mathrm{loc}}_x(g,A).\end{align*}$$

4.4 The trace distribution as a characteristic class

Let S be a geometric point, and let G be a locally pro-p group as in Example 4.2.2 so that $[S/G_S]$ is a decent v-stack and $f\colon [S/G_S]\to S$ is fine and cohomologically smooth. As in that example, we freely identify $D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ with the derived category of $\Lambda $ -modules with a smooth G-action. For an object $M\in D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ , we let $M^{\vee } = \underline {\mathrm {RHom}}(M,\Lambda )$ ; by [FS21, Corollary V.1.4], this is just the usual (derived) smooth dual.

For a compact open subgroup $K\subset G$ , we let $M^K$ be the complex of derived K-invariants. If K is pro-p, the map of complexes $M^K\to M$ admits a section, namely, averaging over K with respect to a normalized Haar measure. Thus, $M^K$ is naturally a summand of M.

Proposition 4.4.1. Let M be an object of $D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ . The following are equivalent:

1. 1. The object $([S/G_S],M)$ of $\operatorname {\mathrm {CoCorr}}_S$ is dualizable, with dual $([S/G_S],\mathbf {D} M)$ , where $\mathbf {D} M=\mathrm {Haar}(G,\Lambda )^* \otimes M^{\vee }$ .

2. 2. The object M is ULA over S.

3. 3. For all compact open pro-p subgroups $K\subset G$ , the (derived) K-invariants $M^K$ are a perfect complex of $\Lambda $ -modules.

Proof. The equivalence between (1) dualizability in $\operatorname {\mathrm {CoCorr}}_S$ and (2) the ULA property is [FS21, Theorem IV.2.23]. For the equivalence between (2) and (3), see [FS21, V.7.1]. (There, the authors work in the more general context of $\operatorname {\mathrm {Bun}}_G$ , but the method of proof can be used in our situation of $[S/G_S]$ .) The Verdier dual of M is $\underline {\mathrm {RHom}}(M,f^!\Lambda )$ , and $f^!\Lambda = \mathrm {Haar}(G,\Lambda )^*$ by Example 4.2.4.

We note that is possible to give a direct proof of the implication (3) $\implies $ (1). Let $\mathfrak {X}=([S/G_S],M)$ and $\mathfrak {X}^{\vee }=([S/G_S],\mathbf {D} M)$ . The evaluation map $\mathfrak {X}^{\vee }\otimes \mathfrak {X} \to 1_{\operatorname {\mathrm {CoCorr}}_S}$ lies over the correspondence $\Delta _f\times f\colon [S/G_S]\to [S/G_S]^2 \times S$ ; on the level of sheaves, it is a twist of the evaluation map $M^{\vee } \otimes M \to \Lambda $ . The coevaluation map $1_{\operatorname {\mathrm {CoCorr}}_S} \to \mathfrak {X}\otimes \mathfrak {X}^{\vee }$ lies over the correspondence $f\times \Delta _f\colon [S/G_S] \to S\times [S/G_S]^2$ . Note that the diagonal map presents G as a direct factor of $G^2$ so that Example 4.2.3 applies to give an explicit description of $\Delta _f^!$ . The result is that the $\Lambda $ -module of cohomological correspondences lying over $f\times \Delta _f$ :

$$\begin{align*}f^*\Lambda\to \Delta_f^!(M\boxtimes \mathbf{D} M)\end{align*}$$

may be identified with the $\Lambda $ -module

$$\begin{align*}\operatorname{\mathrm{Hom}}_{G\times G}(C_c(G,\Lambda)\otimes \mathrm{Haar}(G,\Lambda), M \boxtimes M^{\vee}). \end{align*}$$

In the latter expression, $G\times G$ acts on $C_c(G,\Lambda )$ by left and right translation. We describe the coevaluation map as a $G\times G$ -equivariant function

$$\begin{align*}I\colon C_c(G,\Lambda)\otimes \mathrm{Haar}(G,\Lambda) \to H^0(M\otimes M^{\vee}).\end{align*}$$

Let $h\in C_c(G,\Lambda )$ and $\mu \in \mathrm {Haar}(G,\Lambda )$ ; then integration against $h\;d\mu $ describes an endomorphism $I_{h,\mu }\in \operatorname {\mathrm {End}} M$ :

$$\begin{align*}I_{h,\mu}(v) = \int_{g\in G} h(g)gv \; d\mu(g). \end{align*}$$

The function h is left and right K-invariant for some sufficiently small pro-p open subgroup $K\subset G$ , in which case $I_{h,\mu }$ factors through a map $M^K\to M^K$ . Since $M^K$ is perfect by hypothesis, we have described an element of

$$\begin{align*}\operatorname{\mathrm{Hom}}(M^K,M^K) \cong H^0(\mathrm{RHom}(M^K,M^K))\cong H^0(M^K \otimes (M^K)^{\vee}).\end{align*}$$

Then $I(h\otimes \mu )$ is the image of $I_{h,\mu }$ in $H^0(M\otimes M^{\vee })$ .

Definition 4.4.2. The object M of $D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ is admissible if it satisfies the equivalent conditions of Proposition 4.4.1.

Suppose M is an admissible object of $D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ . The trace distribution of M is a canonical element

(4.4.1) $$ \begin{align} \operatorname{\mathrm{tr.dist}}(M)\in \operatorname{\mathrm{Hom}}_G(C_c(G,\Lambda)\otimes \mathrm{Haar}(G,\Lambda), \Lambda), \end{align} $$

where G is meant to act on $C_c(G,\Lambda )$ by conjugation and on $\mathrm {Haar}(G,\Lambda )$ by the modular character. Namely, $\operatorname {\mathrm {tr.dist}}(M)$ sends $h\otimes \mu $ (where $h\in C_c(G,\Lambda )$ and $\mu \in \mathrm {Haar}(G,\Lambda )$ to the Euler characteristic of the operator $I_{h,\mu }$ described in the proof of Proposition 4.4.1.

On the other hand, we have the inertia stack $\operatorname {\mathrm {In}}_S([S/G_S])$ and the characteristic class $\operatorname {\mathrm {cc}}_{[S/G_S]/S}(M)$ as in Definition 4.3.6. We have

the stack of conjugacy classes of G. (Reasoning: For a perfectoid space $Y\to S$ , a Y-point of $\operatorname {\mathrm {In}}_S([S/G_S])$ is a $G_S$ -torsor $\widetilde {Y}\to Y$ together with a $G_S$ -equivariant automorphism $i\colon \widetilde {Y}\to \widetilde {Y}$ . Such an automorphism arises as $i(y)=f(y).y$ , where $f\colon \widetilde {Y}\to G_S$ is a morphism satisfying $f(gy)=gf(y)g^{-1}$ . The pair $(\widetilde {Y},f)$ then constitutes a Y-point of

.) For its part, the characteristic class $\operatorname {\mathrm {cc}}_{[S/G_S]/S}(M)$ lies in $H^0(\operatorname {\mathrm {In}}_S([S/G_S]), K_{\operatorname {\mathrm {In}}_S([S/G_S])/S})$ , and by Example 4.2.5 we have an isomorphism

$$\begin{align*}H^0(\operatorname{\mathrm{In}}_S([S/G_S]), K_{\operatorname{\mathrm{In}}_S([S/G_S])/S})\cong \operatorname{\mathrm{Hom}}_G(C_c(G,\Lambda) \otimes \mathrm{Haar}(G,\Lambda), \Lambda) \end{align*}$$

onto the same module appearing in equation (4.4.1).

Proposition 4.4.3. Let G be a locally pro-p group satisfying the hypotheses of Example 4.2.2. Let M be an admissible object of $D_{\mathrm {\acute {e}t}}([S/G_S],\Lambda )$ . Then

$$\begin{align*}\operatorname{\mathrm{cc}}_{[S/G_S]/S}(M) = \operatorname{\mathrm{tr.dist}}(M). \end{align*}$$

Proof. The characteristic class of M is the categorical trace of the identity on the object $\mathfrak {X}=([S/G_S],M)$ , which is $\operatorname {\mathrm {ev}}_{\mathfrak {X}}\circ \operatorname {\mathrm {coev}}_{\mathfrak {X}}$ . This equals the image of the identity map through the left side of the following commutative diagram:

whereas on the right side of the diagram, the map labeled $\operatorname {\mathrm {coev}}_M$ carries the identity to the integration map I described in the proof of Proposition 4.4.1, and then $\operatorname {\mathrm {ev}}_M$ carries I onto $\operatorname {\mathrm {tr.dist}}(M)$ by definition of the latter.

4.5 A Künneth theorem for characteristic classes

The goal of this section is to prove the compatibility of the categorical trace with fiber products to get an analogue of the relation $\operatorname {\mathrm {tr}}(A\otimes B)=\operatorname {\mathrm {tr}}(A) \operatorname {\mathrm {tr}}(B)$ for square matrices. Again, throughout this section we fix a decent base v-stack S.

As an example of what we will do, suppose $X_1,X_2\to S$ are two fine morphisms of decent v-stacks, and suppose $A_i\in D_{\mathrm {\acute {e}t}}(X_i,\Lambda )$ is ULA over S for $i=1,2$ . Then $A_1\boxtimes _S A_2$ is ULA over S, so we may define the characteristic class $\operatorname {\mathrm {cc}}_{X_1\times _S X_2/S}(A_1\boxtimes _S A_2)$ in $H^0(\operatorname {\mathrm {In}}_S(X_1\times _S X_2),K_{\operatorname {\mathrm {In}}_S(X_1\times _S X_2)})$ . We have an isomorphism $\operatorname {\mathrm {In}}_S(X_1\times _S X_2)\cong \operatorname {\mathrm {In}}_S(X_1)\times _S \operatorname {\mathrm {In}}_S(X_2)$ and therefore a Künneth map

$$ \begin{align*} \kappa_S \colon K_{\operatorname{\mathrm{In}}_S(X_1)/S} \otimes K_{\operatorname{\mathrm{In}}_S(X_2)/S} \to K_{\operatorname{\mathrm{In}}_S(X_1\times_S X_2)/S},\\[-15pt] \end{align*} $$

which we notate as $\mu _1\otimes \mu _2\mapsto \mu _1\boxtimes _S \mu _2$ for global sections $\mu _i$ of $K_{\operatorname {\mathrm {In}}_S(X_i)/S}$ . Then it is straightforward to show (and a corollary of Theorem 4.5.3 below) that

$$\begin{align*}\operatorname{\mathrm{cc}}_{X_1/S}(A_1)\boxtimes_S \operatorname{\mathrm{cc}}_{X_2/S}(A_2)=\operatorname{\mathrm{cc}}_{X_1\times_S X_2/S}(A_1\boxtimes_S A_2).\\[-15pt]\end{align*}$$

For our applications, we need a more general result involving fiber products over bases other than S. First, we need a modification of the above Künneth map in a general setting.

Definition 4.5.1 Modified Künneth map

Let $U\to T$ be a cohomologically smooth morphism of decent S-v-stacks. Suppose we are given a 2-commutative diagram of decent S-v-stacks

for $i=1,2$ such that $f_1$ and $f_2$ are fine. Let $f\colon Y_1\times _U Y_2\to X_1\times _T X_2$ and $g\colon Y_1\times _U Y_2 \to U$ be the induced product maps. Let $A_i$ be an object of $D_{\mathrm {\acute {e}t}}(X_i,\Lambda )$ for $i=1,2$ . We define a map

$$\begin{align*}\kappa_{U/T}\colon f_1^!A_1 \boxtimes_U f_2^!A_2 \to f^!(A_1\boxtimes_T A_2) \otimes g^*K_{U/T} \\[-15pt]\end{align*}$$

as follows. There is a Cartesian diagram

The map $\kappa _{U/T}$ is defined as the composition

$$ \begin{align*} & f_1^!A_1\boxtimes_U f_2^!A_2 \\[2pt] \cong & (\Delta_{U/T}')^*(f_1^!A_1 \boxtimes_T f_2^!A_2) \\[2pt] \to & (\Delta_{U/T}')^*(f_1\times_T f_2)^!(A_1\boxtimes_T A_2) \\[2pt] \stackrel{(4.1.3)}{\cong} & (\Delta_{U/T}')^*(f_1\times_T f_2)^!(A_1\boxtimes_T A_2) \otimes g^*(\Delta_{U/T})^!\Lambda_T \otimes g^*K_{U/T} \\[2pt] \to & (\Delta_{U/T}')^*(f_1\times_T f_2)^! (A_1\boxtimes_T A_2) \otimes (\Delta_{U/T}')^!\Lambda_{Y_1\times_T Y_2} \otimes g^*K_{U/T} \\[2pt] \to & (\Delta_{U/T}')^!(f_1\times_T f_2)^!(A_1\boxtimes_T A_2) \otimes g^*K_{U/T} \\[2pt] \cong & f^!(A_1\boxtimes_T A_2) \otimes g^*K_{U/T}. \end{align*} $$

In particular, the case $X_1=X_2=T=S$ yields a map

(4.5.1) $$ \begin{align} \kappa_{U/S} \colon K_{Y_1/S} \boxtimes_U K_{Y_2/S} \to K_{Y_1\times_U Y_2/S} \otimes g^*K_{U/S}. \end{align} $$

We can now introduce the setup of the main theorem of this section. We consider bases $T\to S$ satisfying two hypotheses: (1) $T\to S$ is cohomologically smooth, and (2) $\Delta _{T/S}\colon T\to T\times _S T$ is cohomologically smooth. These are satisfied for instance when $T=[S/G]$ , where G is a cohomologically smooth locally spatial group diamond over S. Considering the diagram

in which both squares are Cartesian, we see that $\operatorname {\mathrm {In}}_S(T)\to S$ is also cohomologically smooth. Furthermore, we can trivialize the dualizing complex $K_{\operatorname {\mathrm {In}}_S(T)/S}$ .

Lemma 4.5.2. Let T be a decent S-v-stack such that the structure map $\pi \colon T\to S$ and diagonal $\Delta _{T/S} \colon T\to T\times _S T$ are both cohomologically smooth. Then the object $\mathfrak {T}=(T,\Lambda _T)\in \operatorname {\mathrm {CoCorr}}_S$ is dualizable, and its characteristic class $\operatorname {\mathrm {cc}}_{T/S}(\Lambda _T)$ , considered as a morphism $\Lambda _{\operatorname {\mathrm {In}}_S(T)} \to K_{\operatorname {\mathrm {In}}_S(T)/S}$ , is an isomorphism.

In the case $T=[S/G]$ , where G is a cohomologically smooth locally spatial group diamond over S, the inertia stack is , the stack of conjugacy classes of G. This is a cohomologically smooth stack of dimension 0, so perhaps it is unsurprising that it has trivial dualizing complex; the lemma states that the trivialization is in fact canonical.

Proof. Let $\operatorname {\mathrm {pr}}_1,\operatorname {\mathrm {pr}}_2\colon T\times _S T\to T$ be the projection morphisms. The morphism in $D_{\mathrm {\acute {e}t}}(T,\Lambda )$ associated with the cohomological correspondence $\operatorname {\mathrm {coev}}_{\mathfrak {T}}\colon (S,\Lambda _S)\to (T\times _S T, \operatorname {\mathrm {pr}}_2^*K_{T/S})$ is an isomorphism:

$$\begin{align*}\pi^*\Lambda_S\cong\Delta_{T/S}^!\operatorname{\mathrm{pr}}_1^!\pi^*\Lambda_S \stackrel{\alpha}{\cong} \Delta_{T/S}^!\operatorname{\mathrm{pr}}_2^*\pi^!\Lambda_S = \Delta_{T/S}^!\operatorname{\mathrm{pr}}_2^*K_{T/S}. \end{align*}$$

Here, we the use cohomological smoothness of $\pi $ to get the isomorphism $\alpha $ , as in Theorem 4.1.6. The morphism $\Delta _{T/S}^*\operatorname {\mathrm {pr}}_2^*K_{T/S}\to \pi ^!\Lambda _S$ associated with $\operatorname {\mathrm {ev}}_{\mathfrak {T}}\colon (T\times _S T,\operatorname {\mathrm {pr}}_2^*K_{T/S})\to (S,\Lambda _S)$ is also an isomorphism (this is true without any hypotheses on $\pi $ or $\Delta _{T/S}$ ). Referring to the diagram

we may describe the characteristic class $\operatorname {\mathrm {cc}}_{T/S}(\Lambda _T)$ as the composition

$$ \begin{align*} h^*\pi^*\Lambda_S &\cong h^*\Delta_{T/S}^!\operatorname{\mathrm{pr}}_2^*K_{T/S}\\ &\stackrel{\beta}{\cong} h^!\Delta_{T/S}^*\operatorname{\mathrm{pr}}_2^* K_{T/S}\cong h^!K_{T/S}\cong K_{\operatorname{\mathrm{In}}_S(T)/S}, \end{align*} $$

which is an isomorphism. Here, we once again applied Theorem 4.1.6, using the cohomological smoothness of $\Delta _{T/S}$ .

Now suppose $p_i\colon X_i\to T$ ( $i=1,2$ ) is a fine morphism of decent S-v-stacks, with fiber product $p\colon X_1\times _T X_2\to T$ . Then there is an isomorphism

$$\begin{align*}\operatorname{\mathrm{In}}_S(X_1\times_T X_2) \cong \operatorname{\mathrm{In}}_S(X_1)\times_{\operatorname{\mathrm{In}}_S(T)} \operatorname{\mathrm{In}}_S(X_2), \end{align*}$$

and therefore by the discussion above we have a modified Künneth map

$$\begin{align*}\kappa_{\operatorname{\mathrm{In}}_S(T)/S}\colon K_{\operatorname{\mathrm{In}}_S(X_1)/S} \boxtimes_{\operatorname{\mathrm{In}}_S(T)} K_{\operatorname{\mathrm{In}}_S(X_2)/S} \to K_{\operatorname{\mathrm{In}}_S(X_1\times_T X_2)/S} \otimes \operatorname{\mathrm{In}}(p)^*K_{\operatorname{\mathrm{In}}_S(T)/S}. \end{align*}$$

Using the trivialization $\Lambda _{\operatorname {\mathrm {In}}_S(T)/S}\cong K_{\operatorname {\mathrm {In}}_S(T)/S}$ from Lemma 4.5.2, we obtain a map

(4.5.2) $$ \begin{align} K_{\operatorname{\mathrm{In}}_S(X_1)/S} \boxtimes_{\operatorname{\mathrm{In}}_S(T)} K_{\operatorname{\mathrm{In}}_S(X_2)/S} \to K_{\operatorname{\mathrm{In}}_S(X_1\times_T X_2)/S}, \end{align} $$

which on global sections we notate as $\mu _1\otimes \mu _2\mapsto \mu _1\boxtimes _{\operatorname {\mathrm {In}}_S(T)} \mu _2$ .

Finally, we can state the main theorem of the section.

Theorem 4.5.3. Let T be a decent S-v-stack such that the structure map $\pi \colon T\to S$ and the diagonal $\Delta _{T/S}\colon T\to T\times _S T$ are both cohomologically smooth. Let $X_1,X_2\to T$ be two fine morphisms of decent v-stacks (so also the induced morphisms $X_1,X_2 \to S$ are fine), and let $A_i\in D_{\mathrm {\acute {e}t}}(X_i,\Lambda )$ be a sheaf which is ULA over S. Then $A_1\boxtimes _T A_2$ is ULA over S, and

$$\begin{align*}\operatorname{\mathrm{cc}}_{X_1/S}(A_1)\boxtimes_{\operatorname{\mathrm{In}}_S(T)} \operatorname{\mathrm{cc}}_{X_2/S}(A_2) = \operatorname{\mathrm{cc}}_{X_1\times_T X_2/S}(A_1\boxtimes_T A_2). \end{align*}$$

In order to prove Theorem 4.5.3, we need to enhance the category $\operatorname {\mathrm {CoCorr}}_S$ to include data coming from a smooth base v-stack, which we allow to vary. At a first pass, one might think that such a category would have objects $(X\to T,A)$ , where $T\to S$ is cohomologically smooth, $X\to T$ is a morphism of v-stacks and $A\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ . The morphisms $(X\to T,A)\to (X'\to T',A')$ would be pairs $(q^{\natural },\alpha )$ , where $q^{\natural } =(p,q,p')$ is a morphism of correspondences as in a 2-commutative diagram:

(4.5.3)

and $\alpha \colon c^*A\to (c')^!A'$ is a morphism. We assume that $u'$ is cohomologically smooth. Given pairs $\mathfrak {X}_1=(X_1\to T,A_1)$ and $\mathfrak {X}_2=(X_2\to T,A_2)$ with common base T, we could then define $\mathfrak {X}_1\boxtimes _T \mathfrak {X}_2 = (X_1\times _T X_2\to T,A_1\boxtimes _T A_2)$ .

However, it turns out that this definition is not functorial in $\mathfrak {X}_1$ and $\mathfrak {X}_2$ . That is, given a morphism $e\colon T\to T'$ in $\operatorname {\mathrm {Corr}}_S$ and morphisms $f_i\colon \mathfrak {X}_i\to \mathfrak {X}_i'$ lying over e, one cannot in general define a map $f_1\otimes _e f_2\colon \mathfrak {X}_1\boxtimes _T \mathfrak {X}_2\to \mathfrak {X}_1'\boxtimes _{T'} \mathfrak {X}_2'$ . Essentially, this is due to the appearance of the invertible sheaf $K_{U/S}$ appearing in the modified Künneth map (4.5.1).

To obtain a functorial definition of $\mathfrak {X}_1\boxtimes _T \mathfrak {X}_2$ , we need to define an enhancement of the category which keeps track of an invertible sheaf living on the base.

Definition 4.5.4 (The category of based cohomological correspondences)

We define a symmetric monoidal 2-category $\operatorname {\mathrm {BCoCorr}}_S$ . The objects of $\operatorname {\mathrm {BCoCorr}}_S$ are triples $(X\to T,A,B)$ , where $X\to T$ is a (fine) morphism of decent S-v-stacks whose structure maps to S are fine, where A is an object of $D_{\mathrm {\acute {e}t}}(X,\Lambda )$ and where B is an invertible object of $D_{\mathrm {\acute {e}t}}(T,\Lambda )$ . We assume that the structure map $T\to S$ is cohomologically smooth.

