3.1 The stack of vector bundles on the Fargues-Fontaine curve

Let

$$\begin{align*}\mathrm{Bun}_n \end{align*}$$

be the small v-stack sending $S\in \mathrm {Perf}_k$ to the groupoid of vector bundles of rank n on the Fargues–Fontaine curve $X_S=X_{S,E}$ relative to S (and the local field E). The geometry of the stack $\mathrm {Bun}_n$ has been extensively studied in [Reference Fargues and Scholze11]. Let us recall in particular the following important facts:

1. (1) There is a map $G(\breve {E})\to \mathrm {Bun}_{n}(k),\ b\mapsto \mathcal {E}_b$ inducing a bijection

   $$\begin{align*}B(G):=G(\breve{E})/{\sigma-\text{conjugacy}}\cong |\mathrm{Bun}_n|. \end{align*}$$

   Here, $\mathrm {Bun}_n(k)$ denotes groupoid of the global sections of $\mathrm {Bun}_n$ (i.e., the groupoid of morphisms $\mathrm {Spd}(k)\to \mathrm {Bun}_n$ of stacks over $\mathrm {Perf}_k$ ).

2. (2) For $b\in B(G)$ , write

   $$\begin{align*}\mathrm{Bun}_n^b \end{align*}$$

   for the (locally closed) substack of vector bundles, which are v-locally isomorphic to $\mathcal {E}_b$ . The locally closed inclusions

   $$\begin{align*}j_b\colon \mathrm{Bun}_n^b\hookrightarrow \mathrm{Bun}_n \end{align*}$$

   induce a stratification of $\mathrm {Bun}_n$ .

3. (3) The vector bundle $\mathcal {E}_b$ is semistable if and only if b is basic. The semistable locus $\mathrm {Bun}_n^{\mathrm {ss}}$ is open in $\mathrm {Bun}_n$ and has the description

   $$\begin{align*}\coprod\limits_{d\in \mathbb{Z}}[\ast/\underline{G_b(E)}]\cong \coprod\limits_{b\in B(G) \text{ basic }} \mathrm{Bun}_n^b=\mathrm{Bun}_n^{\mathrm{ss}}, \end{align*}$$
   where $\kappa (b)=\mathrm {deg}(\mathcal {E}_b)=d$ (with $\kappa $ refers to the Kottwitz map, [Reference Fargues8, Section 8]), $G_b$ is the $\sigma $ -stabilizer of b (an inner form of G) and
   $$\begin{align*}\underline{G_b(E)}\cong \mathrm{Aut}(\mathcal{E}_b) \end{align*}$$
   is the sheaf on $\mathrm {Perf}_k$ associated to the topological group $G_b(E)$ . We denote by
   $$\begin{align*}j\colon \mathrm{Bun}_n^{\mathrm{ss}} \hookrightarrow \mathrm{Bun}_n \end{align*}$$
   the open immersion of the semistable locus in $\mathrm {Bun}_n$ .

For each perfectoid space S, each $\mathcal {E}\in \mathrm {Bun}_m(S),\ m\geq 0$ , gives rise to a small v-sheafFootnote ²

$$\begin{align*}\mathrm{BC}(\mathcal{E})\colon \mathrm{Perf}_S\to (E-v.s.),\ T\mapsto H^0(X_{T,E},\mathcal{E}_T). \end{align*}$$

Set

$$\begin{align*}\mathrm{Div}^d:=(\mathrm{BC}(\mathcal{O}(d))\setminus \{0\})/\underline{E^\times}, \end{align*}$$

where $\mathcal {O}(d)\in \mathrm {Bun}_1(k)$ is the line bundle associated to $b=\pi ^{-d}\in \mathrm {GL}_1(\breve {E})$ . The v-sheaf $\mathrm {Div}^d$ parametrizes ‘relative Cartier effective divisors of $X_{E}$ of degree d’. As v-sheaves,

$$\begin{align*}\mathrm{Div}^d=(\mathrm{Div}^1)^d/{S_d}, \end{align*}$$

where $S_d$ denotes the symmetric group on d letters. When $d=1$ , the correspondence between classical points on the Fargues-Fontaine curve and untilts provide an explicit description of $\mathrm {Div}^1$ , cf. [Reference Fargues and Scholze11, Definition II.1.19]. Namely,

$$\begin{align*}\mathrm{Div}^1=\mathrm{Spd}(\breve{E})/\varphi_{\mathrm{Spd}(\breve{E})}^{\mathbb{Z}}. \end{align*}$$

