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 [Reference Scholze and WeinsteinSW20, §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
$\mathcal {M}_{G,b,\mu ,K}$
[Reference Scholze and WeinsteinSW20, §24]. The latter is a local Shimura variety, whose general existence was conjectured in [Reference Rapoport and ViehmannRV14]. The theory of Rapoport–Zink spaces [Reference Rapoport and ZinkRZ96] 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 [Reference RapoportRap95, Conjecture 5.1], [Reference Rapoport and ViehmannRV14, 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 [Reference Fargues and ScholzeFS21, 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 [Reference ShinShi11], 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 [Reference MantovanMan04]) 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 [Reference KalethaKal16a, 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 [Reference KalethaKal16a, Conjecture G] and the construction of [Reference Fargues and ScholzeFS21], i.e., we do not require that the construction of [Reference Fargues and ScholzeFS21] satisfy any portion of [Reference KalethaKal16a, Conjecture G].
We take this opportunity to give a brief summary of the status of [Reference KalethaKal16a, Conjecture G]. In short, the full conjecture is known for regular supercuspidal parameters [Reference KalethaKal19a, 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 [Reference KalethaKal19a, §5.3] and [Reference Fintzen, Kaletha and SpiceFKS19, §4.4]. However, various parts of that conjecture are known under less restrictive assumptions. To describe this, we remind the reader that [Reference KalethaKal16a, Conjecture G] consists of the following assertions:
-
1. The existence of a finite set
$\Pi _{\phi }$
of representations of rigid inner forms of G for each tempered L-parameter
$\phi $
. -
2. The existence and uniqueness of a generic constituent of
$\Pi _{\phi }$
for a fixed Whittaker datum. -
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. The character identities of ordinary endoscopy, as recalled in Appendix A.
At the moment, a set
$\Pi _{\phi }$
has been constructed in [Reference KalethaKal19b, §§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 [Reference KalethaKal19b] works under weaker assumptions on p). A bijection between
$\Pi _{\phi }$
and
$\mathrm {Irr}(\pi _0(S_{\phi }^+))$
has been constructed in [Reference KalethaKal19b, §§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 [Reference Fintzen, Kaletha and SpiceFKS19, §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 [Reference KottwitzKot84a, 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
$\operatorname {\mathrm {Groth}}(G(F)\times W_E)$
:
$$ \begin{align} \operatorname{\mathrm{Mant}}_{b,\mu}(\rho) = \sum_{\pi \in \Pi_{\phi}(G)} \pi \boxtimes \mathrm{Hom}_{S_{\phi}}(\delta_{\pi,\rho},r_{\mu} \circ \phi_E). \end{align} $$
This conjecture is more general than the formulation of Kottwitz’s conjecture in [Reference RapoportRap95] and [Reference Rapoport and ViehmannRV14], 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 [Reference KalethaKal16a, Conjecture G] based on the cohomology sets
$H^1(u \to W,Z \to G)$
of [Reference KalethaKal16b]. 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 [Reference KalethaKal16a, 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 [Reference Fargues and ScholzeFS21, §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. [Reference ImaiIma] 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 [Reference Fargues and ScholzeFS21, §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 [Reference Scholze and WeinsteinSW13]. 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 [Reference HarrisHar15, Chap. 9] and as far as we know first appears in [Reference FaltingsFal94]. 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 [Reference StrauchStr05], [Reference StrauchStr08], [Reference MiedaMie12], [Reference MiedaMie14a] and also in the case of a basic Rapoport–Zink space for GSp(4) [Reference MiedaMie].
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 [Reference MiedaMie14b] 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 [Reference ShenShe14] 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 [Reference MiedaMie14b].
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 [Reference FarguesFar], followed by the work [Reference Fargues and ScholzeFS21]. 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 [Reference Fargues and ScholzeFS21] 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 [Reference Fargues and ScholzeFS21] 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 [Reference Lu and ZhengLZ22] 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, [Reference Lu and ZhengLZ22] 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 [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
$$ \begin{align} \delta_{\pi,\rho} := \check\tau_{b^*,\mathfrak{w},\pi} \otimes\tau_{b^*,\mathfrak{w},\rho}. \end{align} $$
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
$$ \begin{align} g^{-1} z_b(w)w(g) = b \cdot \sigma(b) \cdots \sigma^{|w|-1}(b) \qquad \forall w \in W_F \to \mathcal{E}, \end{align} $$
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
$\widehat G_{\mathrm {sc}} \to \widehat G_{\mathrm {der}}$
.
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 [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)]
$$ \begin{align} S_{\phi}^+ \to S_{\phi},\qquad (a,(b_n)) \mapsto \frac{a_{\mathrm{der}} \cdot b_1}{N_{E/F}(b_{[E:F]})}. \end{align} $$
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
$\operatorname {\mathrm {Rep}}(\pi _0(S_{\phi }^+),\lambda _{z_b}) \to \operatorname {\mathrm {Rep}}(S_{\phi },\lambda _b)$
.
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 [Reference KalethaKal18, §4.2].
Lemma 2.3.3. Assume [Reference KalethaKal16a, 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 [Reference KalethaKal16a, 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 [Reference KalethaKal13], the proof of which is valid for a general G that satisfies [Reference KalethaKal16a, 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 [Reference KalethaKal18, Lemma 6.2].
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. S-points of
$\operatorname {\mathrm {Spd}} F$
, -
2. Untilts
$S^{\sharp }$
of S over F, -
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 $
.
Remark 2.4.2. According to [Reference Scholze and WeinsteinSW20, 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 [Reference Scholze and WeinsteinSW20] to denote the infinite level structure.
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 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 [Reference KalethaKal14].
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,
$H^1(\breve {F},T)=0$
[Reference SteinbergSte65, 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
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