Given objects $\mathfrak {X}=(X\to T,A,B)$ and $\mathfrak {X}'=(X'\to T',A',B')$ , an object of the category $\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {BCoCorr}}_S}(\mathfrak {X},\mathfrak {X}')$ is a triple $(q^{\natural },\alpha ,\beta )$ . The first element in the triple is a morphism $q^{\natural }=(p,q,p')$ between correspondences as in equation (4.5.3), where $u'$ is cohomologically smooth. The second element is a morphism $\alpha \colon c^*A \to (c')^!A'$ , and the third is an isomorphism $\beta \colon u^*B\stackrel {\cong }{\longrightarrow } (u')^! B'$ . Compositions of morphisms are defined similarly as in Definition 4.3.4. (Note that the cohomological smoothness of $u'$ is preserved under composition of correspondences.) Given objects $(X\to T,A,B)$ and $(X'\to T',A',B')$ , and morphisms between them represented by $C\to X\times _U X'$ (lying over $U\to T\times _S T'$ ) and $D\to X\times _V X'$ (lying over $V\to T\times _S T'$ ), a 2-morphism is an equivalence class of 2-commutative diagrams as in equation (4.3.3), together with a similar one involving morphisms $U\to V$ .

The monoidal structure on $\operatorname {\mathrm {BCoCorr}}_S$ is defined by

$$\begin{align*}(X\to T,A,B)\otimes (X'\to T',A',B')=(X\times_S X'\to T\times_S T',A\boxtimes_S A',B\boxtimes_S B'). \end{align*}$$

The unit object is $(S \overset {\mathrm {id}}{\to } S,\Lambda _S,\Lambda _S)$ .

Finally, we introduce the obvious monoidal functors $\mathcal {B},\mathcal {S}\colon \operatorname {\mathrm {BCoCorr}}_S\to \operatorname {\mathrm {CoCorr}}_S$ , with $\mathcal {B}(X\to T,A,B)=(T,B)$ (the base) and $\mathcal {S}(X\to T,A,B)=(X,A)$ (the source).

So far, the objects A and B in Definition 4.5.4 have nothing to do with each other. They only begin to interact when we talk about fiber products of objects of $\operatorname {\mathrm {BCoCorr}}_S$ over common bases. Given objects $\mathfrak {X}_i=(X_i\stackrel {p_i}{\to } T,A_i,B)$ for $i=1,2$ with common base $\mathfrak {T}=(T,B)$ , we define

$$\begin{align*}\mathfrak{X}_1\boxtimes_{\mathfrak{T}} \mathfrak{X}_2 = (X_1\times_T X_2\stackrel{p}{\to} T, (A_1\boxtimes_T A_2)\otimes p^*B^{-1},B). \end{align*}$$

We claim that $\boxtimes _{\mathfrak {T}}$ defines a monoidal functor

(4.5.4) $$ \begin{align} \operatorname{\mathrm{BCoCorr}}_S \times_{\mathcal{B},\operatorname{\mathrm{CoCorr}}_S} \operatorname{\mathrm{BCoCorr}}_S \to \operatorname{\mathrm{BCoCorr}}_S. \end{align} $$

A morphism in the category $\operatorname {\mathrm {BCoCorr}}_S \times _{\mathcal {B},\operatorname {\mathrm {CoCorr}}_S} \operatorname {\mathrm {BCoCorr}}_S$ is a morphism $e\colon \mathfrak {T}\to \mathfrak {T}'$ in $\operatorname {\mathrm {CoCorr}}_S$ together with a pair of morphisms $\mathfrak {f}_i\colon \mathfrak {X}_i\to \mathfrak {X}_i'$ ( $i=1,2$ ) lying over e. We may represent this state of affairs with a diagram

(4.5.5)

for $i=1,2$ , with $u'$ cohomologically smooth, together with morphisms $\alpha _i\colon c_i^*A_i\to (c_i')^!A_i'$ for $i=1,2$ and an isomorphism $\beta \colon u^*B\stackrel {\cong }{\longrightarrow } (u')^! B'$ . Taking fiber products over the base correspondence, we obtain a morphism of correspondences $q^{\natural }=(p,q,p')$ fitting into a diagram

(4.5.6)

The required morphism

(4.5.7) $$ \begin{align} c^*\left((A_1\boxtimes_T A_2)\otimes p^*B^{-1}\right) \to (c')^!\left((A_1'\boxtimes_{T'} A_2') \otimes (p')^*(B')^{-1}\right) \end{align} $$

is defined as the composition

$$ \begin{align*} \begin{array}{lcl} c^*\left((A_1\boxtimes_T A_2)\otimes p^*B^{-1}\right) &\cong &\!\! (c_1^*A_1 \boxtimes_U c_2^* A_2)\otimes q^*u^*B^{-1} \\[2pt] &\stackrel{(\alpha_1\boxtimes_U\alpha_2)\otimes \beta}{\to} &\!\! (c_1')^!A_1' \boxtimes_U (c_2')^!A_2' \otimes q^*((u')^!B')^{-1} \\[2pt] &\stackrel{\kappa_{U/T'}}{\to} &\!\! (c')^!(A_1' \boxtimes_{T'} A_2') \otimes q^*(K_{U/T'}\otimes ((u')^!B')^{-1}) \\[2pt] &\cong &\!\! (c')^!(A_1' \boxtimes_{T'} A_2') \otimes q^*(u')^*(B')^{-1} \\[2pt] &\cong &\!\! (c')^!\left((A_1'\boxtimes_{T'} A_2') \otimes (p')^*(B')^{-1}\right), \end{array} \end{align*} $$

where in the last step we used Lemma 4.1.10.

To completely justify that equation (4.5.4) is a functor, one must also produce a 2-isomorphism

(4.5.8) $$ \begin{align} (\mathfrak{f}_1'\boxtimes_{e'} \mathfrak{f}_2') \circ (\mathfrak{f}_1\boxtimes_e \mathfrak{f}_2) \cong (\mathfrak{f}_1'\circ \mathfrak{f}_1)\boxtimes_{e'\circ e} (\mathfrak{f}_2'\circ \mathfrak{f}_2) \end{align} $$

whenever all compositions are defined. Furthermore, one must also show that equation (4.5.4) is a monoidal functor; that is, we have an isomorphism

(4.5.9) $$ \begin{align} (\mathfrak{X}_1 \boxtimes_{\mathfrak{T}} \mathfrak{X}_2) \otimes (\mathfrak{X}_1'\boxtimes_{\mathfrak{T}'} \mathfrak{X}_2') \cong (\mathfrak{X}_1\otimes \mathfrak{X}_1')\boxtimes_{\mathfrak{T}\otimes \mathfrak{T}'} (\mathfrak{X}_2\otimes\mathfrak{X}_2'). \end{align} $$

The details are straightforward but tedious.

We can now prove Theorem 4.5.3. Let $X_1,X_2\to T$ be two morphisms satisfying the assumptions of that theorem, and let $A_i\in D_{\mathrm {\acute {e}t}}(X_i,\Lambda )$ be two sheaves which are ULA over S. Assume that the structure map $\pi \colon T\to S$ and the diagonal $\Delta _{T/S}$ are both cohomologically smooth.

Let $\mathfrak {X}_i=(X_i\to T, A_i,\Lambda _T)\in \operatorname {\mathrm {BCoCorr}}_S$ for $i=1,2$ so that $\mathcal {B}(\mathfrak {X}_1)=\mathcal {B}(\mathfrak {X}_2)=\mathfrak {T}=(T,\Lambda _T)$ . Then $\mathfrak {X}_i$ is dualizable, with dual $\mathfrak {X}_i^{\vee }=(X_i\to T,\mathbf {D}_{X_i/S}A_i,K_{T/S})$ , as witnessed by $\operatorname {\mathrm {coev}}_{\mathfrak {X}_i}\colon 1_{\operatorname {\mathrm {BCoCorr}}_S}\to \mathfrak {X}_i\otimes \mathfrak {X}_i^{\vee }$ and $\operatorname {\mathrm {ev}}_{\mathfrak {X}_i}\colon \mathfrak {X}_i^{\vee } \otimes \mathfrak {X}_i \to 1_{\operatorname {\mathrm {BCoCorr}}_S}$ . Note that $\mathcal {B}(\operatorname {\mathrm {coev}}_{\mathfrak {X}_i})=\operatorname {\mathrm {coev}}_{\mathfrak {T}}$ and $\mathcal {S}(\operatorname {\mathrm {coev}}_{\mathfrak {X}_i})=\operatorname {\mathrm {coev}}_{\mathcal {S}(\mathfrak {X}_i)}$ , and similarly for $\operatorname {\mathrm {ev}}$ . Then the categorical trace of $1_{\mathfrak {X}_i}$ is $\operatorname {\mathrm {tr}}(1_{\mathfrak {X}_i})=\operatorname {\mathrm {ev}}_{\mathfrak {X}_i}\circ \operatorname {\mathrm {coev}}_{\mathfrak {X}_i}$ so that $\mathcal {S}(\operatorname {\mathrm {tr}}(1_{\mathfrak {X}_i}))=\operatorname {\mathrm {tr}}(1_{\mathcal {S}(\mathfrak {X}_i)})=\operatorname {\mathrm {cc}}_{X_i/S}(A_i)$ .

Now consider $\mathfrak {X}=\mathfrak {X}_1\boxtimes _{\mathfrak {T}} \mathfrak {X}_2=(X_1\boxtimes _T X_2\to T, A_1\boxtimes _T A_2,\Lambda _T)$ . Define an object $\mathfrak {X}^{\vee } = \mathfrak {X}_1^{\vee }\boxtimes _{\mathfrak {T}^{\vee }} \mathfrak {X}_2^{\vee }$ , and define morphisms $\operatorname {\mathrm {coev}}_{\mathfrak {X}}$ and $\operatorname {\mathrm {ev}}_{\mathfrak {X}}$ via the diagrams

and

Then $\mathfrak {X}^{\vee }$ , $\operatorname {\mathrm {coev}}_{\mathfrak {X}}$ and $\operatorname {\mathrm {ev}}_{\mathfrak {X}}$ witness the dualizability of $\mathfrak {X}$ . It follows that $\mathcal {S}(\mathfrak {X})=(X_1\times _T X_2,A_1\boxtimes _T A_2)$ is dualizable so that $A_1\boxtimes _T A_2$ is ULA over S. Now consider $\operatorname {\mathrm {tr}}(1_{\mathfrak {X}})=\operatorname {\mathrm {ev}}_{\mathfrak {X}}\circ \operatorname {\mathrm {coev}}_{\mathfrak {X}}$ , an endomorphism of $1_{\operatorname {\mathrm {BCoCorr}}_S}$ . On the one hand, $\mathcal {S}(\operatorname {\mathrm {tr}}(1_{\mathfrak {X}}))=\operatorname {\mathrm {tr}}(1_{\mathcal {S}(\mathfrak {X})})\in \operatorname {\mathrm {End}} 1_{\operatorname {\mathrm {CoCorr}}_S}$ is the datum of the inertia stack $\operatorname {\mathrm {In}}_S(X_1\times _T X_2)$ together with the characteristic class $\operatorname {\mathrm {cc}}_{X_1\times _T X_2/S}(A)\in H^0(\operatorname {\mathrm {In}}_S(X_1\times _T X_2),K_{\operatorname {\mathrm {In}}_S(X_1\times _T X_2)/S})$ . On the other hand, equation (4.5.8) gives a 2-isomorphism

$$\begin{align*}\operatorname{\mathrm{tr}}(1_{\mathfrak{X}}) \cong (\operatorname{\mathrm{ev}}_{\mathfrak{X}_1}\boxtimes_{\operatorname{\mathrm{ev}}_{\mathfrak{T}}} \operatorname{\mathrm{ev}}_{\mathfrak{X}_2}) \circ (\operatorname{\mathrm{coev}}_{\mathfrak{X}_1}\boxtimes_{\operatorname{\mathrm{coev}}_{\mathfrak{T}}} \operatorname{\mathrm{coev}}_{\mathfrak{X}_2}) \cong \operatorname{\mathrm{tr}}(1_{\mathfrak{X_1}}) \boxtimes_{\operatorname{\mathrm{tr}}(1_{\mathfrak{T}})} \operatorname{\mathrm{tr}}(1_{\mathfrak{X}_2}). \end{align*}$$

The source of this morphism is the correspondence $\operatorname {\mathrm {In}}_S(X_1)\times _{\operatorname {\mathrm {In}}_S(T_1)} \operatorname {\mathrm {In}}_S(X_2)\to S\times _S S \cong S$ together with a global section of $K_{\operatorname {\mathrm {In}}_S(X_1\times _T X_2)/S}$ . Reviewing the definition of $\boxtimes $ for morphisms in $\operatorname {\mathrm {BCoCorr}}_S$ as in equation (4.5.7), we see that this section is the image of $\operatorname {\mathrm {cc}}_{X_1/S}(A_1)\otimes \operatorname {\mathrm {cc}}_{X_2/S}(A_2)$ under

$$ \begin{align*} \begin{array}{lcl} K_{\operatorname{\mathrm{In}}_S(X_1)/S}\boxtimes_{\operatorname{\mathrm{In}}_S(T)} K_{\operatorname{\mathrm{In}}_S(X_2)/S} &\stackrel{\kappa_{\operatorname{\mathrm{In}}_S(T)/S}}{\to} &\!\! K_{\operatorname{\mathrm{In}}_S(X_1\times_T X_2)/S} \otimes \operatorname{\mathrm{In}}(p)^*K_{\operatorname{\mathrm{In}}_S(T)} \\[2pt] &\cong &\!\! K_{\operatorname{\mathrm{In}}_S(X_1\times_T X_2)/S}, \end{array} \end{align*} $$

where the last isomorphism is induced from the inverse to $\operatorname {\mathrm {cc}}_{T/S}(\Lambda _T)\colon \Lambda _{\operatorname {\mathrm {In}}_S(T)} \to K_{\operatorname {\mathrm {In}}_S(T)/S}$ . The result is exactly $\operatorname {\mathrm {cc}}_{X_1/S}(A_1)\boxtimes _{\operatorname {\mathrm {In}}_S(T)} \operatorname {\mathrm {cc}}_{X_2/S}(A_2)$ as defined in Theorem 4.5.3.

4.6 The case of $[X/G]$ for G smooth

Let X be a nice diamond over S which is equipped with an action of a cohomologically smooth S-group diamond G. Let $\alpha \colon X\times _S G\to X$ be the action map $(x,g)\mapsto g(x)$ . Let $Y=[X/G]$ be the stack quotient; this is a decent S-v-stack whose structure map to S is fine. The point of this section is to compare two contexts for the Lefschetz–Verdier trace formula: one for the identity correspondence on $[X/G]$ and the other for the morphism $g\colon X\to X$ for an individual $g\in G(S)$ .

Let $A\in D_{\mathrm {\acute {e}t}}(Y,\Lambda )$ be ULA over S. Then the pair $(Y,A)$ is dualizable in $\operatorname {\mathrm {CoCorr}}_S$ , and we obtain a characteristic class

$$\begin{align*}\mathrm{cc}_{Y/S}(A)\in H^{0}(\mathrm{In}_{S}(Y),K_{\mathrm{In}_{S}(Y)/S}). \end{align*}$$

On the other hand, the pullback $A_X$ of A along $X\to Y$ is also ULA over S (because $G\to S$ is cohomologically smooth). For each element $g\in G(S)$ , we have an isomorphism $u_g\colon A_X\to g^*A_X$ lying over $g\colon X\to X$ . The pair $(g,u_g)$ constitutes an endomorphism of the dualizable object $(X,A_X)$ in $\operatorname {\mathrm {CoCorr}}_S$ , so we may define the categorical trace $\operatorname {\mathrm {tr}}(g,u_g)\in H^0(\operatorname {\mathrm {Fix}}(g),K_{\operatorname {\mathrm {Fix}}(g)/S})$ . Here, $\operatorname {\mathrm {Fix}}(g)=X\times _{g,X\times _S X,\Delta _{X/S}} X$ is the fixed-point locus of g on X. The object is to show how the $\operatorname {\mathrm {tr}}(g,u_g)$ can be derived from $\mathrm {cc}_{Y/S}(A)$ .

First, we give a concrete presentation of $\mathrm {In}_{S}(Y)$ . Define a correspondence c on X by

$$ \begin{align*} c=\operatorname{\mathrm{pr}}_X\times_S \alpha:X\times_S G&\to X\times_S X\\ (x,g)&\mapsto (x,g(x)). \end{align*} $$

Then the fixed-point locus $\operatorname {\mathrm {Fix}}(c)\subset X\times _S G$ is G-stable for the G-action on $X\times _S G$ given by $h(x,g)=(h(x),hgh^{-1})$ , and then $\operatorname {\mathrm {In}}_S(Y)\cong [\operatorname {\mathrm {Fix}}(c)/G]$ . With respect to this isomorphism, the canonical map is the quotient by G of the projection map $\operatorname {\mathrm {Fix}}(c)\to G$ .

The G-equivariance of $A\vert _X$ may be expressed an isomorphism $u\colon A\vert _{X\times G} \to \alpha ^*A\vert _X$ . This is not a cohomological correspondence in general (as $\alpha ^*\neq \alpha ^!$ ). To obtain a cohomological correspondence on nonstacky objects, we work over the base G. Let $X_G=X\times _S G$ , and consider the correspondence $\widetilde {c}$ defined by the diagram of diamonds over G:

By design, the fiber of this correspondence over $g\in G(S)$ is automorphism $g\colon X\to X$ . Moreover, there is a natural isomorphism $\operatorname {\mathrm {Fix}}(c)\cong \operatorname {\mathrm {Fix}}(\widetilde {c})$ , and the fiber of $\mathrm {Fix}(\widetilde {c})$ over any $g\in G(S)$ is exactly $\mathrm {Fix}(g)$ . The G-equivariance of $A\vert _{X}$ is encoded by an isomorphism $\widetilde {u}\colon A\vert _{X_G}\to \widetilde {\alpha }^*A\vert _{X_G}$ . Since $\widetilde {\alpha }$ is an isomorphism, we have $\widetilde {\alpha }^*\cong \widetilde {\alpha }^!$ , and therefore, the pair $(\widetilde {c},\widetilde {u})$ constitutes an endomorphism of the dualizable object $(X_G,A\vert _{X_G})$ of $\operatorname {\mathrm {CoCorr}}_G$ . The categorical trace of $(\widetilde {c},\widetilde {u})$ is an element

$$\begin{align*}\operatorname{\mathrm{tr}}(\widetilde{c},\widetilde{u}) \in H^0(\operatorname{\mathrm{Fix}}(\widetilde{c}),K_{\operatorname{\mathrm{Fix}}(\widetilde{c})/G}). \end{align*}$$

This is the ‘universal local term’ for the action of G on X, in the sense that, for any $g\in G(S)$ , the restriction map

$$\begin{align*}H^0(\operatorname{\mathrm{Fix}}(\widetilde{c}),K_{\operatorname{\mathrm{Fix}}(\widetilde{c})/G}) \to H^0(\operatorname{\mathrm{Fix}}(g), K_{\operatorname{\mathrm{Fix}}(g)/S}) \end{align*}$$

carries $\operatorname {\mathrm {tr}}(\widetilde {c},\widetilde {u})$ onto $\operatorname {\mathrm {tr}}(g,u_g)$ .

We want to compare the characteristic class $\mathrm {cc}_{Y/S}(A)$ with the universal local term $\operatorname {\mathrm {tr}}(\widetilde {c},\widetilde {u})$ . To do this, we first observe that from the Cartesian square

we obtain a canonical map

, and thus a canonical pullback map

(4.6.1)

Next, Lemma 4.5.2 applied to $T=[S/G]$ shows that $\operatorname {\mathrm {cc}}_{T/S}(\Lambda _T)$ is an isomorphism

. This induces an isomorphism

(4.6.2)

Combining equations (4.6.1) and (4.6.2), we obtain a canonical map

(4.6.3) $$ \begin{align} \iota:H^{0}(\mathrm{In}_{S}(Y),K_{\mathrm{In}_{S}(Y)/S})\to H^{0}(\mathrm{Fix}(\widetilde{c}),K_{\mathrm{Fix}(\widetilde{c})/G}). \end{align} $$

The main result of this section is the following theorem.

Theorem 4.6.1. Notation and assumptions as above, we have an equality

$$\begin{align*}\iota\left(\mathrm{cc}_{Y/S}(A)\right)=\mathrm{tr}_{\widetilde{c}}(\widetilde{u},A|_{X_{G}}). \end{align*}$$

Proof. We restate the theorem in the language of based cohomological correspondences. The main players are

• • $T=([S/G],\Lambda _{[S/G]})$ , a dualizable object of $\operatorname {\mathrm {CoCorr}}_S$ .

• • $\mathfrak {Y}=(Y\to [S/G],A,\Lambda _{[S/G]})$ , a dualizable object of $\operatorname {\mathrm {BCoCorr}}_S$ with base T.

• • $\mathfrak {X}_G=(X_G,A_{X_G})$ , a dualizable object of $\operatorname {\mathrm {CoCorr}}_G$ with base $1_{\operatorname {\mathrm {CoCorr}}_G}$ .

• • $\alpha \in \operatorname {\mathrm {End}} \mathfrak {X}_G$ , the endomorphism described by the pair $(\widetilde {c},\widetilde {u})$ .

We would like to relate $\operatorname {\mathrm {tr}}(\operatorname {\mathrm {id}}_{\mathfrak {Y}})$ to $\operatorname {\mathrm {tr}}(\alpha )$ . The idea is to promote $\alpha $ to an endomorphism of based cohomological correspondences which lies over $\operatorname {\mathrm {id}}_{T}$ . To this end, we introduce some more objects in $\operatorname {\mathrm {BCoCorr}}_S$ :

• • $\mathfrak {G}=(G\to S,\Lambda _G,\Lambda _S)$ with base $1_{\operatorname {\mathrm {CoCorr}}_S}$ ,

• • $\mathfrak {G}'=(G\to [S/G]\times _S [S/G],\Lambda _G,\Lambda _{[S/G]}\boxtimes _S K_{[S/G]/S})$ , with base $T\otimes T^{\vee }$ , where the morphism $G\to [S/G]\times _S [S/G]$ is defined as the trivial $G\times _S G$ -torsor $G_{G\times _S G}=G\times _S G\times _S G$ .

We also define morphisms in $\operatorname {\mathrm {BCoCorr}}_S$ :

• • $\operatorname {\mathrm {coev}}_G \colon \mathfrak {G}\to \mathfrak {G}'$ , which has base $\operatorname {\mathrm {coev}}_T$ and source $1_{(G,\Lambda _G)}$ ,

• • $\operatorname {\mathrm {ev}}_G\colon \mathfrak {G}'\to \mathfrak {G}$ , which has base $\operatorname {\mathrm {ev}}_T$ and source $1_{(G,\Lambda _G)}$ .

• • An automorphism $\alpha _0\in \operatorname {\mathrm {Aut}} \mathfrak {G}'$ lying over the identity on both base and source, coming from a 2-isomorphism of $G\to [S/G]^2$ corresponding to the automorphism of the trivial torsor $G_{G\times G}\to G_{G\times G}$ defined by $(x,g,h)\mapsto (x,gx,h)$ .