This description shows in particular that the category of finite rank L-local systems on $\mathrm {Div}^1$ is equivalent to the category of finite degree continuous L-representations of the Weil group $W_E$ (here, continuity is understood with respect to the topology on L defined as the colimit topology of the $\ell $ -adic topology on its finitely generated $\mathbb {Z}_\ell $ -submodules); see [Reference Fargues10, Proposition 3.1]. For this reason, we will largely not differentiate in the notation between representations of $W_E$ and the associated sheaves on $\mathrm {Div}^1$ .

3.2 The category of lisse $\ell $ -adic sheaves

Defining the correct category of sheaves on the geometric objects we just introduced is a delicate problem. For a small Artin v-stack Y,Footnote ³ Fargues-Scholze define full subcategories

$$\begin{align*}D_{\text{lis}}(Y,L)\subseteq D_{\blacksquare}(Y,L) \subseteq D(Y_v,L) \end{align*}$$

and various operations on them. The first one will be the more relevant to us, but is constructed through the intermediate category $D_{\blacksquare }(Y,L)$ of solid sheaves of L-modules on Y. Here again, we refer the reader to [Reference Fargues and Scholze11] – in particular, [Reference Fargues and Scholze11, Chapter VII] – for a detailed discussion. Let us only recall the following key properties:

1. (1) If $\ell ^mL=0$ for some $m\geq 1$ and Y admits a separated $\ell $ -cohomologically smooth surjection from a locally spatial diamond whose étale site has bounded $\ell $ -cohomological dimension, then

   $$\begin{align*}D_{\text{lis}}(Y,L)=D_{\mathrm{{\acute{e}t}}}(Y,L), \end{align*}$$
   with the right-hand side the category defined in [Reference Scholze24].

2. (2) Excision holds on $\mathrm {Bun}_n$ (i.e., $D_{\text {lis}}(\mathrm {Bun}_n,L)$ admits a(n infinite) semi-orthogonal decomposition by the subcategories $D_{\text {lis}}(\mathrm {Bun}_n^b,L), b\in B(G)$ ).

3. (3) For $b\in B(G)$ , there are equivalences

   $$\begin{align*}\begin{matrix} D(\mathrm{Rep}_{L}^\infty G_b(E)) & \cong & D_{\text{lis}}([\ast/\underline{G_b(E)}],L)&\cong & D_{\text{lis}}(\mathrm{Bun}_n^b,L) \\ \pi& &\mapsto & & \mathcal{F}_\pi \end{matrix} \end{align*}$$
   with $\mathrm {Rep}_{L}^\infty G_b(E)$ the category of smooth L-representations of $G_b(E)$ . Hence, the category $D_{\text {lis}}(\mathrm {Bun}_n,L)$ provides a natural way to geometrize the category of smooth representations on L-modules of $G(E)$ and its inner forms.

4. (4) Let $f: Y^{\prime } \to Y$ be a morphism of small Artin v-stacks. The usual pullback functor $f_v^\ast $ attached to the morphism $f_v\colon Y^{\prime }_v\to Y_v$ induces functors

   $$\begin{align*}f^\ast\colon D_\blacksquare(Y,L)\to D_\blacksquare(Y^{\prime},L), ~ f^\ast\colon D_{\text{lis}}(Y,L)\to D_{\text{lis}}(Y^{\prime},L). \end{align*}$$

5. (5) Let $f: Y^{\prime } \to Y$ be a morphism of small v-stacks. The functor $f^\ast \colon D_\blacksquare (Y,L)\to D_\blacksquare (Y^{\prime },L)$ has a left adjoint

   $$ \begin{align*}f_\natural \colon D_\blacksquare(Y^{\prime},L)\to D_\blacksquare(Y,L). \end{align*} $$

