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 [Reference KottwitzKot97, §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 [Reference Rapoport and ZinkRZ96, 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 [Reference KottwitzKot85, §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 [Reference KottwitzKot97, §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 [Reference KottwitzKot85, Proposition 5.6]. Assuming the validity of the refined local Langlands conjecture [Reference KalethaKal16a, 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 [Reference RapoportRap95, Conjecture 5.1] and [Reference Rapoport and ViehmannRV14, 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 [Reference KalethaKal16a, Conjecture F], which is implied by the rigid formulation [Reference KalethaKal16a, Conjecture G] according to [Reference KalethaKal18], 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 [Reference KottwitzKot85, 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 [Reference KalethaKal16a, 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 [Reference KalethaKal16a, 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 [Reference KalethaKal16b] and [Reference KalethaKal18], some of which is summarized in [Reference KalethaKal16a]. First, we will need the cohomology set $H^1(u \to W,Z \to G^*)$ defined in [Reference KalethaKal16b, §3] for any finite central subgroup $Z \subset G^*$ defined over F. As in [Reference KalethaKal18, §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 [Reference KottwitzKot], 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 [Reference KalethaKal18, §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 [Reference KottwitzKot97, 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 [Reference KottwitzKot97, §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 [Reference KalethaKal18, §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 [Reference KalethaKal16b, 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 [Reference KalethaKal16a, 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 [Reference KalethaKal18] and work with a canonical infinite cover, namely the preimage of $S_{\phi }$ in the universal cover of $\widehat G$ . Following [Reference KalethaKal18, §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 $\widehat G_{\mathrm {sc}} \to \widehat G_{\mathrm {der}}$ .

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 [Reference KalethaKal16b, 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 [Reference KalethaKal16a, 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 [Reference KalethaKal18, (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 [Reference KalethaKal18, 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 $\operatorname {\mathrm {Rep}}(\pi _0(S_{\phi }^+),\lambda _{z_b}) \to \operatorname {\mathrm {Rep}}(S_{\phi },\lambda _b)$ .

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 [Reference KalethaKal18, §4.2].

2.4 Spaces of local shtukas and their cohomology

We recall here some material from [Reference Scholze and WeinsteinSW20] and [Reference FarguesFar] 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 $X_S$ [Reference Fargues and FontaineFF18], 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 [Reference FarguesFar20].

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 [Reference Scholze and WeinsteinSW20, §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 $ .

We will use the cohomology theory developed in [Reference ScholzeSch17]. 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 [Reference Scholze and WeinsteinSW20, §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 [Reference Rapoport and ViehmannRV14].

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 $\mathcal {M}_{G,b,\mu ,K}$ [Reference Scholze and WeinsteinSW20, §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 [Reference Rapoport and ViehmannRV14], $H^*(G,b,\mu )[\rho ]$ 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 [Reference Fargues and ScholzeFS21]. 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 [Reference Rapoport and ViehmannRV14, Conjecture 7.3] has been absorbed into the normalization of $\operatorname {\mathrm {Mant}}_{b,\mu }$ .

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.

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 [Reference KalethaKal14].

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 3.2.4. 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.

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.

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., [Reference KalethaKal16a, 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 [Reference KottwitzKot86, 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).

4.1 Decent v-stacks and the six-functor formalism

We recall here some material from [Reference ScholzeSch17] and [Reference Gulotta, Hansen and WeinsteinGHW22] 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 [Reference ScholzeSch17, 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 [Reference ScholzeSch17, Proposition 11.9]. A particularly well-behaved class of diamonds is locally spatial diamonds [Reference ScholzeSch17, 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 [Reference ScholzeSch17, 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.

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 $D_{\mathrm {\acute {e}t}}(X,\Lambda )$ [Reference ScholzeSch17, 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 [Reference ScholzeSch17, Proposition 14.15].

The familiar six functors of Grothendieck have analogues in the world of small v-stacks [Reference ScholzeSch17, 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.

If $f\colon Y\to X$ is representable in nice diamonds, then there is an adjoint pair of functors $Rf_!$ and $Rf^!$ [Reference ScholzeSch17, Sections 22 and 23].

We also need the following base change results.

Theorem 4.1.5 [Reference ScholzeSch17, 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 [Reference ScholzeSch17, 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 [Reference ScholzeSch17, 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.

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 [Reference ScholzeSch17]. In the companion paper [Reference Gulotta, Hansen and WeinsteinGHW22], we have extended the $!$ -functor formalism to certain stacky maps between certain small v-stacks, using the $\infty $ -categorical machinery of [Reference Liu and ZhengLZ]. Here, we briefly recall the main results from [Reference Gulotta, Hansen and WeinsteinGHW22], referring the reader to that paper for a more detailed discussion.

Definition 4.1.7 [Reference Gulotta, Hansen and WeinsteinGHW22, 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 [Reference Gulotta, Hansen and WeinsteinGHW22, 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 [Reference Gulotta, Hansen and WeinsteinGHW22, 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 [Reference Gulotta, Hansen and WeinsteinGHW22], 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 [Reference Fargues and ScholzeFS21].

• • 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.

Again, we refer the reader to [Reference Gulotta, Hansen and WeinsteinGHW22] 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 sectionsFootnote ²

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

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

We record the following lemmas for convenience.

Lemma 4.1.11. 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.