Now observe that $\operatorname {\mathrm {tr}}_G:=\operatorname {\mathrm {ev}}_G\circ \alpha _0\circ \operatorname {\mathrm {coev}}_G$ is an endomorphism of $\mathfrak {G}$ with base $\operatorname {\mathrm {tr}}(\operatorname {\mathrm {id}}_T)$ and source $\operatorname {\mathrm {id}}_{\mathcal {S}(\mathfrak {G})}$ , whose underlying based correspondence is shown in the diagram:

where the central horizontal arrow sends g to its own conjugacy class. (If we had omitted $\alpha _0$ from the definition of $\operatorname {\mathrm {tr}}_G$ , the central horizontal arrow would send everything to the identity of G.)

Recall that $\mathcal {S}\colon \operatorname {\mathrm {BCoCorr}}_S\to \operatorname {\mathrm {CoCorr}}_S$ takes a based cohomological corresponce onto its source. Let $\mathcal {F}\colon \operatorname {\mathrm {CoCorr}}_G\to \operatorname {\mathrm {CoCorr}}_S$ be the functor which forgets the base G. We have a diagram in $\operatorname {\mathrm {CoCorr}}_S$ :

where all squares are filled in with 2-isomorpisms. The functoriality property of $\boxtimes $ from equation (4.5.8) now gives a 2-isomorphism

(4.6.4) $$ \begin{align} \mathcal{S}\left(\operatorname{\mathrm{cc}}_{Y/S}(A)\boxtimes_{\operatorname{\mathrm{cc}}_{[S/G]/S}(\Lambda_{[S/G]})} \operatorname{\mathrm{tr}}_G(\alpha_0) \right)\stackrel{\cong}{\longrightarrow} \mathcal{F}(\operatorname{\mathrm{tr}}(\alpha)). \end{align} $$

The isomorphism of v-stacks implicit in equation (4.6.4) is expressed by the fact that we have a Cartesian diagram:

On the level of cohomology classes, equation (4.6.4) tells us that $\operatorname {\mathrm {tr}}(\alpha )$ , considered as an element of $H^0(\operatorname {\mathrm {Fix}}(\widetilde {c}),K_{\operatorname {\mathrm {Fix}}(\widetilde {c})/S})$ , can be derived from $\operatorname {\mathrm {cc}}_{Y/S}(A)$ in the manner described by the theorem.

We conclude the section with a remark about isolated fixed points. Assume there exists a conjugacy-invariant open subset $U\subset G$ whose elements act on X with only isolated fixed points. (This is the case for the action of the positive loop group on the affine Grassmannian. We study that scenario in the next section.) Write $\operatorname {\mathrm {In}}_S([X/G])_U$ for the pullback of under . Then is étale over ; as such, we have a canonical trivialization $K_{\operatorname {\mathrm {In}}_S([X/G])_U/S}\cong \Lambda _{\operatorname {\mathrm {In}}_S([X/G])_U}$ . Therefore, the restriction over U of the characteristic class of A is an element

$$\begin{align*}cc_{[X/G]/S}(A)_U\in H^0(\operatorname{\mathrm{In}}_S([X/G])_U,\Lambda); \end{align*}$$

that is, it is a continuous function on the space of pairs $(x,g)\in X\times _S U$ with $g(x)=x$ . Theorem 4.6.1 implies that this function is $(x,g)\mapsto \operatorname {\mathrm {loc}}_x(g,A)$ .

5 Local terms on the $B_{\mathrm {dR}}$ -affine Grassmannian

The goal of this chapter is to explicitly compute certain local terms on the $B_{\operatorname {\mathrm {dR}}}$ -affine Grassmannian in terms of the geometric Satake equivalence.

5.1 The main result

To explain the main result, let us fix some notation. Let $F/\mathbf {Q}_{p}$ be a finite extension with residue field $\mathbf {F}_q$ . Let C be the completion of an algebraic closure of F. Let $G/F$ be a connected reductive group, and let $\mathrm {Gr}_{G}=LG/L^{+}G$ be the associated $B_{\mathrm {dR}}$ -affine Grassmannian over $\operatorname {\mathrm {Spd}} C$ . We explain the notation: For a perfectoid C-algebra R, we have the loop group $LG(R)=B_{\operatorname {\mathrm {dR}}}(R)$ and its positive subgroup $L^+G(R)=B_{\operatorname {\mathrm {dR}}}^+(R)$ . Then $\operatorname {\mathrm {Gr}}_G$ is an ind-spatial diamond admitting an action of $L^+G$ and in particular its subgroup $G(F)$ . For a cocharacter $\mu $ of $G_{\overline {F}}$ , we let $\operatorname {\mathrm {Gr}}_{G,\leq \mu }$ be the corresponding closed Schubert cell; this is a proper diamond. Finally, define the local Hecke stack by

$$\begin{align*}\operatorname{\mathrm{Hecke}}_G^{\operatorname{\mathrm{loc}}} = [L^+G\backslash \operatorname{\mathrm{Gr}}_G ] = [L^+G \backslash LG/L^+G].\end{align*}$$

We remark that there are versions of these objects living over $\operatorname {\mathrm {Spd}} F$ , but we will not need these for our results.

Fix a coefficient ring $\Lambda \in \left \{ \mathbf {Z}/\ell ^{n}\mathbf {Z}[\sqrt {q}],\mathbf {Z}_{\ell }[\sqrt {q}\right \} $ . The Satake category

$$\begin{align*}\operatorname{\mathrm{Sat}}_G(\Lambda) \subset D_{\mathrm{\acute{e}t}}(\operatorname{\mathrm{Hecke}}^{\operatorname{\mathrm{loc}}}_{G,C}, \Lambda) \end{align*}$$

is the subcategory of objects which are perverse, $\Lambda $ -flat and ULA over $\operatorname {\mathrm {Spd}} C$ [FS21, Definition I.6.2]. It is a symmetric monoidal category under the convolution product.

Theorem 5.1.1 [FS21, Theorem I.6.3]

There is an equivalence of symmetric monoidal categories:

$$ \begin{align*} \operatorname{\mathrm{Rep}}_{\widehat{G}}(\Lambda) &\stackrel{\cong}{\longrightarrow} \operatorname{\mathrm{Sat}}_G(\Lambda) \\ V &\;\mapsto \ \mathcal{S}_V, \end{align*} $$

where $\widehat {G}$ is the Langlands dual group (considered over $\Lambda $ ), and $\operatorname {\mathrm {Rep}}_{\widehat {G}}(\Lambda )$ is the category of representations of $\widehat {G}$ on finite projective $\Lambda $ -modules.

We continue to write $\mathcal {S}_V$ for the pullback of this object along the quotient $\operatorname {\mathrm {Gr}}_G\to [L^+G \backslash \operatorname {\mathrm {Gr}}_G]$ .

Our next order of business is to determine, for $g\in G(F)_{\operatorname {\mathrm {sr}}}$ , the fixed point locus $\operatorname {\mathrm {Gr}}_G^g$ . The answer is the same regardless of which sort of affine Grassmannian we consider (classical, Witt vector, $B_{\operatorname {\mathrm {dR}}}$ ), as the following proposition shows.

Proposition 5.1.2. Let $K^+$ be a discrete valuation ring with algebraically closed residue field k and fraction field K. Let G be a reductive group over $K^+$ . Let $g\in G(K^+)$ be an element whose image in $G(k)$ is strongly regular, and let $T=\operatorname {\mathrm {Cent}}(g,G)$ . The inclusion $T\subset G$ induces a bijection

$$\begin{align*}T(K)/T(K^+)\cong (G(K)/G(K^+))^g,\end{align*}$$

so that the fixed point locus of g may be identified with $X_*(T)$ .

Consequently, if $\operatorname {\mathrm {Gr}}_G$ is any incarnation of the affine Grassmannian, then $\operatorname {\mathrm {Gr}}_{G,\leq \mu }^g$ is finite over its base with underlying set $X_*(T)_{\leq \mu }$ .

Proof. Let $\mathcal {B}$ be the (reduced) Bruhat–Tits building of the split reductive group $G_K$ over the discretely valued field K. Thus, $\mathcal {B}$ is a locally finite simplicial complex admitting an action of $LG=G(K)$ . We will identify the $LG$ -set $\operatorname {\mathrm {Gr}}_G$ with a piece of this building.

By [BT84, 5.1.40], there exists a hyperspecial point $\bar o \in \mathcal {B}$ corresponding to $L^+G=G(K^+)$ . The point $\bar o$ can be characterized by [BT84, 4.6.29] as the unique fixed point of $L^+G$ . Let $\mathcal {B}^{\mathrm {ext}}$ be the extended Bruhat–Tits building of $G_K$ . Recall that $\mathcal {B}^{\mathrm {ext}}=\mathcal {B} \times X_*(A_G)_{\mathbf {R}}$ , where $A_G$ is the connected center of G. The group $LG$ acts on $X_*(A_G)_{\mathbf {R}}$ via the isomorphism $X_*(A_G)_{\mathbf {R}} \to X_*(A_G')_{\mathbf {R}}$ , where $A_G'$ is the maximal abelian quotient of G. Let $o=(\bar o,z)$ be any point in $\mathcal {B}^{\mathrm {ext}}$ lying over $\bar o$ . Then $L^+G$ can be characterized as the full stabilizer of o in $G(K)$ : It is clear that $L^+G$ stabilizes o, and the reverse inclusion follows from the Cartan decomposition $LG = L^+G \cdot X_*(T) \cdot L^+G$ (which relies on $\bar o$ being hyperspecial) and the fact that $X_*(T)$ acts on the apartment of T in $\mathcal {B}^{\mathrm {ext}}$ by translations. It follows that the action of $LG$ on $\mathcal {B}^{\mathrm {ext}}$ provides an $LG$ -equivariant bijection from $\operatorname {\mathrm {Gr}}_G$ to the orbit of $LG$ through o.

Now suppose $x\in \operatorname {\mathrm {Gr}}_G$ is fixed by a strongly regular element $g\in L^+T_{\operatorname {\mathrm {sr}}}$ . Then its image in $\mathcal {B}^{\mathrm {ext}}$ is a g-fixed point belonging to the orbit of o, and we can write $x=ho$ for some $h\in LG$ . For every root $\alpha \colon T\to \mathbf {G}_{\mathrm {m}}$ , the element $\alpha (g)$ does not lie in the kernel of $L^+\mathbf {G}_{\mathrm {m}}\to \mathbf {G}_{\mathrm {m}}$ . According to [Tit79, 3.6.1], the image of x in $\mathcal {B}$ belongs to the apartment $\mathcal {A}$ of T. At the same time, $g\in L^+G$ also fixes $\bar o$ , so for the same reason, $\bar o \in \mathcal {A}$ . Thus, $\bar o$ belongs to both apartments $\mathcal {A}$ and $h^{-1}\mathcal {A}$ . Since $L^+G$ acts transitively on the apartments containing $\bar o$ [BT84, 4.6.28], we can multiply h on the right by an element of $L^+G$ to ensure that $h^{-1}\mathcal {A}=\mathcal {A}$ . By [BT72, 7.4.10], we then have $h \in L^+N(T,G)$ . Since $\bar o$ is hyperspecial, every Weyl reflection is realized in $L^+G$ , and hence, we may again modify h on the right to achieve $h \in LT$ . We see now that $x=ho$ is fixed by all of $LT$ and that furthermore the coset $x=hL^+G$ is the image of the coset $hL^+T$ .

Proposition 5.1.2 shows that, if we fix a split maximal torus $\widehat {T}\subset \widehat {G}$ , there is a natural finite-to-one map

$$ \begin{align*} \mathrm{Gr}_{G}^{g} & \to X_{+}^{\ast}(\widehat{T})\\ x & \mapsto \,\nu_{x}. \end{align*} $$

Note that $\nu _{x}$ simply records which open Schubert cell of $\mathrm {Gr}_{G}$ contains the point x.

Now, for any $V\in \mathrm {Rep}(\widehat {G})$ and any $x\in \mathrm {Gr}_{G}^{g}$ , there is an associated local term $\mathrm {loc}_{x}(g,\mathcal {S}_{V})\in \Lambda $ . The main result of this chapter is the following theorem, giving an explicit computation of these local terms.

Theorem 5.1.3. Let $V\in \mathrm {Rep}(\widehat {G})$ be an object of the Satake category, and let $g\in G(F)_{\mathrm {sr}}$ be a strongly regular semisimple element. Then for any $x\in \mathrm {Gr}_{G}^{g}$ , there is an equality in $\Lambda $ :

$$\begin{align*}\mathrm{loc}_{x}(g,\mathcal{S}_{V})=(-1)^{\left\langle 2\rho,\nu_{x}\right\rangle }\mathrm{rank}_{\Lambda}V[\nu_{x}]. \end{align*}$$

Note that, since V is (by hypothesis) a finite projective $\Lambda $ -module and tori are reductive in the strongest sense, the weight space $V[\nu _{x}]$ is a finite projective $\Lambda $ -module, so the right-hand side of this equality is well-defined.

Due to the highly inexplicit nature of local terms, the proof of Theorem 5.1.3 is rather indirect. Indeed, we would be able to give a simple proof of Theorem 5.1.3 if we knew the equality between ‘true’ and ‘naive’ local terms on $\mathrm {Gr}_{G}$ . Unfortunately, this equality seems to be a very difficult problem. Even for schemes, the problem of comparing true and naive local terms was only settled very recently by Varshavsky. Instead, our strategy reduces the computation of the local terms in Theorem 5.1.3 to an analogous computation on the Witt vector affine Grassmannian, where a global-to-local argument can be pushed through. The key theme in the proof is the idea that local terms are constant in families.

For our applications, the following restatement of the main results of this section in terms of characteristic classes on the quotient $[\operatorname {\mathrm {Gr}}_{G,\leq \mu }/L_m^+G]$ will be useful.

Theorem 5.1.4. Let V be such that $\mathcal {S}_V$ is supported on some Schubert cell $\operatorname {\mathrm {Gr}}_{G,\leq \mu }$ . Choose some large m such that the $L^+ G$ -action on this cell factors through the quotient $L_m^+G$ , and set $X=[\operatorname {\mathrm {Gr}}_{G,\leq \mu }/L_m^+G]$ .

Then the set of connected components of $\operatorname {\mathrm {In}}_S(X)_{\operatorname {\mathrm {sr}}}$ may be identified with $X_*^+(T)_{\leq \mu }/W$ , and the dualizing complex of $\operatorname {\mathrm {In}}_S(X)_{\operatorname {\mathrm {sr}}}$ has a canonical trivialization. With respect to those identifications, the restriction of $\operatorname {\mathrm {cc}}_{X/S}(\mathcal {S}_V)$ to $\operatorname {\mathrm {In}}_S(X)_{\operatorname {\mathrm {sr}}}$ is the function sending $\lambda \in X_*(T)$ to $(-1)^{\left < 2\rho _G,\lambda \right>}\operatorname {\mathrm {rank}}_{\Lambda } V[\lambda ]$ .

Proof. The first claim is proved in §5.4 below. Since $L_m^+G$ is a cohomologically smooth group diamond, Theorem 4.6.1 applies to the quotient $X=[\operatorname {\mathrm {Gr}}_{G,\leq \mu }/L_m^+G]$ . The remark following the proof of that theorem applies to the locus $L_m^+G_{\operatorname {\mathrm {sr}}}$ so that we may relate $\operatorname {\mathrm {cc}}_{X/S}(\mathcal {S}_V)$ to the local terms $\operatorname {\mathrm {loc}}_x(g,\mathcal {S}_V)$ . The latter have been computed by Theorem 5.1.3.

5.2 Strategy of proof

In this section, we reduce Theorem 5.1.3 to four auxiliary propositions stated below. The proofs of these propositions will occupy the remainder of this chapter.

As a preliminary observation, note that all of the objects appearing in Theorem 5.1.3 depend on G only through its base change to $\overline {F}$ , so we may enlarge F whenever convenient in the argument. In particular, we can and do assume that G admits a split reductive model $\mathcal {G} / \mathcal {O}_F$ , and that $\mathcal {G}(\mathcal {O}_F)$ contains elements of finite prime-to-p order with strongly regular semisimple image in $\mathcal {G}(\mathbf {F}_q)$ .

Now we begin the argument. First, we show that the local terms appearing in Theorem 5.1.3 are essentially independent of g.

Proposition 5.2.1. In the notation and setup of Theorem 5.1.3, $\mathrm {loc}_{x}(g,\mathcal {S}_{V})$ depends on g and x only through the cocharacter $\nu _{x}$ . More precisely, if $g,g'\in G(F)_{\mathrm {sr}}$ are two strongly regular semisimple elements and $x\in \mathrm {Gr}_{G}^{g}$ , resp., $x'\in \mathrm {Gr}_{G}^{g'}$ are fixed points such that $\nu _{x}=\nu _{x'}$ , then

$$\begin{align*}\mathrm{loc}_{x}(g,\mathcal{S}_{V})=\mathrm{loc}_{x'}(g',\mathcal{S}_{V}). \end{align*}$$

Next, we are going to degenerate from characteristic zero into characteristic p. For this, fix a split reductive model $\mathcal {G}/\mathcal {O}_{F}$ of G, and let $\mathrm {Gr}_{\mathcal {G}}$ be the associated Beilinson–Drinfeld affine Grassmannian over $S=\operatorname {\mathrm {Spd}} \mathcal {O}_C$ . Recall that this is a small v-sheaf which interpolates between the $B_{\mathrm {dR}}$ -affine Grassmannian $\mathrm {Gr}_{G}$ and the Witt vector affine Grassmannian $\mathrm {Gr}_{\mathcal {G}}^{W}$ , in the sense that we have a commutative diagram

with Cartesian squares. We will crucially use the fact that all of these Grassmannians satisfy compatible forms of geometric Satake, in the sense that there are natural monoidal functors

such that the vertical arrows are equivalences of categories on the essential images of $\mathrm {Rep}(\widehat {G})$ .

Proposition 5.2.2. Let $g\in \mathcal {G}(\mathcal {O}_{F})$ be an element such that $\overline {g}\in \mathcal {G}(\mathbf {F}_q)$ is strongly regular semisimple. Then $\mathcal {T}=\mathrm {Cent}(g,\mathcal {G})$ is a maximal torus, and there is a natural isomorphism

$$\begin{align*}\mathrm{Gr}_{\mathcal{G}}^{g}\cong\mathrm{Gr}_{\mathcal{T}}\cong X_{\ast}(\mathcal{T})_S. \end{align*}$$

In particular, if $\beta \simeq S\subset \mathrm {Gr}_{\mathcal {G}}^{g}$ is any connected component, then $\beta _{\eta }$ and $\beta _{s}$ are isolated fixed points for the g-action in the generic and special fiber, respectively.

Proposition 5.2.3. Let $g\in \mathcal {G}(\mathcal {O}_{F})$ be an element such that $\overline {g}\in \mathcal {G}(\mathbf {F}_q)$ is strongly regular semisimple. Then for any $V\in \mathrm {Rep}(\widehat {G})$ and any connected component $\beta \subset \mathrm {Gr}_{\mathcal {G}}^{g}$ , we have the equality

$$\begin{align*}\mathrm{loc}_{\beta_{\eta}}(g,j^{\ast}\mathcal{S}_{V})=\mathrm{loc}_{\beta_{s}}(g,i^{\ast}\mathcal{S}_{V}) \end{align*}$$

of local terms.

Finally, we compute the local terms on the Witt vector affine Grassmannian by a direct argument.

Proposition 5.2.4. Let $g\in \mathcal {G}(\mathcal {O}_{F})$ be an element with finite prime-to-p order such that $\overline {g}\in \mathcal {G}(\mathbf {F}_q)$ is strongly regular semisimple. Then for any $x\in \mathrm {Gr}_{\mathcal {G}}^{W,g}$ and any $V\in \mathrm {Rep}(\widehat {G})$ ,

$$\begin{align*}\mathrm{loc}_{x}(g,\mathcal{S}_{V})=(-1)^{\left\langle 2\rho,\nu_{x}\right\rangle }\mathrm{rank}_{\Lambda}V[\nu_{x}], \end{align*}$$

where $\nu _{x}\in X_{+}^{\ast }(\widehat {T})$ is as before.

5.3 Local terms and base change

In this section, we prove two key technical results, namely that formation of local terms commutes with any base change and with passage from perfect schemes to v-sheaves.

In order to fix notation, we briefly recall the key definitions concerning local terms; we apologize for the overlap with Chapter 4. Let S be a small v-sheaf, which will be our base object. Let $f:X\to S$ be a map of v-sheaves representable in nice diamonds. Consider a correspondence $c=(c_1,c_2)\colon C\to X\times _{S}X$ given by a map of v-sheaves representable in nice diamonds. This gives rise to a Cartesian diagram

of small v-sheaves. We will sometimes assume that $c_{1}$ is proper and that $\mathrm {Fix}(c)$ is a disjoint union of open-closed subspaces which are proper over S. These conditions will hold, e.g., if f and c are proper.

Let $\mathcal {F}\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ be an f-ULA object. Recall that a cohomological correspondence over c is a map $u:Rc_{2!}c_{1}^{\ast }\mathcal {F}\to \mathcal {F}$ , i.e., an element $u\in \mathrm {Hom}(c_{1}^{\ast }\mathcal {F},Rc_{2}^{!}\mathcal {F})$ . If $c_{1}$ is proper, then applying $Rf_{!}$ induces an endomorphism

$$ \begin{align*} Rf_{!}u:Rf_{!}\mathcal{F} & \to Rf_{!}Rc_{1\ast}c_{1}^{\ast}\mathcal{F}=Rf_{!}Rc_{1!}c_{1}^{\ast}\mathcal{F}\\ & \;\;\;\cong Rf_{!}Rc_{2!}c_{1}^{\ast}\mathcal{F}\overset{u}{\to}Rf_{!}\mathcal{F} \end{align*} $$

of $Rf_{!}\mathcal {F}$ . On the other hand, there is a natural map

$$\begin{align*}\mathrm{Hom}(c_{1}^{\ast}\mathcal{F},Rc_{2}^{!}\mathcal{F})\to H^{0}(\mathrm{Fix}(c),K_{\mathrm{Fix}(c)/S}), \end{align*}$$

cf. the discussion immediately before Definition 4.3.5, and we write $\mathrm {tr}_{c}(u,\mathcal {F})\in H^{0}(\mathrm {Fix}(c),K_{\mathrm {Fix}(c)/S})$ for the image of u under this map.