   Moreover, if f is $\ell $ -cohomologically smooth and representable in locally spatial diamonds, then

   $$\begin{align*}f_\natural(D_{\text{lis}}(Y^{\prime},L))\subseteq D_{\text{lis}}(Y,L). \end{align*}$$

   In this case, we set

   $$\begin{align*}f_!(-):=f_{\natural}(-\otimes_L (f^{!}L)^\vee)\colon D_\blacksquare(Y^{\prime},L)\to D_\blacksquare(Y,L), \end{align*}$$
   where $f^{!}L:=L{\otimes}_{{\mathbb{Z}}_\ell} \varprojlim\limits_{m}f^{!}{\mathbb{Z}}/\ell^m\in D_\blacksquare(Y^{\prime},L)$ denotes the dualizing complex. If L is $\ell ^m$ -torsion for some $m\geq 1$ (and f representable in locally spatial diamonds), then this recovers the functor $f_{!}$ constructed in [Reference Scholze24, Section 22].

6. (6) If $f\colon [\ast /\underline {G_b(E)}]\to \ast $ , then $f_\natural $ identifies with group homology of smooth L-representations of $G_b(E)$ .

3.3 Statement of Fargues’ conjecture for $\mathrm {GL}_n$

After these preliminaries, we can now state Fargues’ conjecture, in the special case relevant to us.

Remark 3.2. Note that (2), (3) force

(*) $$ \begin{align} \mathrm{Aut}_{\mathbb{L}} \cong \bigoplus\limits_{b\in B(G) \text{ basic}} j_{b,!}(\mathcal{F}_{\mathrm{LL}_b({\mathbb{L}})}); \end{align} $$

hence, $\mathrm {Aut}_{\mathbb {L}}$ is unique up to isomorphism. One could take this formula as a possible definition of $\mathrm {Aut}_{\mathbb {L}}$ , but this would make the Hecke eigensheaf property (1) hard to verify.

Let us make more precise the meaning of the Hecke eigensheaf property. Fargues’ conjecture is formulated for $L=\overline {\mathbb {Q}}_\ell $ , but the Hecke property makes sense for any $\mathbb {Z}_l$ -algebra L as ring of coefficients, so let us formulate it in this generality. Recall that $\hat {G}=\mathrm {GL}_{n,L}$ . By the geometric Satake equivalence, the category

$$\begin{align*}\mathrm{Rep}(\hat{G}) \end{align*}$$

of finite dimensional (algebraic) representations of $\hat {G}$ acts on $D_{\text {lis}}(\mathrm {Bun}_n,L)$ (cf. [Reference Fargues and Scholze11, Proposition IX.2.1.]). Even better, for any finite set I, there exists a functor

$$\begin{align*}T^I\colon \mathrm{Rep}(\hat{G})^{I}\times D_{\text{lis}}(\mathrm{Bun}_n,L)\to D_{\blacksquare}(\mathrm{Bun}_n\times (\mathrm{Div}^1)^I,L),\ (V,\mathcal{F})\mapsto T_{V}^I(\mathcal{F}), \end{align*}$$

depending functorially on the finite set I.

For an n-dimensional $\ell $ -adic representation $\mathbb {L}$ of $W_{E}$ and a(n algebraic) representation $\hat {G}_L \to \mathrm {GL}(V)$ , set $r_{V,\ast } (\mathbb {L})$ as the composition

$$\begin{align*}W_{E}\to \mathrm{GL}(\mathbb{L})\cong \hat{G}(L)\to \mathrm{GL}(V). \end{align*}$$

Now, $\mathcal {F}\in D_{\text {lis}}(\mathrm {Bun}_n,\overline {\mathbb {Q}}_\ell )$ is a Hecke eigensheaf with eigenvalue $\mathbb {L}$ if for any finite set I and any collection $(V_i)_{i\in I}$ , we are given an isomorphism

$$\begin{align*}\eta_{(V_i)_{i\in I}}\colon T_{(V_i)_{i\in I}}^I(\mathcal{F})\cong \mathcal{F}\boxtimes (\boxtimes_{i\in I} r_{V_i,\ast}(\mathbb{L})), \end{align*}$$

which is natural in I, $(V_i)_{i\in I}$ , and compatible with composition/exterior tensor product. For example, if $I=\{1\}$ and $V_{\mathrm {std}}$ the standard representation, then

$$\begin{align*}\eta_{V_{\mathrm{st}}}\colon T_{V_{\mathrm{st}}}(\mathcal{F})\cong \mathcal{F}\boxtimes \mathbb{L}. \end{align*}$$

The meaning of the cuspidality condition in condition (2) of the conjecture will be clarified below (see Definition 4.3).