If $\beta \subset \mathrm {Fix}(c)$ is a closed-open subspace with proper structure map $g:\beta \to S$ , then $H^{0}(\beta ,K_{\beta /S})=H^{0}(\beta ,Rg^{!}\Lambda )$ is canonically a direct summand of $H^{0}(\mathrm {Fix}(c),K_{\mathrm {Fix}(c)/S})$ , and we can further consider the image of $\mathrm {tr}_{c}(u,\mathcal {F})$ under the map

$$ \begin{align*} H^{0}(\mathrm{Fix}(c),K_{\mathrm{Fix}(c)/S}) & \to H^{0}(\beta,Rg^{!}\Lambda)\cong H^{0}(S,Rg_{\ast}Rg^{!}\Lambda)\\ & \;\;\;\;\;\cong H^{0}(S,Rg_{!}Rg^{!}\Lambda)\to H^{0}(S,\Lambda). \end{align*} $$

By definition, this is the local term $\mathrm {loc}_{\beta }(u,\mathcal {F})$ . In most situations we care about, S is connected, so $H^{0}(S,\Lambda )=\Lambda $ , and we simply regard $\mathrm {loc}_{\beta }(u,\mathcal {F})$ as an element of $\Lambda $ . Note that local terms are additive in the sense that if $\beta =\beta _{1}\coprod \beta _{2}$ , then $\mathrm {loc}_{\beta }(u,\mathcal {F})=\mathrm {loc}_{\beta _{1}}(u,\mathcal {F})+\mathrm {loc}_{\beta _{2}}(u,\mathcal {F})$ . If S is a geometric point, f and c are proper, and $\pi _{0}(\mathrm {Fix}(c))$ is a discrete (and therefore finite) set, the usual Lefschetz trace formula holds, and says that

$$\begin{align*}\mathrm{tr}(u|R\Gamma_{c}(X,\mathcal{F}))=\sum_{\beta\in\pi_{0}(\mathrm{Fix}(c))}\mathrm{loc}_{\beta}(u,\mathcal{F}). \end{align*}$$

We need to understand how local terms interact with base change on S. More precisely, assume we are given a morphism $a:T\to S$ . Then all objects and morphisms above naturally base change to objects over T. Note that $\mathrm {Fix}(c)_{T}=\mathrm {Fix}(c_{T})$ . We write $a_{X}:X_{T}\to X$ , $a_{C}:C_{T}\to C$ , etc. for the base changes of a. We naturally get a cohomological correspondence $u_{T}$ on $\mathcal {F}_{T}=a_{X}^{\ast }\mathcal {F}$ over $c_{T}$ by taking the image of u under the map

$$ \begin{align*} \mathrm{Hom}(c_{1}^{\ast}\mathcal{F},Rc_{2}^{!}\mathcal{F}) & \to\mathrm{Hom}(a_{C}^{\ast}c_{1}^{\ast}\mathcal{F},a_{C}^{\ast}Rc_{2}^{!}\mathcal{F})\cong\mathrm{Hom}(c_{1,T}^{\ast}a_{X}^{\ast}\mathcal{F},a_{C}^{\ast}Rc_{2}^{!}\mathcal{F})\\ & \;\;\;\;\to\mathrm{Hom}(c_{1,T}^{\ast}a_{X}^{\ast}\mathcal{F},Rc_{2,T}^{!}a_{X}^{\ast}\mathcal{F}). \end{align*} $$

The final arrow here is induced by the canonical map $a_{C}^{\ast }Rc_{2}^{!}\mathcal {F}\to Rc_{2,T}^{!}a_{X}^{\ast }\mathcal {F}$ . This map is a special case of the natural transformation $\beta _{f,g}:\widetilde {f}^{\ast }Rg^{!}\to R\widetilde {g}^{!}f^{\ast }$ which exists for any Cartesian diagram

with g representable in nice diamonds. The transformation in question is adjoint to the map $R\widetilde {g}_{!}\widetilde {f}^{\ast }Rg^{!}\cong f^{\ast }Rg_{!}Rg^{!}\to f^{\ast }$ (it is also adjoint to the map $Rg^{!}\to Rg^{!}Rf_{\ast }f^{\ast }\cong R\widetilde {f}_{\ast }R\widetilde {g}^{!}f^{\ast }$ ).

In this setup, the next proposition says that formation of local terms commutes with base change along $T\to S$ .

Proposition 5.3.1. For any given $\beta \subset \mathrm {Fix}(c)$ as above, the natural map

$$\begin{align*}H^{0}(S,\Lambda)\to H^{0}(T,\Lambda) \end{align*}$$

sends $\mathrm {loc}_{\beta }(u,\mathcal {F})$ to $\mathrm {loc}_{\beta _{T}}(u_{T},\mathcal {F}_{T})$ . In particular, if S and T are connected, then $\mathrm {loc}_{\beta }(u,\mathcal {F})=\mathrm {loc}_{\beta _{T}}(u_{T},\mathcal {F}_{T})$ as elements of $\Lambda $ .

Proof. By a straightforward argument, this reduces to showing that there is a natural map

$$\begin{align*}H^{0}(\mathrm{Fix}(c),K_{\mathrm{Fix}(c)/S})\to H^{0}(\mathrm{Fix}(c)_{T},K_{\mathrm{Fix}(c)_{T}/T}) \end{align*}$$

compatible with the map $H^{0}(S,\Lambda )\to H^{0}(T,\Lambda )$ and sending $\mathrm {tr}_{c}(u,\mathcal {F})$ to $\mathrm {tr}_{c_{T}}(u_{T},\mathcal {F}_{T})$ . To obtain the map itself, apply $H^{0}(\mathrm {Fix}(c),-)$ to the composition

$$ \begin{align*} K_{\mathrm{Fix}(c)/S} & =R(f\circ c')^{!}\Lambda\to Ra_{\mathrm{Fix}(c)\ast}a_{\mathrm{Fix}(c)}^{\ast}R(f\circ c')^{!}\Lambda\\ & \;\overset{\beta_{a,f\circ c'}}{\longrightarrow}Ra_{\mathrm{Fix}(c)\ast}R(f_{T}\circ c^{\prime}_{T})^{!}a^{\ast}\Lambda=Ra_{\mathrm{Fix}(c)\ast}K_{\mathrm{Fix}(c)_{T}/T}. \end{align*} $$

The claim about the relation between $\mathrm {tr}_{c}$ and $\mathrm {tr}_{c_{T}}$ now follows from the fact that the base change functor $\mathrm {CoCorr}_{S}\to \mathrm {CoCorr}_{T}$ is symmetric monoidal and therefore preserves dualizable objects and traces of endomorphisms thereof.

We will also need to compare local terms associated with perfect schemes and with v-sheaves. More precisely, fix a perfect field $k/\mathbf {F}_{p}$ , and let $\mathrm {PSch}_{k}$ be the category of perfect schemes over k. There is a natural functor $X\mapsto X^{\lozenge }$ from $\mathrm {PSch}_{k}$ to small v-sheaves over $\mathrm {Spd}\,k$ , characterized by $(\mathrm {Spec}\,R)^{\lozenge }(A,A^{+})=\mathrm {Hom}_{k}(R,A)$ . Said differently, $X^{\lozenge }$ sends $\operatorname {\mathrm {Spec}} R$ to $\operatorname {\mathrm {Spa}}(R,R^{+})^{\lozenge }$ where $R^{+}$ is the integral closure of k in R. This functor commutes with finite limits. Moreover, if $f:X\to Y$ is separated and perfectly of finite type, then $f^{\lozenge }$ is representable in locally spatial diamonds and compactifiable with finite dim.trg. By [Sch17, §27], for any X there is a fully faithful symmetric monoidal functor $c_{X}^{\ast }:D_{\mathrm {\acute {e}t}}(X,\Lambda )\to D_{\mathrm {\acute {e}t}}(X^{\lozenge },\Lambda )$ compatible with $f^{\ast }$ and $Rf_{!}$ in the evident senses. Moreover, one has canonical natural transformations

$$\begin{align*}c_{X}^{\ast}R\mathscr{H}\mathrm{om}(-,-)\to R\mathscr{H}\mathrm{om}(c_{X}^{\ast}-,c_{X}^{\ast}-) \end{align*}$$

and $c_{X}^{\ast }Rf^{!}\to Rf^{\lozenge !}c_{Y}^{\ast }$ for f separated and perfectly of finite type.

Now let $\mathrm {PSch}_{k}^{\mathrm {ft}}$ be the full subcategory of schemes separated and perfectly of finite type over k. Fix $X\in \mathrm {PSch}_{k}^{\mathrm {ft}}$ with structure map $f:X\to \operatorname {\mathrm {Spec}} k$ , and let $c:C\to X\times _{k}X$ be a correspondence in $\mathrm {PSch}_{k}^{\mathrm {ft}}$ such that $c_{1}$ and $f\circ c'$ are perfectly proper. Let $\mathcal {F}\in D_{\mathrm {\acute {e}t}}(X,\Lambda )$ be an f-ULA object equipped with a cohomological correspondence u lying over c, so we get local terms $\mathrm {loc}_{\beta }(u,\mathcal {F})\in H^{0}(\mathrm {Spec}k,\Lambda )=\Lambda $ by the schematic version of the recipe recalled above.

On the other hand, applying $(-)^{\lozenge }$ and using commutation with finite limits, we get a correspondence $c^{\lozenge }:C^{\lozenge }\to X^{\lozenge }\times _{\mathrm {Spd}\,k}X^{\lozenge }$ of v-sheaves over $S=\mathrm {Spd}\,k$ with $\mathrm {Fix}(c)^{\lozenge }=\mathrm {Fix}(c^{\lozenge })$ , satisfying all of our assumptions from above. Moreover, u naturally induces a cohomological correspondence $u^{\lozenge }$ on $c_{X}^{\ast }\mathcal {F}$ lying over $c^{\lozenge }$ , by taking the image of u under the natural map

$$ \begin{align*} \mathrm{Hom}(c_{1}^{\ast}\mathcal{F},Rc_{2}^{!}\mathcal{F}) & \to\mathrm{Hom}(c_{C}^{\ast}c_{1}^{\ast}\mathcal{F},c_{C}^{\ast}Rc_{2}^{!}\mathcal{F})\cong\mathrm{Hom}(c_{1}^{\lozenge\ast}c_{X}^{\ast}\mathcal{F},c_{C}^{\ast}Rc_{2}^{!}\mathcal{F})\\ & \;\;\;\;\to\mathrm{Hom}(c_{1}^{\lozenge\ast}c_{X}^{\ast}\mathcal{F},Rc_{2}^{\lozenge!}c_{X}^{\ast}\mathcal{F}). \end{align*} $$

Proposition 5.3.2. Maintain the previous setup and notation. Then $c_{X}^{\ast }\mathcal {F}$ is $f^{\lozenge }$ -ULA, and for any open-closed $\beta \subset \mathrm {Fix}(c)$ , we have an equality

$$\begin{align*}\mathrm{loc}_{\beta}(u,\mathcal{F})=\mathrm{loc}_{\beta^{\lozenge}}(u^{\lozenge},c_{X}^{\ast}\mathcal{F}) \end{align*}$$

of local terms.

Proof. This is formally identical to the proof of Proposition 5.3.1, using the fact that $(-)^{\lozenge }$ induces a symmetric monoidal functor on the appropriate categories of cohomological correspondences.

5.4 Independence of g

In this section, we prove Proposition 5.2.1. In this section only, we set $S=\operatorname {\mathrm {Spd}} C$ .

Fix V as in the proposition. Decomposing V into isotypic summands for the action of $Z(\widehat {G})^{\circ }$ , we can assume that $\mathcal {S}_{V}$ is supported on a single connected component of $\mathrm {Gr}_{G}$ . We can then pick some $\mu $ such that $\mathcal {S}_{V}$ is supported on the Schubert cell $\mathrm {Gr}_{G,\leq \mu }$ . Choose some large m such that the $L^{+}G$ action on $\mathrm {Gr}_{G,\leq \mu }$ factors over the truncated loop group $L_{m}^{+}G$ . The sheaf $\mathcal {S}_{V}$ is naturally the pullback of a sheaf again denoted $\mathcal {S}_{V}$ on the quotient stack $X=[\mathrm {Gr}_{G,\leq \mu }/L_{m}^{+}G]$ , so we can consider the characteristic class $\mathrm {cc}_{X/S}(\mathcal {S}_{V})$ .

To analyze this class, we need to understand the inertia stack of X. For this, we need some notation. Let $L_{m}^{+}G_{\mathrm {sr}}$ be the preimage of the strongly regular semisimple locus $G_{\mathrm {sr}}\subset G$ under the theta map $L_{m}^{+}G\to G$ . Pick any maximal torus $T\subset G$ with Weyl group W, and set $L_{m}^{+}T_{\mathrm {sr}}=L_{m}^{+}T\cap L_{m}^{+}G_{\mathrm {sr}}$ .

Proposition 5.4.1. 1. The open substack

is canonically identified with $[L_{m}^{+}T_{\mathrm {sr}}/(W\ltimes L_{m}^{+}T)]$ via the natural map.

2. The open substack

is canonically identified with

such that the natural map $\mathrm {In}_{S}(X)_{\mathrm {sr}}\to \mathrm {In}_{S}([S/L_{m}^{+}G])_{\mathrm {sr}}$ coincides via the identification in part (1) with the evident projection onto $[L_{m}^{+}T_{\mathrm {sr}}/(W\ltimes L_{m}^{+}T)]$ .

Proof. The idea behind (1) is that any $g\in L^+G_{\operatorname {\mathrm {sr}}}$ is conjugate to an element of $L^+T_{\operatorname {\mathrm {sr}}}$ , which is well-defined up to the action of the normalizer of this group, which is $W\ltimes L^+T$ .

For (2), we observe that an object of $\operatorname {\mathrm {In}}_S(X)_{\operatorname {\mathrm {sr}}}$ is a pair $(x,g)$ , where $g\in L_m^+G_{\operatorname {\mathrm {sr}}}$ fixes $x\in \operatorname {\mathrm {Gr}}_{G,\leq \mu }$ ; the automorphisms of this object are $L^+_mG$ . The g can be conjugated to lie in $L^+T_{\operatorname {\mathrm {sr}}}$ , and then by Proposition 5.1.2, the x can be identified with an element of $X_*(T)_{\leq \mu }$ , which is well-defined up to an element of W.

Corollary 5.4.2. There is a natural isomorphism

$$\begin{align*}H^{0}(\mathrm{In}_{S}(X)_{\mathrm{sr}},K_{\mathrm{In}_{S}(X)_{\mathrm{sr}}/S})\cong C(X_{\ast}(T)_{\leq\mu},\Lambda)^{W} \end{align*}$$

which sends $\operatorname {\mathrm {cc}}_{X/S}(\mathcal {S}_V)$ to the function sending $\lambda \in X_*(T)_{\leq \mu }$ to $\operatorname {\mathrm {loc}}_{x_{\lambda }}(g,\mathcal {S}_V)$ , where $x_{\lambda }\in \operatorname {\mathrm {Gr}}_{G,\leq \mu }$ is the T-fixed point corresponding to $\lambda $ , and $g\in L_m^+T_{\operatorname {\mathrm {sr}}}$ . In particular, $\operatorname {\mathrm {loc}}_{x_{\lambda }}(g,\mathcal {S}_V)$ does not depend on the choice of $g\in L_m^+T_{\operatorname {\mathrm {sr}}}$ .

Proof. Combine Theorem 4.6.1 with the description of $\operatorname {\mathrm {In}}_S(X)_{\operatorname {\mathrm {sr}}}$ from Proposition 5.4.1.

5.5 Degeneration to characteristic p

In this section, we prove Propositions 5.2.2 and 5.2.3.

Proof of Proposition 5.2.2

The isomorphism $\mathrm {Gr}_{\mathcal {T}}\cong {X_{\ast }(\mathcal {T})}_S$ is [SW20, Proposition 21.3.1]. There is an evident map $f: \mathrm {Gr}_{\mathcal {T}}\to \mathrm {Gr}_{\mathcal {\mathcal {G}}}^{g}$ , and it remains to see that f is an isomorphism. For this, we first note that f is a closed immersion. This follows from the observation the source and target of f are both closed subfunctors of $\mathrm {Gr}_{\mathcal {G}}$ . For the source, this follows from [SW20, Proposition 20.3.7], while for the target this follows from the fact that $\mathrm {Gr}_{\mathcal {G}} \to S$ is separated.

Since f is a closed immersion, it is both qcqs and specializing. By [Sch17, Lemma 12.5], it is enough to check that f is a bijection on rank one geometric points. This can be checked separately on the generic and special fibers. Both cases are handled by Proposition 5.1.2.

Proof of Proposition 5.2.3

By two applications of (the connected case of) Proposition 5.3.1, applied to the maps $\eta \to S$ and $s \to S$ , we get equalities

$$\begin{align*}\mathrm{loc}_{\beta_{\eta}}(g,j^{\ast}\mathcal{S}_{V})= \mathrm{loc}_{\beta}(g,\mathcal{S}_{V}) = \mathrm{loc}_{\beta_{s}}(g,i^{\ast}\mathcal{S}_{V}), \end{align*}$$

and the result follows.

5.6 Local terms on the Witt vector affine Grassmannian

Proof of Proposition 5.2.4

Fix g and V as in the statement, and let $\mathcal {T}\subset \mathcal {G}$ be the connected centralizer of g. For every $\nu \in X_{\ast }(\mathcal {T})$ , let $S_{\nu }\subset \mathrm {Gr}_{\mathcal {G}}^{W}$ be the associated semi-infinite orbit, with closure $\overline {S_{\nu }}=\cup _{\nu '\leq \nu }S_{\nu }$ .

Let $X\subset \mathrm {Gr}_{\mathcal {G}}^{W}$ be a finite union of closed Schubert cells containing the support of $\mathcal {S}_{V}$ , so X is a perfectly projective k-scheme by the results in [BS17]. Write $X_{\nu }=X\cap S_{\nu }$ , $X_{\leq \nu }=X\cap \overline {S_{\nu }}$ , and $\partial X_{\leq \nu }=X_{\leq \nu }\smallsetminus X_{\nu }$ . Note that all of these spaces are stable under g and in fact under $\mathcal {T}$ . Note also that each $X_{\nu }$ contains a unique g-fixed point $x_{\nu }$ .

Proposition 5.6.1. The compactly supported Euler characteristic of $\mathcal {S}_V$ on $X_{\nu }$ is

$$\begin{align*}\chi_c(X_{\nu},\mathcal{S}_V) = (-1)^{\left\langle 2\rho,\nu\right\rangle }\operatorname{\mathrm{rank}} V[\nu]. \end{align*}$$

Proof. This is a consequence of the integral-coefficients version of the geometric Satake equivalence for the Witt vector affine Grassmannian given in [Yu19]. There it is shown (Proposition 4.2) that $H^d_c(X_{\nu }, \mathcal {S}_V)$ is zero unless $d=\left < 2\rho ,\nu \right>$ , and in that degree it corresponds exactly to the $\nu $ -weight functor in the Satake category.

Any $t\in \mathcal {T}$ must act trivially on $H^d_c(X_{\nu },\mathcal {S}_V)$ , so $\chi _{c}(U,\mathcal {S}_{V})$ coincides with the trace of g on $R\Gamma _{c}(X_{\nu },\mathcal {S}_{V})$ . The same is true for $X_{\leq \nu }$ and $\partial X_{\leq \nu }$ . We compute:

$$ \begin{align*} (-1)^{\left\langle 2\rho,\nu\right\rangle }\operatorname{\mathrm{rank}} V[\nu] & =\chi_{c}(X_{\nu},\mathcal{S}_{V})\\ & =\chi_{c}(X_{\leq\nu},\mathcal{S}_{V})-\chi_{c}(\partial X_{\leq\nu},\mathcal{S}_{V})\\ & =\mathrm{tr}(g|R\Gamma_{c}(X_{\leq\nu},\mathcal{S}_{V}))-\mathrm{tr}(g|R\Gamma_{c}(\partial X_{\leq\nu},\mathcal{S}_{V}))\\ & =\sum_{\nu'\leq\nu}\mathrm{loc}_{x_{\nu'}}(g,\mathcal{S}_{V}|_{X_{\leq\nu}})-\sum_{\nu'\leq\nu,\nu'\neq\nu}\mathrm{loc}_{x_{\nu'}}(g,\mathcal{S}_{V}|_{\partial X_{\leq\nu}})\\ & =\sum_{\nu'\leq\nu}\mathrm{loc}_{x_{\nu'}}(g,\mathcal{S}_{V})-\sum_{\nu'\leq\nu,\nu'\neq\nu}\mathrm{loc}_{x_{\nu'}}(g,\mathcal{S}_{V})\\ & =\mathrm{loc}_{x_{\nu}}(g,\mathcal{S}_{V}). \end{align*} $$

The penultimate equality is the key technical fact and follows from Proposition 5.6.2 below together with the assumptions on g.

Proposition 5.6.2. Let $k/\mathbf {F}_{p}$ be an algebraically closed field, and let X be a perfectly finite type k-scheme with an automorphism $g:X\to X$ of finite prime-to-p order. Let $A\in D_{c}^{b}(X,\mathbf {Z}_{\ell })$ be an object equipped with a morphism $u:g^{\ast }A\to A$ . Then for every isolated g-fixed point x, the true local term $\mathrm {loc}_{x}(g,A)$ equals the naive local term $\mathrm {tr}(g|A_{x})$ .

In particular, if $Z\subset X$ is a g-stable closed subscheme, then $\mathrm {loc}_{x}(g,A)=\mathrm {loc}_{x}(g,A|_{Z})$ .

Proof. With the word ‘perfectly’ deleted, this is a recent result of Varshavsky [Var20] (combine Theorem 4.10(b) and Corollary 5.4(b)). We will reduce to Varshavsky’s result by deperfecting.

Precisely, since $\mathrm {loc}_{x}(g,A)$ is insensitive to replacing $X,A$ by $U,A|U$ for $U\subset X$ any g-invariant open neighborhood of x, we can assume that X is affine, so $X=\operatorname {\mathrm {Spec}} R$ with R perfectly of finite type. Let $R_{0}\subset R$ be a finite type k-algebra with $R_{0}^{\mathrm {perf}}=R$ , and let $R_1\subset R$ be the k-algebra generated by $g^{i}R_{0}$ for all $1\leq i\leq \operatorname {\mathrm {ord}}(g)$ . Then $R_1\subset R$ is a finite-type k-algebra stable under g, with $R_1^{\mathrm {perf}}=A$ , so $X_{1}=\operatorname {\mathrm {Spec}} R_1$ is a deperfection of X equipped with an automorphism $g_{1}$ deperfecting g; since $X\to X_{1}$ is a homeomorphism, there is a unique $g_{1}$ -fixed point $x_{1}$ under x. Next, $g^{\ast }A \to A$ deperfects uniquely to a complex $A_{1}$ on $X_1$ equipped with a map $g_{1}^{\ast }A_{1}\to A_{1}$ , using the equivalence of categories $D(X_{\mathrm {et}},\Lambda )\cong D(X_{1,\mathrm {\acute {e}t}},\Lambda )$ . Finally, we compute that

$$\begin{align*}\mathrm{loc}_{x}(g,A)=\mathrm{loc}_{x_{1}}(g_{1},A_{1})=\mathrm{tr}(g_{1}|A_{1,x_{1}})=\mathrm{tr}(g|A_{x}), \end{align*}$$

where the first equality is formal nonsense (the six functors on k-varieties and on perfectly finite type k-schemes are compatible under perfection), the second equality is Varshavsky’s theorem and the third equality is trivial.