3.4 Fargues-Scholze’s construction of the spectral action

In this short subsection, we recall one of the main results of [Reference Fargues and Scholze11] (specialized to the case $G=\mathrm {GL}_n$ ), since we will use it later.

As in the last section, we denote $\hat {G}=\mathrm {GL}_{n,L}$ . For each finite set I and each $V \in \mathrm {Rep}(\hat {G})^{I}$ , the essential image of the Hecke operator

$$ \begin{align*}T_V^I: D_{\text{lis}}(\mathrm{Bun}_n,L) \to D_\blacksquare(\mathrm{Bun}_n \times (\mathrm{Div}^1)^I, L) \end{align*} $$

introduced above lies inside the essential image of $D_\blacksquare (\mathrm {Bun}_n \times [\ast /W_E^I], L)$ (recall the map $\mathrm {Div}^1=[\mathrm {Spd}(C)/W_E] \to [\ast /W_E]$ ) and its pullback to $D_\blacksquare (\mathrm {Bun}_n,L)$ lies in $D_{\text {lis}}(\mathrm {Bun}_n,L)$ . In other words, $T_V^I$ can be seen as a functor

$$ \begin{align*}T_V^I: D_{\text{lis}}(\mathrm{Bun}_n,L) \to D_{\text{lis}}(\mathrm{Bun}_n,L)^{W_E^I} \end{align*} $$

into the category of objects in $D_{\text {lis}}(\mathrm {Bun}_n,L)$ endowed with a $W_E^I$ -action. This follows by induction from the key invariance

$$ \begin{align*}D_{\text{lis}}(\mathrm{Bun}_n,L) \cong D_{\text{lis}}(\mathrm{Bun}_n \times \mathrm{Spd}(C),L), \end{align*} $$

which is itself a consequence of the semi-orthogonal decomposition of $D_{\text {lis}}(\mathrm {Bun}_n,L)$ by the $D_{\text {lis}}(\mathrm {Bun}_n^b,L)$ , $b \in B(G)$ (point (2) of Section 3.2) and the analogous invariance statement for the classifying stacks $[\ast /G_b(E)]$ (via point (2) of Section 3.2).

The existence of such a functorial collection of operators $T_V^I$ , which is already enough to provide the construction of semi-simple L-parameters attached to smooth irreducible representations of the groups $G_b(E)$ – see [Reference Fargues and Scholze11, Chapter VIII.4.] – can be reformulated in a different, more geometric, manner.

Let

$$\begin{align*}X_{\hat{G}} \end{align*}$$

be the algebraic stack over L of n-dimensional continuous L-representations of $W_E$ , defined and studied in [Reference Fargues and Scholze11, VIII.1] or [Reference Dat, Helm, Kurinczuk and Moss4], [Reference Zhu27]. The algebraic stack $X_{\hat {G}}$ comes with a map

$$ \begin{align*}f\colon X_{\hat{G}} \to [\mathrm{Spec}(L)/\hat{G}]. \end{align*} $$

Note that $X_{\hat {G}}$ is an infinite disjoint union of finite type Artin stacks. Therefore, it could be seen as a stack locally of finite type, but we rather want to see it as an ind-finite type stack. In particular, coherent sheaves on it are required to have support on finitely many connected components, and thus, we see, for example, the structure sheaf of $X_{\hat {G}}$ not as a vector bundle, but as an ind-vector bundle (cf. [Reference Zhu27, Remark 3.1.7]).

The following is [Reference Fargues and Scholze11, Theorem X.0.1].