6 Application to the Hecke stacks

In this final, chapter we prove Theorem 1.0.2 by applying the technology of the Lefschetz–Verdier trace formula to the Hecke stacks over $\operatorname {\mathrm {Bun}}_G$ .

6.1 $\operatorname {\mathrm {Bun}}_G$ , the local and global Hecke stacks and their relation to shtuka spaces

Let $F/\mathbf {Q}_p$ be a finite extension, and let $G/F$ be a connected reductive group. Let k be an algebraically closed perfectoid field containing the residue field of F.

For an algebraically closed perfectoid field $C/k$ , there is a bijection [Far20]

$$\begin{align*}b \mapsto \mathcal{E}^b \end{align*}$$

between Kottwitz’ set $B(G)$ and isomorphism classes of G-bundles on the Fargues–Fontaine curve $X_C$ . Therefore, the moduli stack of G-bundles is some geometric version of the set $B(G)$ .

Definition 6.1.1 [FS21, Definition III.0.1 and Theorem III.0.2]

Let $\operatorname {\mathrm {Bun}}_G$ be the v-stack which assigns to a perfectoid space $S/k$ the groupoid of G-bundles on $X_S$ . Given a class $b\in B(G)$ , let $i_b\colon \operatorname {\mathrm {Bun}}_G^b\to \operatorname {\mathrm {Bun}}_G$ be the locally closed substack classifying G-bundles which are isomorphic to $\mathcal {E}^b$ at every geometric point.

Then $\operatorname {\mathrm {Bun}}_G$ is a cohomologically smooth Artin v-stack over $\operatorname {\mathrm {Spd}} k$ [FS21, Theorem I.4.1(vii)]. Central to its study are the substacks $\operatorname {\mathrm {Bun}}_G^b$ . For each $b\in B(G)$ , we have an isomorphism $\operatorname {\mathrm {Bun}}_G^b\cong [\operatorname {\mathrm {Spd}} k/\widetilde {G}_b]$ , where

$$\begin{align*}\widetilde{G}_b=\underline{\operatorname{\mathrm{Aut}}}\; \mathcal{E}_b\end{align*}$$

is a group diamond over $\operatorname {\mathrm {Spd}} k$ . This fits in an exact sequence of group diamonds over $\operatorname {\mathrm {Spd}} k$ :

$$\begin{align*}0 \to \widetilde{G}_b^{\circ} \to \widetilde{G}_b \to G_b(F)_{\operatorname{\mathrm{Spd}} k} \to 0\end{align*}$$

Here, the neutral component $\widetilde {G}_b^{\circ }\subset \widetilde {G}_b$ is a cohomologically smooth group diamond over $\operatorname {\mathrm {Spd}} k$ , and $G_b$ is the automorphism group of the isocrystal b. The group $G_b$ is an inner form of a Levi subgroup of the quasisplit inner form of G. If b is basic, then $i_b$ is an open immersion, and $\widetilde {G}_b=G_b(F)_{\operatorname {\mathrm {Spd}} k}$ .

We next recall the Hecke correspondence on $\operatorname {\mathrm {Bun}}_G$ and its relation to the local shtuka spaces $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ . Since our main result on the cohomology of local Shimura varieties does not concern the action of a Weil group, all objects in this discussion will live over the base $S=\operatorname {\mathrm {Spd}} C$ , where C is an algebraically closed perfectoid field containing F, whose residue field contains k. In particular, we have $\operatorname {\mathrm {Bun}}_{G,C}=\operatorname {\mathrm {Bun}}_G \times _{\operatorname {\mathrm {Spd}} k} S$ . If T is a perfectoid space over C, the Fargues–Fontaine curve $X_T$ comes equipped with a degree 1 Cartier divisor $D_T$ , corresponding to the untilt T of $T^{\flat }$ .

We introduce now a diagram of v-stacks over $\operatorname {\mathrm {Spd}} C$ containing both local and global Hecke correspondences:

(6.1.1)

We explain below the objects and morphisms appearing in equation (6.1.1). Let $T=\operatorname {\mathrm {Spa}}(R,R^+)$ be an affinoid perfectoid space over $\operatorname {\mathrm {Spa}} C$ .

• • The T-points of the stack $\operatorname {\mathrm {Hecke}}_{G,C}$ classify triples $(\mathcal {E}_1,\mathcal {E}_2,f)$ , where $\mathcal {E}_1$ and $\mathcal {E}_2$ are G-bundles on $X_T$ , and

  $$\begin{align*}f\colon \mathcal{E}_1\vert_{X_T\backslash D_T} \cong \mathcal{E}_2\vert_{X_T\backslash D_T} \end{align*}$$
  is an isomorphism which is meromorphic along $D_T$ .

• • The morphism $h_i$ sends a triple as above to $\mathcal {E}_i$ for $i=1,2$ .

• • The T-points of $\operatorname {\mathrm {Bun}}_{G,C}^{\operatorname {\mathrm {loc}}}$ classify G-bundles on $\operatorname {\mathrm {Spec}} B_{\operatorname {\mathrm {dR}}}^+(R)$ , this being the completion of $X_T$ along $D_T$ . Such G-bundles are v-locally trivial on T so that we have an isomorphism

  $$\begin{align*}\operatorname{\mathrm{Bun}}_{G,C}^{\operatorname{\mathrm{loc}}} \cong [\operatorname{\mathrm{Spd}} C / L^+G], \end{align*}$$
  where $L^+G = G(B_{\operatorname {\mathrm {dR}}}^+)$ is the positive loop group.

• • The T-points of $\operatorname {\mathrm {Hecke}}^{\operatorname {\mathrm {loc}}}_{G,C}$ classify triples $(\mathcal {E}_1,\mathcal {E}_2,f)$ , where $\mathcal {E}_1$ and $\mathcal {E}_2$ are G-bundles on $\operatorname {\mathrm {Spec}} B_{\operatorname {\mathrm {dR}}}^+(R)$ , and f is an isomorphism between their restrictions to $\operatorname {\mathrm {Spec}} B_{\operatorname {\mathrm {dR}}}(R)$ , meromorphic along $D_T$ . We have an isomorphism

  $$\begin{align*}\operatorname{\mathrm{Hecke}}_{G,C}^{\operatorname{\mathrm{loc}}} \cong [L^+G \backslash LG / L^+G ],\end{align*}$$
  where $LG=G(B_{\operatorname {\mathrm {dR}}})$ is the full loop group. Put another way, we have the $B_{\operatorname {\mathrm {dR}}}$ -affine Grassmannian $\operatorname {\mathrm {Gr}}_{G,C}=LG/L^+G$ and then $\operatorname {\mathrm {Hecke}}_{G,C}^{\operatorname {\mathrm {loc}}}=[L^+G \backslash \operatorname {\mathrm {Gr}}_{G,C}]$ .

• • The morphism $h_i^{\operatorname {\mathrm {loc}}}$ sends such a triple to $\mathcal {E}_i$ for $i=1,2$ .

• • The vertical maps send an object to its completion along $D_T$ in the evident manner.

The squares in equation (6.1.1) are Cartesian by Beauville–Laszlo gluing.

It is a basic fact that $\operatorname {\mathrm {Bun}}_{G,C}$ is a decent v-stack, and the structure map $\operatorname {\mathrm {Bun}}_{G,C} \to S=\operatorname {\mathrm {Spd}} C$ is fine. With some care, it is possible to ‘truncate’ some of the other objects appearing in equation (6.1.1) to obtain decent S-v-stacks with fine structure maps to S. In particular, let $\mu $ be a dominant cocharacter of G, and let $\operatorname {\mathrm {Hecke}}_{G,\leq \mu ,C}$ be the substack of $\operatorname {\mathrm {Hecke}}_{G,\mu }$ consisting of triples $(\mathcal {E}_1,\mathcal {E}_2,f)$ , where the meromorphy of f is fiberwise bounded by $\mu $ . Then $\operatorname {\mathrm {Hecke}}_{G,\leq \mu ,C}$ is decent, and the maps to $\operatorname {\mathrm {Bun}}_{G,C}$ induced by restricting $h_1$ and $h_2$ are fine. We may define $\operatorname {\mathrm {Hecke}}_{G,\leq \mu ,C}^{\operatorname {\mathrm {loc}}}$ analogously; this is isomorphic to $[\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}/L^+G]$ , where $\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}$ is the bounded Grassmannian. This is not quite a decent v-stack. However, if we instead form the quotient $[\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}/L^{+}_{m}G]$ for some sufficiently large truncation as in Theorem 5.1.4, we do obtain a decent S-v-stack with fine structure map. This is sufficient for our purposes.

Now let $b\in B(G,\mu )$ be basic. We explain the relation between Hecke stacks and local shtuka spaces. It will be helpful to refer to the commutative diagram of stacks

(6.1.2)

in which all squares are Cartesian, the morphisms labeled with i are open immersions and the morphisms $h_1$ and $h_2$ are proper.

The top row of equation (6.1.2) can be described via the diagram:

Explanation: $\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}^1$ assigns to $T=\operatorname {\mathrm {Spa}}(R,R^+)$ the set of pairs $(\mathcal {E},f)$ , where $\mathcal {E}$ is a G-bundle on $X_S$ , and $f\colon \mathcal {E}^1\vert _{X_T\backslash D_T}\cong \mathcal {E}\vert _{X_T\backslash D_T}$ is an isomorphism, which is bounded by $\mu $ along $D_T$ . The bundle $\mathcal {E}^1$ can be canonically trivialized over $\operatorname {\mathrm {Spa}} B_{\operatorname {\mathrm {dR}}}^+(R)$ , and in so doing, we obtain an isomorphism $\operatorname {\mathrm {Gr}}^1_{G,\leq \mu ,C}\cong \operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}$ . Within $\operatorname {\mathrm {Gr}}_{G,\leq \mu }^1$ , we have the open locus $\operatorname {\mathrm {Gr}}_{G,\leq \mu }^{1,\operatorname {\mathrm {adm}}}$ , consisting of those pairs $(\mathcal {E},\gamma )$ , where $\mathcal {E}$ is everywhere isomorphic to $\mathcal {E}^b$ .

Similarly, the leftmost column of equation (6.1.2) can be described via the diagram:

Explanation: $\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}^b$ assigns to $T=\operatorname {\mathrm {Spa}}(R,R^+)$ the set of pairs $(\mathcal {E},f)$ , where $\mathcal {E}$ is a G-bundle, and $f\colon \mathcal {E}\vert _{X_T\backslash D_T}\cong \mathcal {E}^b\vert _{X_T\backslash D_T}$ is an isomorphism, which is bounded by $\mu $ along $D_T$ . We have an isomorphism $\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}^b\cong \operatorname {\mathrm {Gr}}_{G,\leq -\mu ,C}$ . Within $\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}^b$ , we have the open admissible locus $\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}^{b,\operatorname {\mathrm {adm}}}$ consisting of pairs $(\mathcal {E},f)$ , where $\mathcal {E}$ is everywhere isomorphic to $\mathcal {E}^1$ .

The moduli space of local shtukas $\operatorname {\mathrm {Sht}}_{G,b,\mu ,C}$ appears as the fiber product:

(6.1.3)

where the right vertical morphism corresponds to $\mathcal {E}^b\times \mathcal {E}^1$ . This is evident from the definition of $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ : Its S-points are morphisms $f\colon \mathcal {E}^1_{X_S\backslash D_S}\cong \mathcal {E}^b_{X_S\backslash D_S}$ which are bounded by $\mu $ on $D_S$ .

We also have the period morphisms:

(6.1.4)

The morphism $\pi _1$ is a $G_b(F)_S$ -equivariant $G(F)_S$ -torsor over the admissible locus $\operatorname {\mathrm {Gr}}^{b,\operatorname {\mathrm {adm}}}_{G,\leq \mu ,C}$ . Similarly, $\pi _2$ is a $G(F)_S$ -equivariant $G_b(F)_S$ -torsor over the admissible locus $\operatorname {\mathrm {Gr}}^{1,\operatorname {\mathrm {adm}}}_{G,\leq \mu ,C}$ .

6.2 The inertia stack of the Hecke stack; admissibility of elliptic fixed points

We continue to put $S=\operatorname {\mathrm {Spd}} C$ . Here, we investigate the inertia stack $\operatorname {\mathrm {In}}_S(\operatorname {\mathrm {Hecke}}_{G,\leq \mu ,C})$ , or at least the part of it lying over the strongly regular locus in .

It will help to introduce some notation. Suppose $\mathcal {E}$ is a G-bundle on $X_C$ equipped with a trivialization over the completion at $\infty =D_C$ . Let $T\subset G$ be a maximal torus. We have seen in Proposition 5.1.2 that there is a bijection $\lambda \mapsto L_{\lambda }$ between $X_*(T)$ and the set of T-fixed points of $\operatorname {\mathrm {Gr}}_{G}$ . Given $\lambda \in X_*(T)$ , we let $\mathcal {E}[\lambda ]$ be the modification of $\mathcal {E}$ corresponding to $L_{\lambda }$ .

Lemma 6.2.1 [CS17, Lemma 3.5.5], see also [CFS21, §2.2], but note that we use the opposite convention concerning Schubert cells

Let $\mathcal {E}$ be a G-bundle on $X_{C}$ equipped with a trivialization at $\infty $ , let $T\subset G$ be a maximal torus, let $\lambda \in X_*(T)$ be a cocharacter and let $\widehat {\lambda }\in X^*(\widehat {T})$ be the corresponding character. In the group $X^*(Z(\widehat {G})^{\Gamma })$ , we have

$$\begin{align*}\kappa(\mathcal{E}[\lambda])=\kappa(\mathcal{E})+\widehat{\lambda}\vert_{Z(\widehat{G})^{\Gamma}}.\end{align*}$$

Proposition 6.2.2. Suppose a pair $(g,g')\in G(F)_{\operatorname {\mathrm {sr}}}\times G_b(F)_{\operatorname {\mathrm {sr}}}$ fixes a point $x\in \operatorname {\mathrm {Sht}}_{G,b, \mu }(C)$ . Let $T=\operatorname {\mathrm {Cent}}(g,G)$ and $T'=\operatorname {\mathrm {Cent}}(g',G_b)$ . Then $\pi _1(x)\in \operatorname {\mathrm {Gr}}_{G,\leq -\mu }^{g'}$ and $\pi _2(x)\in \operatorname {\mathrm {Gr}}_{G,\leq \mu }^g$ correspond to cocharacters $\lambda '\in X_*(T')_{\leq -\mu }$ and $\lambda \in X_*(T)_{\leq \mu }$ , respectively.

There exists $y\in G(\breve {F})$ such that $\operatorname {\mathrm {ad}} y$ is an F-rational isomorphism $T\to T'$ , which carries g to $g'$ and $\lambda $ onto $-\lambda '$ . The invariant $\operatorname {\mathrm {inv}}[b](g,g')\in B(T)\cong X_*(T)_{\Gamma }$ agrees with the image of $\lambda $ under $X_*(T)\to X_*(T)_{\Gamma }$ . Therefore, $(g,g',\lambda )$ lies in $\operatorname {\mathrm {Rel}}_{b,\mu }$ .

Proof. The point x corresponds to an isomorphism $\gamma \colon \mathcal {E}^1[\lambda ]\to \mathcal {E}^b$ and also to an isomorphism $\gamma '\colon \mathcal {E}^1\to \mathcal {E}^b[\lambda ']$ . Each of these interlaces the action of g with $g'$ and furthermore $\gamma =\gamma '$ away from $\infty $ . Trivializing $\mathcal {E}^1$ and $\mathcal {E}^b$ away from $\infty $ , we see that g and $g'$ become conjugate over the ring $B_e=H^0(X_C\backslash \left \{ \infty \right \},\mathcal {O}_{X_C})$ , which implies they are conjugate over $\overline {F}$ and (by Lemma 3.2.1) they are even conjugate over $\breve {F}$ . Let $y\in G(\breve {F})$ be an element such that $(\operatorname {\mathrm {ad}} y)(g) = g'$ . Then $\operatorname {\mathrm {ad}} y$ is a $\breve {F}$ -rational isomorphism $T\to T'$ which carries g onto $g'$ . In fact, since there is only one such isomorphism, we can conclude that $\operatorname {\mathrm {ad}} y$ is an F-rational isomorphism $T\to T'$ . Let $\lambda _0=(\operatorname {\mathrm {ad}} y^{-1})(\lambda ')\in X_*(T)$ .

Let $b_0=y^{-1}by^{\sigma }$ . Then (cf. Definition 3.2.2) we have $b_0\in T(F)$ . The element y induces isomorphisms $y\colon \mathcal {E}^{b_0}\to \mathcal {E}^b$ and $y\colon \mathcal {E}^{b_0}[\lambda _0]\to \mathcal {E}^b[\lambda ']$ . Then the isomorphism $y^{-1}\gamma '\colon \mathcal {E}^1\to \mathcal {E}^{b_0}[\lambda _0]$ descends to an isomorphism of T-bundles; comparing this with the isomorphism $\gamma y^{-1}\colon \mathcal {E}^1[\lambda ]\to \mathcal {E}^{b_0}$ shows that $\lambda _0=-\lambda $ . In light of the isomorphism of T-bundles $\mathcal {E}^1[\lambda ]\cong \mathcal {E}^{b_0}$ , Lemma 6.2.1 implies that the identity $\kappa (\mathcal {E}^{b_0})=\lambda $ holds in $B(T)$ . But also $\kappa (\mathcal {E}^{b_0})$ is the class of $[b_0]$ in $B(T)$ , which is $\operatorname {\mathrm {inv}}[b](g,g')$ by definition.

Proposition 6.2.2 shows that if $(g,g')\in G(F)_{\operatorname {\mathrm {sr}}}\times G_b(F)_{\operatorname {\mathrm {sr}}}$ fixes a point of $\operatorname {\mathrm {Sht}}_{G,b,\mu }$ , then g and $g'$ are related. However the converse may fail: If a pair of related strongly regular elements $(g,g')$ is given, it is not necessarily true that $(g,g')$ fixes a point of $\operatorname {\mathrm {Sht}}_{G,b, \mu }$ . Indeed, a necessary condition for this is that the action of $g'$ on $\operatorname {\mathrm {Gr}}_{G,\leq -\mu }^b$ has a fixed point in the admissible locus, and this is not automatic.

This converse result is always true, however, if g (or equivalently, $g'$ ) is an elliptic element.

Theorem 6.2.3. Let $g\in G(F)_{\operatorname {\mathrm {ell}}}$ . Then the fixed points of g acting on $\operatorname {\mathrm {Gr}}_{G,\leq \mu }^1$ lie in the admissible locus $\operatorname {\mathrm {Gr}}_{G,\leq \mu }^{1,\operatorname {\mathrm {adm}}}$ . Similarly, if $g'\in G_b(F)_{\operatorname {\mathrm {ell}}}$ , then the fixed points of $g'$ acting on $\operatorname {\mathrm {Gr}}_{G,\leq -\mu }^b$ lie in the admissible locus $\operatorname {\mathrm {Gr}}_{G,\leq -\mu }^{b,\operatorname {\mathrm {adm}}}$ .

Proof. We prove the first statement; the second is similar. Let $g\in G(F)_{\operatorname {\mathrm {ell}}}$ , and let $T=\operatorname {\mathrm {Cent}}(g,G)$ be the elliptic maximal torus containing g. Suppose we are given a g-fixed point $x\in \operatorname {\mathrm {Gr}}_{G,\leq \mu }(C)$ . Then x corresponds to a cocharacter $\lambda \in X_*(T)$ , which in turn corresponds to a modification $\mathcal {E}^1[\lambda ]$ of the trivial G-bundle $\mathcal {E}^1$ . We wish to show that $\mathcal {E}^1[\lambda ]\cong \mathcal {E}^b$ . First, we will show that it is semistable.

Let $b'\in G(\breve {F})$ be an element whose class in $B(G)$ corresponds to the isomorphism class of $\mathcal {E}^1[\lambda ]$ . We wish to show that $b'$ is basic. We have the algebraic group $G_{b'}/F$ , which is a priori an inner form of a Levi subgroup $M^*$ of $G^*$ , where $G^*$ is the quasi-split inner form of G. Showing that $b'$ is basic is equivalent to showing that $M^*=G^*$ .

We have an isomorphism $\gamma \colon \mathcal {E}^1[\lambda ]\cong \mathcal {E}^{b'}$ . The action of $g\in T(F)$ on $\mathcal {E}^1$ extends to an action on $\mathcal {E}^1[\lambda ]$ , which can be transported via $\gamma $ to obtain an automorphism $g'\in \widetilde {G}_{b'}(C)=\operatorname {\mathrm {Aut}} \mathcal {E}_{b'}$ . Let $\overline {g}'$ be the image of $g'$ under the projection $\widetilde {G}_{b'}(C)\to G_{b'}(F)$ .

The G-bundles $\mathcal {E}^1$ and $\mathcal {E}^{b'}$ may be trivialized over $\operatorname {\mathrm {Spec}} B_{\operatorname {\mathrm {dR}}}^+(C)$ . In doing so, we obtain embeddings of $G(F)=\operatorname {\mathrm {Aut}} \mathcal {E}^1$ and $\widetilde {G}_{b'}(C)=\operatorname {\mathrm {Aut}} \mathcal {E}^{b'}$ into $G(B_{\operatorname {\mathrm {dR}}}^+(C))$ ; we denote both of these by $h\mapsto h_{\infty }$ . We also have the isomorphism $\gamma _{\infty }$ between $\mathcal {E}^1$ and $\mathcal {E}^{b'}$ over $\operatorname {\mathrm {Spec}} B_{\operatorname {\mathrm {dR}}}(C)$ ; we may identify $\gamma _{\infty }$ with an element of $G(B_{\operatorname {\mathrm {dR}}}(C))$ , and then $g^{\prime }_{\infty } =\gamma _{\infty } g_{\infty } \gamma _{\infty }^{-1}$ holds in $G(B_{\operatorname {\mathrm {dR}}}(C))$ .

The element $\bar g^{\prime }_{\infty }$ is conjugate to $g^{\prime }_{\infty }$ , so $\bar g^{\prime }_{\infty }$ is conjugate to $g_{\infty }$ in $G(B_{\operatorname {\mathrm {dR}}})$ . Since g and $\overline {g}'$ are both regular semisimple $\bar F$ -points of G, being conjugate in $G(B_{\operatorname {\mathrm {dR}}})$ is the same as being conjugate in $G(\bar F)$ . Their centralizers, being F-rational tori, are thus isomorphic over F. Thus, $G_{b'}$ contains a maximal torus that is elliptic for G. Elliptic maximal tori transfer across inner forms [Kot86, §10], which means that the Levi subgroup $M^* \subset G^*$ of which $G_{b'}$ is an inner form contains a maximal torus that is elliptic for $G^*$ . Therefore, $M^*=G^*$ .

We have shown that $\mathcal {E}^1[\lambda ]\cong \mathcal {E}_{b'}$ is semistable, implying that $\operatorname {\mathrm {Aut}} \mathcal {E}^{b'}=G_{b'}(F)$ and that $g'\in G_{b'}(F)$ . Lemma 6.2.1 shows that $\kappa ([b'])$ equals the image of $\lambda $ in $\pi _1(G)_{\Gamma }$ ; this is the same as the image of $\mu $ , which in turn is the same as $\kappa ([b])$ because $b\in B(G,\mu )$ . Since $b'$ is basic, we have $[b']=[b]$ by [Kot85, Proposition 5.6].

Recall the locally profinite set $\operatorname {\mathrm {Rel}}_{b,\leq \mu }$ from Definition 3.2.4. This is the set of conjugacy classes of triples $(g,g',\lambda )$ , where $g\in G(F)$ and $g'\in G_b(F)$ are related strongly regular elements, and $\lambda $ is a cocharacter of $T=\operatorname {\mathrm {Cent}}(g,G)$ , bounded by $\mu $ such that $\kappa (\operatorname {\mathrm {inv}}[b](g,g'))$ agrees with the image of $\lambda $ in $X_*(T)_{\Gamma }$ . Let $\operatorname {\mathrm {Rel}}_{b,\leq \mu ,\operatorname {\mathrm {ell}}}$ be the subset, where g (equivalently, $g'$ ) is elliptic.

Theorem 6.2.3 has the following corollary. For a v-stack X, we write $\left \lvert X \right \rvert $ for the underlying topological space.

Corollary 6.2.4. Let $\operatorname {\mathrm {In}}_S(\operatorname {\mathrm {Hecke}}^{b,\ast }_{G,\leq \mu ,S})_{\operatorname {\mathrm {ell}}}$ be the preimage under $\operatorname {\mathrm {In}}_S(h_1)$ of $\operatorname {\mathrm {In}}_S(\operatorname {\mathrm {Bun}}^b_{G,S})_{\operatorname {\mathrm {ell}}}$ . Similarly let $\operatorname {\mathrm {In}}_S(\operatorname {\mathrm {Hecke}}^{\ast ,1}_{G,\leq \mu ,S})_{\operatorname {\mathrm {ell}}}$ be the preimage under $\operatorname {\mathrm {In}}_S(h_2)$ of $\operatorname {\mathrm {In}}_S(\operatorname {\mathrm {Bun}}^1_{G,S})_{\operatorname {\mathrm {ell}}}$ . Then

$$\begin{align*}\operatorname{\mathrm{In}}_S(\operatorname{\mathrm{Hecke}}^{b,\ast}_{G,\leq\mu,S})_{\operatorname{\mathrm{ell}}}= \operatorname{\mathrm{In}}_S(\operatorname{\mathrm{Hecke}}^{\ast,1}_{G,\leq\mu,S})_{\operatorname{\mathrm{ell}}}= \operatorname{\mathrm{In}}_S(\operatorname{\mathrm{Hecke}}^{b,1}_{G,\leq\mu,S})_{\operatorname{\mathrm{ell}}}. \end{align*}$$

There is a homeomorphism $\left \lvert \operatorname {\mathrm {In}}_S(\operatorname {\mathrm {Hecke}}_{G,b,\leq \mu ,S}^{b,1})_{\operatorname {\mathrm {ell}}} \right \rvert \cong \operatorname {\mathrm {Rel}}_{b,\mu ,\operatorname {\mathrm {ell}}}$ .

Proof. The first claim is just the statement that fixed points of elliptic elements on $\operatorname {\mathrm {Gr}}^b_G$ and $\operatorname {\mathrm {Gr}}^1_G$ are admissible. For the second claim: Since $\operatorname {\mathrm {Hecke}}^{\ast ,1}_{G,\leq \mu ,S} \cong [\operatorname {\mathrm {Gr}}_{G,\leq \mu ,S}/G(F)_S]$ , we can think of $\left \lvert \operatorname {\mathrm {In}}_S(\operatorname {\mathrm {Hecke}}_{G,b,\leq \mu }^{b,1})_{\operatorname {\mathrm {ell}}} \right \rvert $ as the set of conjugacy classes of pairs $(g,\lambda )$ , where $g\in G(F)_{\operatorname {\mathrm {ell}}}$ and $\lambda \in X_*(T)_{\leq \mu }$ , where $T=\operatorname {\mathrm {Cent}}(g,G)$ . We have an isomorphism $\mathcal {E}^1[\lambda ]\cong \mathcal {E}^b$ . The element $g\in G(F)\cong \operatorname {\mathrm {Aut}} \mathcal {E}^1$ determines an element $g'\in G_b(F)\cong \operatorname {\mathrm {Aut}}\mathcal {E}^b$ , up to conjugacy. By Proposition 6.2.2, the triple $(g,g',\lambda )$ determines an element of $\operatorname {\mathrm {Rel}}_{b,\mu ,\operatorname {\mathrm {ell}}}$ . Conversely, given such a triple $(g,g',\lambda )$ , the pair $(g,\lambda )$ determines an element $g''\in G_b(F)$ as we have just argued, but then $g'$ and $g''$ are conjugate by Remark 3.2.5.

6.3 Transfer of distributions from $G_b$ to G

We continue to let b be a basic element of $B(G)$ . Let $\Lambda $ be a ring in which p is invertible. Recall the Hecke transfer map

from 3.2.7. As promised, we can now promote this to a transfer of distributions, at least after restriction to elliptic loci (and assuming, as we have been doing all along, that the $\Lambda $ -valued Haar measures on $G(F)$ and $G_b(F)$ are chosen compatibly).

Recall the period morphisms:

$$\begin{align*}\operatorname{\mathrm{Gr}}^b_{G,\leq \mu,C} \stackrel{\pi_1}{\leftarrow} \operatorname{\mathrm{Sht}}_{G,b,\mu,C} \stackrel{\pi_2}{\to} \operatorname{\mathrm{Gr}}^1_{G,\leq\mu,C}, \end{align*}$$

in which $\pi _1$ is a $G(F)_S$ -torsor over its image, and $\pi _2$ is a $G_b(F)_S$ -torsor over its image. Consider the action map on $\operatorname {\mathrm {Sht}}_{G,b,\mu ,C}$ :

$$\begin{align*}\alpha_{\operatorname{\mathrm{Sht}}}\colon G(F)_S \times G_b(F)_S \times \operatorname{\mathrm{Sht}}_{G,b,\mu,C} \to \operatorname{\mathrm{Sht}}_{G,b,\mu,C} \end{align*}$$

and also those on the period domains:

$$ \begin{align*} \alpha_1\colon G(F)_S\times \operatorname{\mathrm{Gr}}^1_{G,\leq\mu,C} &\to \operatorname{\mathrm{Gr}}^1_{G,\leq\mu,C} \\ \alpha_b\colon G_b(F)_S\times \operatorname{\mathrm{Gr}}^b_{G,\leq\mu,C} &\to \operatorname{\mathrm{Gr}}^b_{G,\leq\mu,C}. \end{align*} $$

For $?\in \left \{ \operatorname {\mathrm {Sht}},1,b \right \}$ we can define the elliptic fixed-point locus $\operatorname {\mathrm {Fix}}(\alpha _?)_{\operatorname {\mathrm {ell}}}$ of the corresponding action map, consisting of pairs $(g,x)$ with g elliptic and $g.x=x$ ; let us think of each $\operatorname {\mathrm {Fix}}(\alpha _?)$ as a locally profinite set. For instance, $\operatorname {\mathrm {Fix}}(\alpha _1)_{\operatorname {\mathrm {ell}}}$ is the set of pairs $(g,\lambda )$ , where $g\in G(F)_{\operatorname {\mathrm {ell}}}$ , and $\lambda \in X_*(T_g)$ ( $T_g=\operatorname {\mathrm {Cent}}(g,G)$ ) is bounded by $\mu $ . These fit into a diagram

(6.3.1)

of locally profinite sets, in which $p_1$ is a $G_b(F)$ -equivariant $G(F)$ -torsor, $p_2$ is a $G(F)$ -equivariant $G_b(F)$ -torsor and $q_1$ and $q_2$ are finite étale. (The maps $p_i$ are surjective by Theorem 6.2.3.) Furthermore, let us observe that, for $(g,g',x)\in \operatorname {\mathrm {Fix}}(\alpha _{\operatorname {\mathrm {Sht}}})$ , the image of x in $\operatorname {\mathrm {Gr}}_G^1(C)^g$ may be identified with a cocharacter $\lambda \in X_*(T)$ of $T=\operatorname {\mathrm {Cent}}(g,G)$ , and then the triple $(g,g',\lambda )$ lies in $\operatorname {\mathrm {Rel}}_{b,\mu }$ by Proposition 6.2.2. A key observation is that we have a diagram of stacks in locally profinite sets, in which both squares are Cartesian:

(6.3.2)

Thus, at least over the elliptic locus, we have promoted a correspondence between sets of conjugacy classes to a correspondence between stacks of conjugacy classes. Formally, this is exactly what is required to promote our transfer of functions to a transfer of distributions.

Lemma 6.3.1. Let H be a locally pro-p group, and let $\Lambda $ be a commutative ring in which p is invertible. Choose a $\Lambda $ -valued Haar measure on H. Let $h\colon \widetilde {T}\to T$ be an H-torsor in locally profinite sets. The integration-along-fibers map $C_c(\widetilde {T},\Lambda ) \to C_c(T,\Lambda )$ induces an isomorphism of $C(T,\Lambda )$ -modules

$$\begin{align*}h_*\colon C_c(\widetilde{T},\Lambda)_H\to C_c(T,\Lambda) \end{align*}$$

and, dually, an isomorphism of $C(T,\Lambda )$ -modules

$$\begin{align*}h_*\colon \operatorname{\mathrm{Dist}}(\widetilde{T},\Lambda)^H \to \operatorname{\mathrm{Dist}}(T,\Lambda). \end{align*}$$

Proof. The $C(T,\Lambda )$ -modules $C_c(T,\Lambda )$ and $C_c(\widetilde T,\Lambda )$ are smooth in the sense of Definition B.2.1. Therefore, by Lemma B.2.5 the statement is local on T, so we may assume that the torsor $\widetilde {T}=T\times H$ is split. Then $C_c(\widetilde T,\Lambda )_H=C_c(T,\Lambda ) \otimes _{\Lambda } C_c(H,\Lambda )_H$ . The integration map $C_c(H,\Lambda )_H \to \Lambda $ is an isomorphism so that $C_c(\widetilde {T},\Lambda )_H\cong C_c(T,\Lambda )$ .

Recall from §3.4 that we have chosen compatible Haar measures on $G(F)$ and $G_b(F)$ .

Definition 6.3.2. With notation as in equation (6.3.1), we define a $\Lambda $ -linear map

(6.3.3) $$ \begin{align} \widetilde{T}_{b,\mu}^{G\to G_b} \colon C_c(G(F)_{\operatorname{\mathrm{ell}}},\Lambda)_{G(F)} \to C_c(G_b(F)_{\operatorname{\mathrm{ell}}},\Lambda)_{G_b(F)} \end{align} $$

as the composition

$$ \begin{align*} \begin{array}{lcl} C_c(G(F)_{\operatorname{\mathrm{ell}}},\Lambda)_{G(F)} &\stackrel{q_2^*}{\to} &\!\! C_c(\operatorname{\mathrm{Fix}}(\alpha_1)_{\operatorname{\mathrm{ell}}},\Lambda)_{G(F)} \\[2pt] &\stackrel{(p_2)_*^{-1}}{\stackrel{\cong}{\longrightarrow}} &\!\! C_c(\operatorname{\mathrm{Fix}}(\alpha_{\operatorname{\mathrm{Sht}}})_{\operatorname{\mathrm{ell}}},\Lambda)_{G(F)\times G_b(F)} \\[2pt] &\stackrel{(p_1)_*}{\stackrel{\cong}{\longrightarrow}} &\!\! C_c(\operatorname{\mathrm{Fix}}(\alpha_b)_{\operatorname{\mathrm{ell}}},\Lambda)_{G_b(F)} \\[2pt] &\stackrel{\cdot K_{\mu}}{\to} &\!\! C_c(\operatorname{\mathrm{Fix}}(\alpha_b)_{\operatorname{\mathrm{ell}}},\Lambda)_{G_b(F)} \\[2pt] &\stackrel{q_{1*}}{\to} &\!\! C_c(G_b(F)_{\operatorname{\mathrm{ell}}},\Lambda)_{G_b(F)}, \end{array} \end{align*} $$

where $q_2^*$ means pullback, $q_{1*}$ means pushforward (i.e., sum over fibers), the isomorphisms $(p_i)_*$ are induced by our choices of Haar measures as in Lemma 6.3.1, and finally $K_{\mu }\in C(\operatorname {\mathrm {Fix}}(\alpha _b),\Lambda )^{G_b(F)}$ is the function $(g',\lambda ')\mapsto (-1)^d\operatorname {\mathrm {rank}} V_{\mu }^{\vee }[\lambda ']$ , where $d=\left < \mu ,2\rho _G \right>$ .

Proposition 6.3.3. Assume that $\Lambda = \overline {\mathbf {Q}_{\ell }}$ . Let $\phi \in C_c(G(F)_{\operatorname {\mathrm {ell}}},\Lambda )$ , and let $\phi '\in C_c(G_b(F)_{\operatorname {\mathrm {ell}}},\Lambda )$ be any lift of $\widetilde {T}^{G\to G_b}_{b,\mu }\phi $ . Then the orbital integrals of $\phi $ and $\phi '$ are related by

$$\begin{align*}\phi_{G_b}' = T_{b,\mu}^{G\to G_b}\phi_G. \end{align*}$$

Proof. For $g'\in G_b(F)_{\operatorname {\mathrm {ell}}}$ , with centralizer $T'$ , we have:

$$ \begin{align*} &\; \phi^{\prime}_{G_b}(g') \\ =&\int_{h'\in G_b(F)/T'(F)} \phi'(h'g'(h')^{-1})\;dh'\\ =&\; (-1)^d \sum_{\lambda'\in X_*(T')_{\leq -\mu}} \operatorname{\mathrm{rank}} V_{\mu}^{\vee}[\lambda'] \int_{h'\in \frac{G_b(F)}{T'(F)}} [(p_1)_*(p_2)_*^{-1}q_2^*\phi](h'.(g',\lambda')) dh\;dh'. \end{align*} $$

Let T be a transfer of the elliptic torus $T'$ to G. Since $\operatorname {\mathrm {Fix}}(\alpha _{\operatorname {\mathrm {Sht}}})_{\operatorname {\mathrm {ell}}}\to \operatorname {\mathrm {Fix}}(\alpha _b)_{\operatorname {\mathrm {ell}}}$ is a $G(F)$ -torsor, we may choose for each $\lambda '$ a lift $y_{\lambda '}=(g_{\lambda '},g',x_{\lambda '})$ of $(g',\lambda ')$ to $\operatorname {\mathrm {Fix}}(\alpha _{\operatorname {\mathrm {Sht}}})$ with $g_{\lambda '}\in T(F)$ . Then $\phi ^{\prime }_{G_b}(g')$ equals

$$\begin{align*}(-1)^d\sum_{\lambda'\in X_*(T')_{\leq -\mu}}\operatorname{\mathrm{rank}} V_{\mu}^{\vee}[\lambda'] \int_{h'\in G_b(F)/T'(F)} \int_{h\in G(F)} [(p_2)_*^{-1}q_2^*\phi] \left((h,h').(y_{\lambda'})\right) \; dh\;dh'. \end{align*}$$

We rewrite the inner integral as a nested integral so that our expression for $\phi ^{\prime }_{G_b}(g')$ equals:

$$ \begin{align*} &(-1)^d\!\sum_{\lambda'}\! \operatorname{\mathrm{rank}} V_{\mu}^{\vee}[\lambda'] \!\int_{h'\in G_b(F)/T'(F)} \!\int_{h\in G(F)/T(F)} \!\int_{t\in T(F)} [(p_2)_*^{-1}q_2^*\phi]\!\left((ht^{-1},h'). y_{\lambda'}\right) dt\;dh\;dh' \\ = \; &(-1)^d\! \sum_{\lambda'}\! \operatorname{\mathrm{rank}} V_{\mu}^{\vee}[\lambda']\!\int_{h'\in G_b(F)/T'(F)}\! \int_{h\in G(F)/T(F)}\! \int_{t'\in T'(F)}\! [(p_2)_*^{-1}q_2^*\phi]\!\left((h,h't').y_{\lambda'}\right)dt'\;dh\;dh'. \end{align*} $$

Here, we have used Proposition 6.2.2: There is an isomorphism $\iota \colon t\mapsto t'$ between $T(F)$ and $T'(F)$ satisfying $(t,t').y_{\lambda '}=y_{\lambda '}$ . This induces a bijection $\lambda \mapsto \lambda '=-\iota _*\lambda $ between $X_*(T)_{\leq \mu }$ and $X_*(T')_{\leq -\mu }$ . Given $\lambda \in X_*(T)_{\leq \mu }$ , we let $g_{\lambda }=g_{\lambda '}$ . Then by Proposition 6.2.2, the preimage of $g'$ in $\operatorname {\mathrm {Rel}}_{b,\mu }$ is exactly $\left \{ (g_{\lambda },g,\lambda ) \right \}_{\lambda \in X_*(T)_{\leq \mu }}$ .

Noting that $\operatorname {\mathrm {rank}} V_{\mu }[\lambda ]=\operatorname {\mathrm {rank}} V_{\mu }^{\vee }[\lambda ']$ , we exchange the order of the first two integrals above to obtain

$$ \begin{align*} \phi^{\prime}_{G_b}(g') &= (-1)^d\sum_{\lambda\in X_*(T)_{\leq\mu}} \operatorname{\mathrm{rank}} V_{\mu}[\lambda] \int_{h\in G(F)/T(F)}\int_{h'\in G_b(F)} [(p_2)_*^{-1}q_2^*\phi]\left((h,h')\cdot y_{\lambda'}\right)\;dh'\;dh \\ &= (-1)^d\sum_{\lambda\in X_*(T)_{\leq\mu}} \operatorname{\mathrm{rank}} V_{\mu}[\lambda']\int_{h\in G(F)/T(F)} \phi(hg_{\lambda} h^{-1})\;dh\\ &=(-1)^d\sum_{\lambda\in X_*(T)_{\leq\mu}} \operatorname{\mathrm{rank}} V_{\mu}[\lambda] \phi_G(g_{\lambda}) \\ &= [T_{b,\mu}^{G\to G_b}\phi_G](g').\\[-32pt] \end{align*} $$

Definition 6.3.4. Let

$$\begin{align*}\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} \colon \operatorname{\mathrm{Dist}}(G_b(F)_{\operatorname{\mathrm{ell}}},\Lambda)^{G_b(F)} \to \operatorname{\mathrm{Dist}}(G(F)_{\operatorname{\mathrm{ell}}},\Lambda)^{G(F)} \end{align*}$$

be the $\Lambda $ -linear dual of $\widetilde {T}_{b,\mu }^{G\to G_b}$ .

Proposition 6.3.5. Assume that $\Lambda = \overline {\mathbf {Q}_{\ell }}$ . Then the transfer of distributions $\mathcal {T}^{G_b\to G}_{b,\mu }$ extends the transfer of functions $T^{G_b\to G}_{b,\mu }$ from Definition 3.2.7.

Proof. Let $f\in C(G_b(F)_{\operatorname {\mathrm {ell}}},\Lambda )^{G_b(F)}$ be a conjugation-invariant function. Let $\phi \in C_c(G(F)_{\operatorname {\mathrm {ell}}},\Lambda )$ , and let $\phi '\in C_c(G_b(F)_{\operatorname {\mathrm {ell}}},\Lambda )$ be a lift of $\widetilde {T}^{G\to G_b}_{b,\mu }\phi $ . Using the Weyl integration formula (3.4.2), Lemma 3.4.1 and Proposition 6.3.3, we compute

$$ \begin{align*} \int_{g\in G(F)_{\operatorname{\mathrm{ell}}}} \phi(g)\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G}(f\;dg') &= \int_{g'\in G_b(F)_{\operatorname{\mathrm{ell}}}} f(g')\phi'(g')\; dg' \\ &=\left< f,\phi^{\prime}_{G_b} \right>_{G_b}\\ &= \left< f,T_{b,\mu}^{G\to G_b}\phi_G \right>_{G_b} \\ &= \left< T_{b,\mu}^{G_b\to G} f, \phi_G \right>_{G}\\ &= \int_{g\in G(F)_{\operatorname{\mathrm{ell}}}} \phi(g)(T_{b,\mu}^{G_b\to G} f)\; dg \end{align*} $$

so that

$$\begin{align*}\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G}(f\; dg') = T_{b,\mu}^{G_b\to G}(f) \; dg\end{align*}$$

as desired.

6.4 Hecke operators on $\operatorname {\mathrm {Bun}}_G$ and the cohomology of shtuka spaces

We are finally ready to reap our rewards. For the remainder of this chapter, we fix a prime $\ell \neq p$ and write $\Lambda $ for a $\mathbf {Z}_{\ell }$ -algebra. Let $\widehat {G}$ be the Langlands dual group over $\mathbf {Z}_{\ell }$ .

We begin by quickly reviewing the results of [FS21] on the categories $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_G,\Lambda )$ and $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_G^{b},\Lambda )$ , and the action of Hecke operators on $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_G,\Lambda )$ .

The first key fact is that, for any $b \in B(G)$ , there is a natural equivalence of categories

(6.4.1) $$ \begin{align} D(G_b(F),\Lambda) \cong D_{\operatorname{\mathrm{lis}}}(\operatorname{\mathrm{Bun}}_G^b,\Lambda) \end{align} $$

[FS21, Theorem I.5.1]. For a complex $\rho $ of smooth representations of $G_b(F)$ , we will slightly abusively also write $\rho $ for the corresponding object of $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_G^b,\Lambda )$ .

Next, recall that there is a notion of ULA objects in $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_G,\Lambda )$ . These admit the following concrete characterization.

Theorem 6.4.1 [FS21, Theorem I.5.1(v)]

The following are equivalent for an object $A\in D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_G,\Lambda )$ .

1. 1. A is ULA over $\operatorname {\mathrm {Spd}} k$ .

2. 2. For all $b\in B(G)$ , the restriction $i_b^*A$ , considered as an object of $D(G_b(F),\Lambda )$ via equation (6.4.1), is admissible in the sense that $(i_b^*A)^K$ is a perfect complex for all pro-p open subgroups $K\subset G_b(F)$ .