Theorem 3.3. The geometric Hecke action gives rise to an L-linear monoidal action of the category

$$ \begin{align*}\mathrm{Perf}(X_{\hat{G}}) \end{align*} $$

of perfect complexes on $X_{\hat {G}}$ on the full sucategory $D_{\text {lis}}(\mathrm {Bun}_n,L)^{\omega }$ of $D_{\text {lis}}(\mathrm {Bun}_n,L)$ formed by compact objects, such that for any finite set I, and any $V=(V_i)_{i\in I} \in \mathrm {Rep}(\hat {G})^I$ , the functor $T_V^I$ is given by the action of the perfect complex

$$ \begin{align*}\bigotimes_{i\in I} f^\ast V_i, \end{align*} $$

where for each $i\in I$ , we denoted again by $V_i$ the vector bundle on $[\mathrm {Spec}(L)/\hat {G}]$ attached to $V_i$ .

Passing to ind-completions (recall that $D_{\text {lis}}(\mathrm {Bun}_n,L)$ is compactly generated, [Reference Fargues and Scholze11, Proposition VII.7.4]), one gets an L-linear monoidal action of $\mathrm {IndPerf}(X_{\hat {G}})$ on $D_{\text {lis}}(\mathrm {Bun}_n,L)$ .

We will use the following notation. If $A \in \mathrm {IndPerf}(X_{\hat {G}})$ , and $\mathcal {F} \in D_{\text {lis}}(\mathrm {Bun}_n,L)$ , we will denote by

$$ \begin{align*}A \ast \mathcal{F} \end{align*} $$

the object in $D_{\text {lis}}(\mathrm {Bun}_n,L)$ obtained by letting A act on $\mathcal {F}$ .

We note that if $V\in \mathrm {Rep}(\hat {G})$ , then $f^\ast V\in \mathrm {Perf}(X_{\hat {G}})$ has a canonical $W_E$ -action, and the isomorphism

$$\begin{align*}f^\ast V\ast \mathcal{F}\cong T_V(\mathcal{F}) \end{align*}$$

is $W_E$ -equivariant for each $\mathcal {F}\in D_{\text {lis}}(\mathrm {Bun}_G,L)$ .

4.1 Constant terms and twisting

Let $P\subseteq G$ be a parabolic with Levi quotient M. As the kernel of

$$\begin{align*}\mathrm{det}\colon G\to {\mathbb{G}_m}=\mathrm{GL}_1 \end{align*}$$

contains the unipotent radical of P, we get a canonical factorization

$$\begin{align*}\mathrm{det}_M\colon M\to {\mathbb{G}_m}, \end{align*}$$

whose restriction to P agrees with the restriction $\mathrm {det}_P\colon P\to {\mathbb {G}_m}$ of $\mathrm {det}$ . From this, we can deduce the existence of a commutative diagram

(1)

We obtain the following compatibility of constant term functors with ‘twisting by characters’.

Lemma 4.4. Let $\mathcal {G}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_1,\Lambda )$ . Then for any $\mathcal {F}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ , there is a natural isomorphism

$$\begin{align*}\mathrm{CT}_{P,!}(\mathcal{F}\otimes_\Lambda \mathrm{det}^\ast\mathcal{G})\cong \mathrm{CT}_{P,!}(\mathcal{F})\otimes_\Lambda \mathrm{det}^\ast_M \mathcal{G}. \end{align*}$$

In particular, $D_{\mathrm {{\acute {e}t}},\mathrm {cusp}}(\mathrm {Bun}_n,\Lambda )$ is stable under the functor $-\otimes _{\Lambda } \mathrm {det}^\ast \mathcal {G}$ .

Proof. Using (Equation (1)), we can calculate

$$\begin{align*}\begin{matrix} & \mathrm{CT}_{P,!}(\mathcal{F}\otimes_\Lambda \mathrm{det}^\ast\mathcal{G}) \\ \cong & Rq_!(p^\ast(\mathcal{F}\otimes_\Lambda \mathrm{det}^\ast\mathcal{G})) \\ \cong & Rq_!(p^\ast(\mathcal{F})\otimes_\Lambda \mathrm{det}^\ast_P\mathcal{G})) \\ \cong & Rq_!(p^\ast(\mathcal{F})\otimes_\Lambda q^\ast\mathrm{det}^\ast_M \mathcal{G})) \\ \cong & \mathrm{CT}_{P,!}(\mathcal{F})\otimes_{\Lambda}\mathrm{det}_M^\ast\mathcal{G}, \end{matrix} \end{align*}$$

where the last isomorphism comes from the projection formula.