Moreover, ULA objects are preserved under Verdier duality $\mathbf {D}=\mathbf {D}_{\operatorname {\mathrm {Bun}}_G/\operatorname {\mathrm {Spd}} k}$ and satisfy Verdier biduality.

Corollary 6.4.2. Let $b\in B(G)$ , and let $\rho $ be an admissible complex in $D(G_b(F),\Lambda )$ . The objects $(i_b)_*\rho $ and $(i_b)_!\rho $ of $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_G,\Lambda )$ are ULA over $\operatorname {\mathrm {Spd}} k$ .

Proof. The object $(i_b)_!\rho $ is ULA by the criterion in Theorem 6.4.1. Using Verdier duality (P4.) we have $\mathbf {D}((i_b)_!\rho ^{\vee })\cong (i_b)_* \rho $ so that $(i_b)_* \rho $ is also ULA.

Next, recall that any object V of $\operatorname {\mathrm {Rep}}_{\widehat {G}}(\Lambda )$ gives rise to a Hecke operator $T_V$ , which is an endofunctor of $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_{G,C},\Lambda )$ . When $\Lambda $ is a torsion ring, there is a natural equivalence $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_{G,C},\Lambda ) \cong D_{\mathrm {\acute {e}t}}(\operatorname {\mathrm {Bun}}_{G,C},\Lambda )$ , and the operator $T_V$ is defined concretely as the operation

$$ \begin{align*} T_V\colon D_{\mathrm{\acute{e}t}}(\operatorname{\mathrm{Bun}}_{G,C},\Lambda) &\to D_{\mathrm{\acute{e}t}}(\operatorname{\mathrm{Bun}}_{G,C}, \Lambda) \\ \mathcal{F} &\mapsto h_{2!}(h_1^* \mathcal{F}\otimes \mathcal{S}_V). \end{align*} $$

Here, $\mathcal {S}_V\in D_{\mathrm {\acute {e}t}}(\operatorname {\mathrm {Hecke}}_{G,C},\Lambda )$ is pulled back from the object $\mathcal {S}_V\in D_{\mathrm {\acute {e}t}}(\operatorname {\mathrm {Hecke}}_{G,C}^{\operatorname {\mathrm {loc}}},\Lambda )$ corresponding to V under the Satake equivalence (Theorem 5.1.1).

Theorem 6.4.3 [FS21, Theorem IX.0.1]

The Hecke operators preserve the subcategories of ULA and compact objects in $D_{\operatorname {\mathrm {lis}}}(\operatorname {\mathrm {Bun}}_{G,C},\Lambda )$ . For any V, $T_V$ has left and right adjoint given by $T_{V^{\vee }}$ , where $V^{\vee }$ is the dual representation of $\widehat {G}$ . The actions of Hecke operators are compatible with extension of scalars along any ring map $\Lambda \to \Lambda '$ .

Next, we explain the relation between the Hecke operators $T_V$ and the cohomology of local shtuka spaces. Let $\mu $ be a dominant cocharacter of G, and let $V_{\mu } \in \operatorname {\mathrm {Rep}}(\widehat {G})$ be the associated Weyl module. For any $\mathbf {Z}_{\ell }$ -algebra $\Lambda $ , we write $V_{\mu ,\Lambda } \in \operatorname {\mathrm {Rep}}(\widehat {G}_{\Lambda })$ for the base change of $V_{\mu }$ . Let $\mathcal {S}_{\mu }=\mathcal {S}_{V_{\mu }}$ be the corresponding object in the Satake category with $\mathbf {Z}_{\ell }$ -coefficients; similarly, if $\Lambda $ is a torsion ring, we write $\mathcal {S}_{\mu ,\Lambda }=\mathcal {S}_{V_{\mu },\Lambda }$ for the corresponding object with $\Lambda $ -coefficients. We will slightly abuse notation by using the same notations for the pullbacks of $\mathcal {S}_{\mu }$ and $\mathcal {S}_{\mu ,\Lambda }$ to various other v-stacks, including $\operatorname {\mathrm {Gr}}_{G,\leq \mu ,C}$ and $\operatorname {\mathrm {Sht}}_{G,b,\mu ,C}$ (along the period morphism $\pi _1$ from equation (6.1.4)).

Lemma 6.4.4. Let $\Lambda $ be a $\mathbf {Z}_{\ell }$ -algebra, let $K\subset G(F)$ be an open compact subgroup and let $I_{K,\Lambda }={\operatorname {\mathrm {cInd}}}_K^{G(F)} \Lambda $ , where $\operatorname {\mathrm {cInd}}$ is compactly supported induction. Then there is a natural isomorphism

$$\begin{align*}R\Gamma_c(\operatorname{\mathrm{Sht}}_{G,b,\mu,K,C},\mathcal{S}_{\mu,\Lambda}) \cong i_b^{\ast} T_{V_{\mu,\Lambda}} (i_1)_! I_{K,\Lambda} \end{align*}$$

in $D(G_b(F),\Lambda )$ .

Proof. When $\Lambda $ is a torsion ring, we can give the following direct argument. The global sections of $i_b^{\ast } T_{V_{\mu ,\Lambda }} (i_1)_!I_{K,\Lambda }$ over $\operatorname {\mathrm {Spd}} C\to \operatorname {\mathrm {Bun}}_G^b$ are

$$\begin{align*}R\Gamma(\operatorname{\mathrm{Spd}} C, i_b^{\ast} h_{2!} (h_1^*(i_1)_! I_{K,\Lambda} \otimes_{\Lambda} \mathcal{S}_{\mu,\Lambda})) \cong R\Gamma_c(\operatorname{\mathrm{Gr}}^{b,1}, I_{K,\Lambda} \vert_{\operatorname{\mathrm{Gr}}^{b,1}} \otimes_{\Lambda} \mathcal{S}_{\mu,\Lambda}). \end{align*}$$

Now use $I_{K,\Lambda } \cong (j_K)_! \Lambda $ along with proper base change to get the result.

The general case follows from the proof of [FS21, Proposition IX.3.2].

Recall that, when $\rho $ is any smooth $G_b(F)$ -representation with $\overline {\mathbf {Q}_{\ell }}$ -coefficients, we defined an object

$$\begin{align*}R\Gamma(G,b,\mu)[\rho] \cong \varinjlim_K R\operatorname{\mathrm{Hom}}_{G_b(F)}(R\Gamma_c(\operatorname{\mathrm{Sht}}_{(G,b,\mu),C}/K,\mathcal{S}_{\mu}) \otimes \overline{\mathbf{Q}_{\ell}}, \rho), \end{align*}$$

in $D(G(F),\overline {\mathbf {Q}_{\ell }})$ , cf. Definition 2.4.3. The association $\rho \mapsto R\Gamma (G,b,\mu )[\rho ]$ clearly extends to a functor $D(G_b(F),\overline {\mathbf {Q}_{\ell }}) \to D(G(F),\overline {\mathbf {Q}_{\ell }})$ . Our next goal is to give an alternative approach to this construction, which is valid for more general coefficient rings and which makes the finiteness properties of this construction transparent.

Proposition 6.4.5. Let $\rho $ be any object of $D(G_b(F),\overline {\mathbf {Q}_{\ell }})$ . Then there is a natural isomorphism

$$\begin{align*}R\Gamma(G,b,\mu)[\rho] \cong i_1^*T_{{V^{\vee}_{\mu,\overline{\mathbf{Q}_{\ell}}}}} (i_b)_*\rho \end{align*}$$

in $D(G(F),\overline {\mathbf {Q}_{\ell }})$ .

If $\rho $ is admissible, then so is $R\Gamma (G,b,\mu )[\rho ]$ . If $\rho $ is of finite length, then so is $R\Gamma (G,b,\mu )[\rho ]$ .

Proof. Let $K\subset G(F)$ be a compact open subgroup. Using Lemma 6.4.4, various adjunctions and the compatibility of Hecke operators with extension of scalars, we have

$$ \begin{align*} R\operatorname{\mathrm{Hom}}_{G_b(F)}(R\Gamma_c(\operatorname{\mathrm{Sht}}_{G,b,\mu,K,C},\mathcal{S}_V) \otimes \overline{\mathbf{Q}_{\ell}},\rho) &\cong R\operatorname{\mathrm{Hom}}_{G_b(F)}(i_b^{\ast} T_{V_{\mu}} (i_1)_! {\operatorname{\mathrm{cInd}}}_K^{G(F)} \mathbf{Z}_{\ell} \otimes \overline{\mathbf{Q}_{\ell}}, \rho) \\ & \cong R\operatorname{\mathrm{Hom}}_{G_b(F)}(i_b^{\ast} T_{V_{\mu,\overline{\mathbf{Q}_{\ell}}}} (i_1)_! {\operatorname{\mathrm{cInd}}}_{K}^{G(F)} \overline{\mathbf{Q}_{\ell}}, \rho) \\ &\cong (i_1^* T_{{V^{\vee}_{\mu,\overline{\mathbf{Q}_{\ell}}}}} (i_b)_*\rho)^K. \end{align*} $$

Taking the colimit over K gives the first claim. The claim about preservation of admissibility now follows from Theorem 6.4.1 combined with Theorem 6.4.3. For the final claim, fix some $\rho $ of finite length. Note that $i_1^*T_{{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}}} (i_b)_*\rho $ is the smooth dual of $i_1^*T_{{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}}} (i_b)_! \rho ^{\vee }$ , so it’s enough to show that $i_1^*T_{{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}}} (i_b)_! \rho ^{\vee }$ is of finite length. But finite length is equivalent to being both compact and admissible, so we conclude by observing that the operation $i_1^*T_{{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}}} (i_b)_!(-)$ preserves compact objects.

Definition 6.4.6. For any $\mathbf {Z}_{\ell }$ -algebra $\Lambda $ , we write

$$\begin{align*}R\Gamma(G,b,\mu)[-]: D(G_b(F),\Lambda) \to D(G(F),\Lambda) \end{align*}$$

for the functor $i_1^* T_{{V^{\vee }_{\mu ,\Lambda }}} (i_b)_*(-)$ .

By the previous discussion, this functor is compatible with extension of scalars along any map $\Lambda \to \Lambda '$ and preserves admissible objects. Moreover, if $\Lambda $ is Artinian and $\rho \in D(G_b(F),\Lambda )$ is admissible of finite length, then $R\Gamma (G,b,\mu )[\rho ]$ is also of finite length by the same argument as in the proof of Proposition 6.4.5.

We now come to the technical heart of this paper. Choose $\mathbf {Z}_{\ell }$ -valued Haar measures on $G(F)$ and $G_b(F)$ , compatibly as in §3.4. These induce $\Lambda $ -valued Haar measures on the same groups compatibly with varying $\Lambda $ . Then for any $\Lambda $ , any admissible representation $\pi $ of $G(F)$ with coefficients in $\Lambda $ has a corresponding $\Lambda $ -valued trace distribution $\operatorname {\mathrm {tr.dist}}(\pi )$ , and similarly for $G_b(F)$ . Recall also that we defined a transfer of $\Lambda $ -valued distributions $\mathcal {T}_{b,\mu }^{\kern3pt G_b\to G}$ , Definition 6.3.4.

Proposition 6.4.7. Let $\Lambda $ be any torsion $\mathbf {Z}_{\ell }$ -algebra, and let $\rho $ be any admissible representation of $G_b(F)$ with coefficients in $\Lambda $ . Then have an equality

$$\begin{align*}\operatorname{\mathrm{tr.dist}} R\Gamma (G,b,\mu)[\rho]_{\operatorname{\mathrm{ell}}} = \mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} \operatorname{\mathrm{tr.dist}}(\rho)_{\operatorname{\mathrm{ell}}} \end{align*}$$

in $\operatorname {\mathrm {Dist}}(G(F)_{\operatorname {\mathrm {ell}}},\Lambda )^{G(F)}$ .

Proof. In the following proof, we set $S=\operatorname {\mathrm {Spd}} C$ and $V=V_{\mu ,\Lambda }$ for brevity.

We have an isomorphism

$$\begin{align*}H^0(\operatorname{\mathrm{In}}_S(\operatorname{\mathrm{Bun}}_{G,C}^1),K_{\operatorname{\mathrm{In}}_S(\operatorname{\mathrm{Bun}}_{G,C}^1)/S})\cong \operatorname{\mathrm{Dist}}(G(F),\Lambda)^{G(F)}\end{align*}$$

and similarly for $G_b(F)$ . With respect those isomorphisms, the left side of the desired equality is the characteristic class

$$\begin{align*}\operatorname{\mathrm{cc}}_{\operatorname{\mathrm{Bun}}_{G,C}^1/S}\left(i_1^*T_{V^{\vee}}(i_b)_*\rho \right)\end{align*}$$

restricted to $\operatorname {\mathrm {Dist}}(G(F)_{\operatorname {\mathrm {ell}}},\Lambda )^{G(F)}$ .

For the remainder of the proof, we introduce the abbreviations $B=\operatorname {\mathrm {Bun}}_{G,C}$ and $H=\operatorname {\mathrm {Hecke}}_{G,\leq \mu ,C}$ and $\operatorname {\mathrm {In}}=\operatorname {\mathrm {In}}_S$ . Let us also use a subscript ‘ell’ to mean restriction to the appropriate elliptic locus. Taking inertia stacks in equation (6.1.2), we obtain a commutative diagram

(6.4.2)

in which all squares are Cartesian, and the morphism labeled $\operatorname {\mathrm {id}}$ is the equality from Corollary 6.2.4. The characteristic class in question is

$$ \begin{align*} \begin{array}{lcl} j_1^*\operatorname{\mathrm{cc}}_{B^1/S}(i_1^*T_{V^{\vee}}(i_b)_*\rho ) &\stackrel{\text{Lem. }4.3.7}{=} &\!\! j_1^*\operatorname{\mathrm{In}}(i_1)^*\operatorname{\mathrm{cc}}_{B/S}\left((h_2)_!(h_1^*(i_b)_*\rho \otimes \mathcal{S}_{V^{\vee}}\right) \\[2pt] &\stackrel{\text{Cor.} 4.3.9}{=} &\!\! j_1^*\operatorname{\mathrm{In}}(i_1)^*\operatorname{\mathrm{In}}(h_2)_*\operatorname{\mathrm{cc}}_{H/S}(h_1^*(i_b)_*\rho \otimes \mathcal{S}_{V^{\vee}})\\[2pt] &= &\!\! (\operatorname{\mathrm{In}}(h_2^{\ast,1})_{\operatorname{\mathrm{ell}}})_*(j_1')^*\operatorname{\mathrm{In}}(i_1')^*\operatorname{\mathrm{cc}}_{H/S}(h_1^*(i_b)_*\rho \otimes\mathcal{S}_{V^{\vee}})\\[2pt] &= &\!\! (\operatorname{\mathrm{In}}(h_2^{\ast,1})_{\operatorname{\mathrm{ell}}})_*(j_b')^*\operatorname{\mathrm{In}}(i_b')^*\operatorname{\mathrm{cc}}_{H/S}(h_1^*(i_b)_*\rho \otimes\mathcal{S}_{V^{\vee}})\\[2pt] &\stackrel{\text{Lem. }4.3.7}{=} &\!\! (\operatorname{\mathrm{In}}(h_2^{\ast,1})_{\operatorname{\mathrm{ell}}})_*(j_b')^*\operatorname{\mathrm{cc}}_{H^{b,\ast}/S}((i_b')^*h_1^*(i_b)_*\rho \otimes\mathcal{S}_{V^{\vee}}) \\[2pt] &= &\!\! (\operatorname{\mathrm{In}}(h_2^{\ast,1})_{\operatorname{\mathrm{ell}}})_*(j_b')^* \operatorname{\mathrm{cc}}_{H^{b,\ast}/S} ((h_1^{b,\ast})^*\rho \otimes \mathcal{S}_{V^{\vee}}). \end{array} \end{align*} $$

Noting that $H^{b,\ast }\cong [\operatorname {\mathrm {Gr}}^b_{G,\leq -\mu ,C}/G_b(F)_S]$ , we have a Cartesian diagram of decent S-v-stacks:

(6.4.3)

Here, $B^b\cong [S/G_b(F)_S]$ , $H^{\operatorname {\mathrm {loc}}}_m\cong [\operatorname {\mathrm {Gr}}_{G,\leq -\mu ,C}/L^+_mG]$ , and $B_m^{\operatorname {\mathrm {loc}}}\cong [S/L^+_mG]$ ; the m here is chosen large enough so that the action of $L^+G$ on $\operatorname {\mathrm {Gr}}_{G,\leq -\mu ,C}$ factors through the quotient $L^+_mG$ . Through this, we can identify $(h_1^{b,\ast })^*\rho \otimes \mathcal {S}_{V^{\vee }}$ with $\rho \boxtimes _{B^{\operatorname {\mathrm {loc}}}_m} \mathcal {S}_{V^{\vee }}$ . It is at this point we apply Theorem 4.5.3, valid because the base $B_m^{\operatorname {\mathrm {loc}}}=[S/L^+_mG]$ satisfies the hypotheses of Lemma 4.5.2. We get

$$ \begin{align*} j_1^*\operatorname{\mathrm{cc}}_{B^1/S}(i_1^*T_{V^{\vee}}(i_b)_*\rho) &= (\operatorname{\mathrm{In}}(h_2^{\ast,1})_{\operatorname{\mathrm{ell}}})_*(j_b')^*\left(\operatorname{\mathrm{cc}}_{B^b/S}\rho \boxtimes_{\operatorname{\mathrm{In}}(B^{\operatorname{\mathrm{loc}}}_m)} \operatorname{\mathrm{cc}}_{H_m^{\operatorname{\mathrm{loc}}}/S}(\mathcal{S}_{V^{\vee}}) \right) \\ &= (\operatorname{\mathrm{In}}(h_2^{\ast,1})_{\operatorname{\mathrm{ell}}})_*\left(\operatorname{\mathrm{cc}}_{B^b/S}\rho_{\operatorname{\mathrm{ell}}} \boxtimes_{\operatorname{\mathrm{In}}(B^{\operatorname{\mathrm{loc}}}_m)_{\operatorname{\mathrm{sr}}}} \operatorname{\mathrm{cc}}_{H_m^{\operatorname{\mathrm{loc}}}/S}(\mathcal{S}_{V^{\vee}})_{\operatorname{\mathrm{sr}}}\right). \end{align*} $$

Considering the diagram

our characteristic class is the image of $\operatorname {\mathrm {cc}}_{B^b/S}(\rho )_{\operatorname {\mathrm {ell}}}$ under the composite vertical map on the left. The diagram is commutative; the hardest thing to check is the commutativity of the top square, which follows from Proposition 6.4.8 below. The composition along the right column is $\mathcal {T}^{G_b\to G}_{b,\mu }$ , giving us the desired equality of distributions.

It remains to justify one step in this computation. Maintain the notation and assumptions of the previous theorem. The v-stack $H^{b,\ast }$ can be expressed as a fiber product as in equation (6.4.3); we have the ULA object $\rho \boxtimes _{B_m^{\operatorname {\mathrm {loc}}}} \mathcal {S}_{V^{\vee }}$ , whose characteristic class can be calculated using Theorem 4.5.3.

Let

$$\begin{align*}\alpha_b \colon G_b(F)_S\times \operatorname{\mathrm{Gr}}_{G,\leq-\mu} \to \operatorname{\mathrm{Gr}}_{G,\leq-\mu} \end{align*}$$

be the action map so that we have an isomorphism

$$\begin{align*}\operatorname{\mathrm{In}}_S(H^{b,\ast}) \cong [\operatorname{\mathrm{Fix}}(\alpha_b)/G_b(F)_S]. \end{align*}$$

Let $\operatorname {\mathrm {Fix}}(\alpha _b)_{\operatorname {\mathrm {sr}}}$ be the open subset lying over $G_b(F)_{\operatorname {\mathrm {sr}}}$ (and use the same convention for other objects); then $\operatorname {\mathrm {Fix}}(\alpha _b)_{\operatorname {\mathrm {sr}}}$ is a locally profinite set, which is finite over $G_b(F)_{\operatorname {\mathrm {sr}}}$ with fibers $X_*(T)_{\leq -\mu }$ .

Proposition 6.4.8. The characteristic class

$$\begin{align*}\operatorname{\mathrm{cc}}_{H/S}(\rho \boxtimes \mathcal{S}_{V^{\vee}})_{\operatorname{\mathrm{sr}}}\in H^0(\operatorname{\mathrm{In}}_S(H)_{\operatorname{\mathrm{sr}}},K_{\operatorname{\mathrm{In}}_S(T)/S})\cong \operatorname{\mathrm{Dist}}(\operatorname{\mathrm{Fix}}(\alpha_b)_{\operatorname{\mathrm{sr}}},\Lambda)^{G_b(F)} \end{align*}$$

equals the image of $\operatorname {\mathrm {tr.dist}}(\rho ) \otimes (-1)^{\left < 2\rho _G,- \right>}\operatorname {\mathrm {rank}} V^{\vee }[-]$ under the evident map

(6.4.4) $$ \begin{align} \operatorname{\mathrm{Dist}}(G_b(F)_{\operatorname{\mathrm{sr}}},\Lambda)^{G_b(F)}\otimes C(X_*(T)_{\leq -\mu},\Lambda)^W \to \operatorname{\mathrm{Dist}}(\operatorname{\mathrm{Fix}}(\alpha_b)_{\operatorname{\mathrm{sr}}},\Lambda)^{G_b(F)}. \end{align} $$

Proof. Take inertia stacks in equation (6.4.3), and restrict to the strongly regular locus in $\operatorname {\mathrm {In}}_S[S/L^+_mG]$ to obtain a Cartesian diagram

The Künneth map (4.5.2) in this situation reduces to equation (6.4.4) on the level of global sections. The result now follows from Theorems 4.5.3 and 5.1.4.

This formally implies the following theorem.

Theorem 6.4.9. Let $\rho $ be any finite length admissible $G_b(F)$ -representation with $\overline {\mathbf {Q}_{\ell }}$ -coefficients. Assume that $\rho $ admits a $\overline {\mathbf {Z}_{\ell }}$ -lattice in the sense of Definition C.2.1. Then for all $\phi \in C_c(G(F)_{\operatorname {\mathrm {ell}}},\overline {\mathbf {Q}_{\ell }})$ , the equality

$$\begin{align*}\operatorname{\mathrm{tr}} (\phi | \operatorname{\mathrm{Mant}}_{b,\mu}(\rho)) = \left[\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} (\operatorname{\mathrm{tr.dist}} \rho)\right](\phi) \end{align*}$$

holds.

Recall that by definition, $\mathcal {T}_{b,\mu }^{\kern3pt G_b\to G} (\operatorname {\mathrm {tr.dist}}\rho )(\phi )$ depends only on $(\operatorname {\mathrm {tr.dist}}\rho )_{\operatorname {\mathrm {ell}}}$ .