Proof. From the proofs of Lemma 4.1 and Lemma 4.2, we can conclude that

$$\begin{align*}\Lambda\to q^!Rq_!\Lambda \end{align*}$$

is an isomorphism. Indeed, by Lemma 4.1, Lemma 4.2 $q^!$ commutes with base change to fibers. Moreover, the fibers are iterated quotients of positive Banach-Colmez spaces by positive Banach-Colmez spaces. Therefore, the claim follows from [Reference Fargues and Scholze11, V.2.1] (and its proof). By descent, we get that for each parabolic $P\subseteq G$ , the object

$$\begin{align*}\mathrm{CT}_{P,!}(\Lambda) \end{align*}$$

is a $\Lambda $ -local system of rank $1$ on $\mathrm {Bun}_M$ . If $n\geq 2$ , we can choose some proper parabolic $P\subseteq G$ and use Lemma 4.4 to see

$$\begin{align*}0=\mathrm{CT}_{P,!}(\mathrm{det}^\ast \mathcal{G})=\mathrm{CT}_{P,!}(\Lambda)\otimes_{\Lambda} \mathrm{det}^\ast_M \mathcal{G}. \end{align*}$$

Thus, we obtain that $\mathrm {det}^\ast _M \mathcal {G}=0$ , which implies $\mathcal {G}=0$ as $\mathrm {Bun}_M\to \mathrm {Bun}_1$ is surjective.

If P is a Borel, the dualizing complex on $\mathrm {Bun}_P$ has been calculated in [Reference Hamannn16, Theorem 1.2], and for general P in [Reference Hamann and Imai17, Theorem 1.6].

4.2 Geometrically cuspidal representations

The next results give some properties of cuspidal objects in $D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ .

Proof. Let $b\in B(G)$ be a non-semistable point and let M be the standard Levi associated to $G_b$ . In particular, we can view $b\in B(M)$ . Let P be the standard parabolic containing M. In fact, we can choose a representative $b\in G(E)$ in such a way that the fiber of $\mathrm {Bun}_P\to \mathrm {Bun}_{M}$ over the M-torsor given by the associated graded of the HN-filtration on $\mathcal {E}_b$ maps isomorphically to $\mathrm {Bun}_{n}$ with image $\mathrm {Bun}^b_n$ as the Harder-Narasimhan filtration is canonical. Moreover, $!$ -pull back along the morphism $\mathrm {Bun}_P^b:=\mathrm {Bun}_P\times _{\mathrm {Bun}_M} \mathrm {Bun}_M^b \to \mathrm {Bun}^b_M$ defines an equivalence

(2) $$ \begin{align} D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_M^b,\Lambda)\cong D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_P^b,\Lambda) \end{align} $$

because the fiber is a classifying stack for a positive Banach-Colmez space (cf. [Reference Fargues and Scholze11, Proposition V.2.1]). This implies that the image of $j_{b,\ast }(D_{\mathrm {{\acute {e}t}}}([\ast /G_b(E)],\Lambda ))$ is contained in the image of $\mathrm {Eis}_{P,\ast }(D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_M,\Lambda ))$ : indeed, let $\mathcal {G} \in D_{\mathrm {{\acute {e}t}}}([\ast /G_b(E)],\Lambda )$ , which by our choice of b, we can see as an object $\mathcal {G}_P \in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_P^b,\Lambda )$ . Let $\mathcal {G}_M \in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_M^b,\Lambda )$ , corresponding to $\mathcal {G}_P$ by (2). Using smoothness of q and proper base change ([Reference Scholze24, Proposition 23.16 (i)]), one sees that the $!$ -pushforward of $\mathcal {G}_M$ to $\mathrm {Bun}_M$ has image $j_{b,\ast }\mathcal {G}$ by $\mathrm {Eis}_{P,\ast }$ .