Proof. Fix $\rho $ and $\phi $ as in the theorem, and fix a $\overline {\mathbf {Z}_{\ell }}$ -lattice $\rho ^{\circ } \subset \rho $ . After rescaling, we may also assume $\phi $ is valued in $\overline {\mathbf {Z}_{\ell }}$ . It is clear from the definitions that $\operatorname {\mathrm {tr}} (\phi | \operatorname {\mathrm {Mant}}_{b,\mu }(\rho )) = \operatorname {\mathrm {tr}} (\phi | R\Gamma (G,b,\mu )[\rho ^{\circ }])$ and

$$\begin{align*}\left[\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} (\operatorname{\mathrm{tr.dist}} \rho)\right](\phi) = \left[\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} (\operatorname{\mathrm{tr.dist}} \rho^{\circ})\right](\phi). \end{align*}$$

For all $n\geq 1$ , set $\rho ^{\circ }_{n} = \rho ^{\circ } \otimes \mathbf {Z}/\ell ^n$ , and write $\phi _n \in C_c(G(F)_{\operatorname {\mathrm {ell}}},\overline {\mathbf {Z}_{\ell }}/\ell ^n)$ for the obvious reductions of $\phi $ . Applying Proposition 6.4.7 with $\Lambda = \overline {\mathbf {Z}_{\ell }}/\ell ^n$ , we get equalities

$$\begin{align*}\operatorname{\mathrm{tr}} (\phi_n | R\Gamma(G,b,\mu)[\rho^{\circ}_n]) = \left[\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} (\operatorname{\mathrm{tr.dist}} \rho^{\circ}_n)\right](\phi_n) \end{align*}$$

for all $n\geq 1$ . The result now follows by taking the inverse limit over n.

6.5 Proof of Theorem 1.0.2

We are finally ready to prove the main theorem of the paper, which we restate for the convenience of the reader.

Theorem 6.5.1. Assume the refined local Langlands correspondence [Kal16a, Conjecture G]. Let $\phi \colon W_F \times \mathrm {SL}_2 \to {^LG}$ be a discrete Langlands parameter with coefficients in $\overline {\mathbf {Q}}_{\ell }$ , and let $\rho \in \Pi _{\phi }(G_b)$ be a member of its L-packet. After ignoring the action of $W_E$ , we have an equality

$$\begin{align*}\operatorname{\mathrm{Mant}}_{b,\mu}(\rho)=\sum_{\pi\in\Pi_{\phi}(G)} \left[\dim \operatorname{\mathrm{Hom}}_{S_{\phi}}(\delta_{\pi,\rho},r_{\mu})\right]\pi + \mathrm{err} \end{align*}$$

in $\operatorname {\mathrm {Groth}}(G(F))$ , where $\mathrm {err} \in \operatorname {\mathrm {Groth}}(G(F))$ is a virtual representation whose character vanishes on the locus of elliptic elements of $G(F)$ .

If the packet $\Pi _{\phi }(G)$ consists entirely of supercuspidal representations and the semisimple L-parameter $\varphi _{\rho }$ associated with $\rho $ as in [FS21, §I.9.6] is supercuspidal, then in fact $\mathrm {err}=0$ .

The main ingredient in the proof is the following extension of Theorem 6.4.9 to its natural level of generality.

Theorem 6.5.2. Let $\rho $ be any finite length admissible $G_b(F)$ -representation with $\overline {\mathbf {Q}_{\ell }}$ -coefficients. Then for all $\phi \in C_c(G(F)_{\operatorname {\mathrm {ell}}},\overline {\mathbf {Q}_{\ell }})$ , the equality

$$\begin{align*}\operatorname{\mathrm{tr}} (\phi | \operatorname{\mathrm{Mant}}_{b,\mu}(\rho)) = \left[\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} (\operatorname{\mathrm{tr.dist}} \rho)\right](\phi) \end{align*}$$

holds.

In particular, the virtual character of $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho )$ restricted to $G(F)_{\operatorname {\mathrm {ell}}}$ is equal to $T_{b,\mu }^{G_b \to G}(\Theta _{\rho })$ .

We will formally deduce this from Theorem 6.4.9 by a continuity argument.

For the proof of this theorem, it will be convenient to use the language of Grothendieck groups. In particular, by the finiteness results mentioned above, $\operatorname {\mathrm {Mant}}_{b,\mu }(-)$ can be regarded as a group homomorphism $\operatorname {\mathrm {Mant}}_{b,\mu }(-): \operatorname {\mathrm {Groth}}(G_b(F)) \to \operatorname {\mathrm {Groth}}(G(F))$ . Recall that any element $\phi \in C_c(G(F),\overline {\mathbf {Q}_{\ell }})$ defines a linear form $\operatorname {\mathrm {tr}}(\phi | -): \operatorname {\mathrm {Groth}}(G(F)) \to \overline {\mathbf {Q}_{\ell }}$ . By definition, a linear form $f: \operatorname {\mathrm {Groth}}(G(F)) \to \overline {\mathbf {Q}_{\ell }}$ is a trace form if it can be written as $\operatorname {\mathrm {tr}}( \phi | -)$ for some $\phi \in C_c(G(F),\overline {\mathbf {Q}_{\ell }})$ . The key ingredient in the proof of Theorem 6.5.2 is the following result, which roughly says that $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho )$ is a continuous function of $\rho $ .

Theorem 6.5.3. For any $\phi \in C_c(G(F),\overline {\mathbf {Q}_{\ell }})$ , the linear form

$$\begin{align*}\operatorname{\mathrm{tr}}(\phi | \operatorname{\mathrm{Mant}}_{b,\mu}(-)) : \operatorname{\mathrm{Groth}}(G_b(F)) \to \overline{\mathbf{Q}_{\ell}} \end{align*}$$

is a trace form.

With future applications in mind, we’ll actually prove the following refined form of this theorem which also accounts for the Weil group action.

Theorem 6.5.4. For any fixed $\phi \in C_{c}(G(F),\overline {\mathbf {Q}_{\ell }})$ and $w\in W_{E}$ , the linear form $\operatorname {\mathrm {Groth}}(G_{b}(F))\to \overline {\mathbf {Q}_{\ell }}$ defined by

$$\begin{align*}\rho\mapsto\mathrm{tr}(\phi\times w | R\Gamma(G,b,\mu)[\rho]) \end{align*}$$

is a trace form.

In the classical setting of Rapoport–Zink spaces, this was conjectured by Taylor, cf. [Shi12, Conjecture 8.3]. Taking $w=1$ , we deduce Theorem 6.5.3.

Proof. For any reductive group $H/F$ , the trace Paley–Wiener theorem of Bernstein–Deligne–Kazhdan, [BDK86], characterizes trace forms among all linear forms on $\operatorname {\mathrm {Groth}}(H(F))$ by the following two conditions:

1. 1. There is some open compact subgroup $K\subset H(F)$ such that $f(\pi )\neq 0$ only if $\pi ^{K}\ne 0$ .

2. 2. For any parabolic $P=MU\subset H$ and any irreducible smooth $M(F)$ -representation $\sigma $ , $f(i_{M}^{H}(\sigma \psi ))$ is an algebraic function of $\psi $ , where $\psi $ varies over the unramified characters of $M(F)$ . Here, $i_{M}^{H}(-)$ denotes normalizes parabolic induction.

We’ll prove the theorem by showing that the linear form $\mathrm {tr}(\phi \times w | R\Gamma (G,b,\mu )[-])$ satisfies the conditions of the trace Paley–Wiener theorem, applied to the group $H=G_b$ .

Verification of Condition 1. Fix a pro-p open compact subgroup $K\subset G(F)$ such that $\phi $ is bi-K-invariant. If $\mathrm {tr}(\phi \times w| R\Gamma (G,b,\mu )[\rho ])\neq 0$ , then $(i_{\mathbf {1}}^{\ast }T_{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}}i_{b \ast }\rho )^{K}\neq 0$ . Therefore, it suffices to see that there is some open compact $K'\subset G_{b}(F)$ such that $(i_{\mathbf {1}}^{\ast }T_{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}}i_{b \ast }\rho )^{K}\neq 0$ only if $\rho ^{K'}\neq 0$ . For this, write

$$ \begin{align*} (i_{\mathbf{1}}^{\ast}T_{V^{\vee}_{\mu,\overline{\mathbf{Q}_{\ell}}}}i_{b \ast}\rho)^{K} & \cong R\mathrm{Hom}(i_{\mathbf{1}!}\operatorname{\mathrm{cInd}}_{K}^{G(F)}\overline{\mathbf{Q}_{\ell}},T_{V^{\vee}_{\mu,\overline{\mathbf{Q}_{\ell}}}}i_{b \ast}\rho)\\ & \cong R\mathrm{Hom}(T_{V_{\mu,\overline{\mathbf{Q}_{\ell}}}}i_{\mathbf{1}!}\operatorname{\mathrm{cInd}}_{K}^{G(F)}\overline{\mathbf{Q}_{\ell}},i_{b \ast}\rho)\\ & \cong R\mathrm{Hom}(i_{b}^{\ast}T_{V_{\mu,\overline{\mathbf{Q}_{\ell}}}}i_{\mathbf{1}!}\operatorname{\mathrm{cInd}}_{K}^{G(F)}\overline{\mathbf{Q}_{\ell}},\rho). \end{align*} $$

But now $i_{b}^{\ast }T_{V_{\mu ,\overline {\mathbf {Q}_{\ell }}}}i_{\mathbf {1}!}\operatorname {\mathrm {cInd}}_{K}^{G(F)}\overline {\mathbf {Q}_{\ell }}$ is compact and hence supported on only finitely many Bernstein components for $G_{b}(F)$ . This shows that the irreducible $\rho $ ’s with $(i_{\mathbf {1}}^{\ast }T_{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}}i_{b \ast }\rho )^{K}\neq 0$ are supported on finitely many Bernstein components for $G_{b}(F)$ . Quite generally, if $\Theta $ is any finite union of Bernstein components for $G_{b}(F)$ , we can choose an open compact subgroup $K'\subset G_{b}(F)$ such that $\rho ^{K'}\neq 0$ if $\rho $ is supported on $\Theta $ . This gives the result.

Verification of Condition 2. Fix $P=MU\subset G_{b}$ and $\sigma $ as in Condition 2. Let $X=\mathrm {Spec}R$ be the smooth affine algebraic variety over $\overline {\mathbf {Q}_{\ell }}$ parametrizing unramified characters of $M(F)$ . Let $\boldsymbol {\psi }:M(F)\to R^{\times }$ be the universal character and form $\Pi =i_{M}^{G_{b}}(\sigma \boldsymbol {\psi })$ . This is an admissible smooth $R[G_{b}(F)]$ -module interpolating the parabolic inductions $i_{M}^{G_{b}}(\sigma \psi )$ over varying unramified characters $\psi $ in the evident sense.⁶ Since $\Pi $ is admissible, the pushforward $i_{b \ast }\Pi \in D_{\mathrm {lis}}(\mathrm {Bun}_{G},R)$ is ULA. Since Hecke operators preserve ULA complexes, we deduce that $i_{\mathbf {1}}^{\ast }T_{V^{\vee }_{\mu ,R}}i_{b \ast }\Pi \in D(G(F),R)^{BW_{E}}$ is an admissible complex of smooth $R[G(F)]$ -modules with $W_{E}$ -action, which interpolates the individual complexes

$$\begin{align*}R\Gamma(G,b,\mu)[i_{M}^{G_{b}}(\sigma\psi)]=i_{\mathbf{1}}^{\ast}T_{V^{\vee}_{\mu,\overline{\mathbf{Q}_{\ell}}}}i_{b \ast}i_{M}^{G_{b}}(\sigma\psi)\end{align*}$$

in the evident sense.

Now fix a pro-p open compact subgroup $K \subset G(F)$ such that $\phi $ is bi-K-invariant, so $\phi \times w$ defines an endomorphism of the perfect complex

$$\begin{align*}(i_{\mathbf{1}}^{\ast}T_{V^{\vee}_{\mu,R}}i_{b \ast }\Pi)^{K}\in\mathrm{Perf}(R). \end{align*}$$

Let $f\in R$ be the trace of this endomorphism. Unwinding definitions, we see that, for any unramified character $\psi :M(F)\to \overline {\mathbf {Q}_{\ell }}^{\times }$ with associated point $x_{\psi }\in X(\overline {\mathbf {Q}_{\ell }})$ , there is an equality

$$\begin{align*}f(x_{\psi})=\mathrm{tr}\left(\phi\times w| R\Gamma(G,b,\mu)[i_{M}^{G_{b}}(\sigma\psi)]\right). \end{align*}$$

This shows that $\mathrm {tr}(\phi \times w|R\Gamma (G,b,\mu )[i_{M}^{G_{b}}(\sigma \psi )])$ is an algebraic function of $\psi $ , as desired.

Let us say a subset $S\subset \mathrm {Irr}_{\overline {\mathbf {Q}_{\ell }}}(G(F))$ is dense if any trace form on $\operatorname {\mathrm {Groth}}(G(F))$ which vanishes on S vanishes identically. For instance, the Langlands classification implies that (for any fixed choice of isomorphism $\mathbf {C}\simeq \overline {\mathbf {Q}_{\ell }}$ ) tempered representations are dense, cf. [Kaz86, Theorem 0].

Lemma 6.5.5. The subset of irreducible representations $\pi \in \mathrm {Irr}_{\overline {\mathbf {Q}_{\ell }}}(G(F))$ admitting $\overline {\mathbf {Z}_{\ell }}$ -lattices is dense.

It seems reasonable to think of this lemma as an $\ell $ -adic analogue of the density of tempered representations.

Proof. Let f be a trace form, and assume that $f(\tau )=0$ for every $\tau \in \mathrm {Irr}_{\overline {\mathbf {Q}_{\ell }}}(G(F))$ admitting a $\overline {\mathbf {Z}_{\ell }}$ -lattice.

By Proposition C.2.2, it’s enough to show that $f( i_{M}^{G}(\sigma \psi ))=0$ for any parabolic $P=MU \subset G$ , any unramified character $\psi $ of $M(F)$ and any $\sigma \in \mathrm {Irr}_{\overline {\mathbf {Q}_{\ell }}}(M(F))$ admitting a $\overline {\mathbf {Z}_{\ell }}$ -lattice. Fix P and $\sigma $ , and consider the function g on unramified characters of $M(F)$ sending $\psi $ to $f \left ( i_{M}^{G}(\sigma \psi ) \right )$ . By the easy direction of the trace Paley–Wiener theorem, g is a regular function on the variety of unramified characters of $M(F)$ .

Let us say an unramified character $\psi $ is integral if it takes values in $\overline {\mathbf {Z}_{\ell }}^{\times }$ . If $\psi $ is integral, then $\sigma \psi $ admits a $\overline {\mathbf {Z}_{\ell }}$ -lattice, and hence also $i_{M}^{G}(\sigma \psi )$ admits a $\overline {\mathbf {Z}_{\ell }}$ -lattice. In particular, if $\psi $ is integral and $i_{M}^{G}(\sigma \psi )$ is irreducible, then $g(\psi )=0$ by combining these observations with our assumption on f. Now integral characters are Zariski-dense in the variety of unramified characters of $M(F)$ , and the subset T of integral characters such that $i_{M}^{G}(\sigma \psi )$ is irreducible is also Zariski-dense (use [Dat05, Theorem 5.1]). Since $g(\psi )=0$ for all $\psi \in T$ , we deduce that $g \equiv 0$ , so in particular

$$\begin{align*}0 = g(\psi) = f \left( i_{M}^{G}(\sigma \psi) \right) \end{align*}$$

for all $\psi $ . This gives the result.

Proof of Theorem 6.5.2

Fix $\phi $ as in the statement of the theorem, and consider the linear form

$$\begin{align*}f(-)=\operatorname{\mathrm{tr}}(\phi | \operatorname{\mathrm{Mant}}_{b,\mu}(-)) - \left[\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} (\operatorname{\mathrm{tr.dist}} - )\right](\phi) \end{align*}$$

on $\operatorname {\mathrm {Groth}}(G_b(F))$ . By Theorem 6.4.9, we know that $f(\rho )=0$ if $\rho $ admits a lattice. We need to show that f vanishes identically.

The key observation is that f is a trace form. Indeed, $\operatorname {\mathrm {tr}}(\phi | \operatorname {\mathrm {Mant}}_{b,\mu }(-))$ is a trace form by Theorem 6.5.3. Moreover, $\left [\mathcal {T}_{b,\mu }^{\kern3pt G_b\to G} (\operatorname {\mathrm {tr.dist}} - )\right ](\phi )$ is a trace form since we can rewrite $\left [\mathcal {T}_{b,\mu }^{\kern3pt G_b\to G} (\operatorname {\mathrm {tr.dist}} \rho )\right ](\phi )$ as the trace of $\widetilde {T}_{b,\mu }^{G\to G_b}(\phi ) \in C_c(G_b(F)_{\operatorname {\mathrm {ell}}},\overline {\mathbf {Q}_{\ell }})_{G_b(F)}$ acting on $\rho $ . Thus, f is a difference of trace forms and hence a trace form. Since $f(\rho )=0$ for any $\rho $ admitting a lattice, Lemma 6.5.5 now implies the desired result.

For the final claim about virtual characters, choose compatible $\overline {\mathbf {Q}_{\ell }}$ -valued Haar measures $dg$ and $dg'$ on $G(F)$ and $G_b(F)$ . Fix some $\rho $ , and let be the virtual character of $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho )$ . Pick any $\phi \in C_c(G(F)_{\operatorname {\mathrm {ell}}},\overline {\mathbf {Q}_{\ell }})$ . Then

$$\begin{align*}\operatorname{\mathrm{tr}}(\phi | \operatorname{\mathrm{Mant}}_{b,\mu}(\rho)) = \int_{G(F)} \Xi (g) \phi(g) dg\end{align*}$$

by definition. On the other hand,

$$\begin{align*}\left[\mathcal{T}_{b,\mu}^{\kern3pt G_b\to G} (\operatorname{\mathrm{tr.dist}} \rho )\right](\phi) = \int_{G(F)} T_{b,\mu}^{G_b \to G}(\Theta_{\rho})(g) \phi(g) dg\end{align*}$$

by compatibility of the Haar measures and Proposition 6.3.5. Combining these observations, we get an equality

$$\begin{align*}\int_{G(F)} T_{b,\mu}^{G_b \to G}(\Theta_{\rho})(g) \phi(g) dg = \int_{G(F)} \Xi (g) \phi(g) dg\end{align*}$$

for any $\phi \in C_c(G(F)_{\operatorname {\mathrm {ell}}},\overline {\mathbf {Q}_{\ell }})$ . The result now follows by varying $\phi $ .

Proof of Theorem 6.5.1

The claimed equality in $\operatorname {\mathrm {Groth}}(G(F))$ is an immediate consequence of Theorem 6.5.2 and Theorem 3.2.9.

For the claim regarding the error term, consider the virtual representation

$$\begin{align*}\mathrm{err}=\operatorname{\mathrm{Mant}}_{b,\mu}(\rho)- \sum_{\pi\in \Pi_{\phi}(G)} \dim \operatorname{\mathrm{Hom}}_{S_{\phi}}(\delta_{\pi,\rho},r_{\mu})\pi. \end{align*}$$

By the first half of the theorem, we know that $\mathrm {err}$ is nonelliptic. By Theorem C.1.1, it thus suffices to show that $\mathrm {err}$ is a virtual sum of supercuspidal representations. Since the packet $\Pi _{\phi }(G)$ is supercuspidal by assumption, we’re reduced to showing that $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho )$ is a virtual supercuspidal representation. By definition, this is the Grothendieck class of the complex $A = i_{\mathbf {1}}^{\ast } T_{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}} i_{b \ast } \rho \in D(G(F),\overline {\mathbf {Q}_{\ell }})$ , so we need to see that any irreducible $\tau $ occurring in the Jordan-Hölder series of $H^{\ast } (A)$ is supercuspidal. Since $\varphi _{\tau } = \varphi _{\rho }$ by the commutation of Hecke operators with excursion operators, the claim now follows from the assumption on $\varphi _{\rho }$ and [FS21, Theorem I.9.6.viii].

6.6 Application to inner forms of $\operatorname {\mathrm {GL}}_n$

We give an application to the local Langlands correspondence. Recall that, for any $G/F$ , any $b\in B(G)$ and any $\tau \in \mathrm {Irr}(G_b(F))$ , the construction in [FS21, Proposition I.9.1] (applied to $A= i_{b!} \tau $ ) gives rise to a semisimple L-parameter $\varphi _{\tau }:W_F \to \phantom {}^L G(\overline {\mathbf {Q}_{\ell }})$ associated with $\tau $ . This construction is canonical and satisfies a long list of desirable properties [FS21, Theorem I.9.6]. However, it is a highly nontrivial problem to compare this construction with ‘previously known’ realizations of the local Langlands correspondence.

Theorem 6.6.1. Let G be any inner form of $\mathrm {GL}_n/F$ , and let $\pi $ be an irreducible smooth representation of $G(F)$ . Then the L-parameter $\varphi _{\pi }$ associated with $\pi $ as in [FS21, §I.9] agrees with the usual semisimplified L-parameter attached to $\pi $ .

Proof. By [FS21, Theorem I.9.6.viii], we can assume $\pi $ is supercuspidal. Pick some basic b with $G_b = \mathrm {GL}_{n}/F$ , and let $\rho \in \mathrm {Irr}(G_b(F))$ be the Jacquet–Langlands transfer of $\pi $ [DKV84], so the (usual) semisimple L-parameters of $\rho $ and $\pi $ agree. By [FS21, Theorem 1.9.6.viii-ix], we know that $\varphi _{\rho }$ agrees with the usual semisimple L-parameter of $\rho $ . To prove the theorem, it thus suffices to show that $\varphi _{\pi } = \varphi _{\rho }$ .

Pick any $\mu $ such that $b \in B(G,\mu )$ . By Theorem 6.5.2 and the usual character relation characterizing the Jacquet–Langlands correspondence, we have an equality $\operatorname {\mathrm {Mant}}_{b,\mu }(\rho ) = \dim V_{\mu } \cdot \pi + e$ in $\operatorname {\mathrm {Groth}}(G(F))$ , where e is a nonelliptic virtual representation. Since $\pi $ is supercuspidal, this implies that $\pi $ occurs as a subquotient of some cohomology group of the complex $A = i_{\mathbf {1}}^{\ast } T_{V^{\vee }_{\mu ,\overline {\mathbf {Q}_{\ell }}}} i_{b \ast } \rho \in D(G(F),\overline {\mathbf {Q}_{\ell }})$ . But Hecke operators commute with excursion operators, so $\varphi _{\tau } = \varphi _{\rho }$ for any irreducible $\tau $ occurring in the Jordan–Holder series of $H^{\ast } (A)$ .