Now let $\mathcal {F}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ be cuspidal. Then by definition,

$$ \begin{align*}\mathrm{CT}_{P,!}(\mathcal{F})=0. \end{align*} $$

Let $\mathcal {G}\in D_{\mathrm {{\acute {e}t}}}([\ast /G_b(E)],\Lambda )$ . As $j_{b,\ast }(D_{\mathrm {{\acute {e}t}}}([\ast /G_b(E)],\Lambda ))$ is in the image of $\mathrm {Eis}_{P,\ast }$ ,

$$ \begin{align*}j_{b,\ast}(\mathcal{G}) = \mathrm{Eis}_{P,\ast}(\mathcal{G}^{\prime}), \end{align*} $$

for some $\mathcal {G}^{\prime } \in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_M,\Lambda )$ . Hence, by adjunction,

$$ \begin{align*}\mathrm{Hom}(j_b^\ast \mathcal{F},\mathcal{G})=\mathrm{Hom}(\mathrm{CT}_{P,!}(\mathcal{F}), \mathcal{G}^{\prime})=0. \end{align*} $$

Since this holds for all $\mathcal {G}\in D_{\mathrm {{\acute {e}t}}}([\ast /G_b(E)],\Lambda )$ , this implies $j_{b}^\ast \mathcal {F}=0$ , as desired.

Let $b\in B(G)$ be a basic class. Recall that a smooth representation $\pi $ of $G_b(E)$ is said to be cuspidal if it has vanishing Jacquet modules for all choices of proper parabolic subgroups. Recall also from Section 3.2 that we denote by

$$ \begin{align*}\mathcal{F}_\pi\in D_{\mathrm{{\acute{e}t}}}([\ast/G_b(E)],\Lambda) \end{align*} $$

the sheaf associated to a representation $\pi \in \mathrm {Rep}^\infty _\Lambda G_b(E)$ .

We introduce the following convenient terminology.

Proof. Let $P_b\subseteq G_b(E)$ be a parabolic subgroup with Levi quotient $M_b$ . Recall that G and $G_b$ are pure inner forms over X. Thus, as G is quasi-split, we can find a parabolic subgroup P with Levi quotient M such that

$$ \begin{align*}\mathrm{Bun}_{P} \cong \mathrm{Bun}_{P_b} ~~ , ~~ \mathrm{Bun}_M \cong \mathrm{Bun}_{M_b}. \end{align*} $$

Let $c\in \mathrm {Bun}_{M}$ be the point determined by the trivial $M_b$ -torsor under the equivalence $\mathrm {Bun}_{M_b}\cong \mathrm {Bun}_M$ . Then

$$ \begin{align*}\mathrm{Bun}_{P}^c:=\mathrm{Bun}_{P}\times_{\mathrm{Bun}_{M}} \mathrm{Bun}_{M}^c\cong[\ast/P_b]\to \mathrm{Bun}_n \end{align*} $$

has image in $\mathrm {Bun}_n^b$ . The resulting diagram identifies with

$$ \begin{align*}[\ast/G_b(E)]\xleftarrow{p} [\ast/P_b] \xrightarrow{q} [\ast/M_b], \end{align*} $$

and thus, $\mathrm {CT}_{P,!}$ restricted to c coincides with the Jacquet functor for $P_b$ . This finishes the proof.

We end with a result in the special case $n=2$ .

Proof. Assume first that $\pi $ is supercuspidal and let us prove that $\pi $ is geometrically cuspidal. For a fixed integer $m\geq d/2$ , consider the space

$$\begin{align*}Z_{{m}}:=\mathcal{H}om(\mathcal{E}_b,\mathcal{O}({m})) \end{align*}$$

of homomorphisms from $\mathcal {E}_b$ , and its open subspace $Z_{{m}}^\circ \subseteq Z_{{m}}$ parametrizing surjections. Over $Z_{{m}}^\circ $ there is the natural $E^\times $ -torsor $\widetilde {Z_{{m}}^\circ }$ parametrizing isomorphisms of the kernel with $\mathcal {O}(d-{m})$ . We have to prove that

$$\begin{align*}R\Gamma_{\natural}(G_b(E),R\Gamma_c(\widetilde{Z_{{m}}^\circ},\Lambda)\otimes_{{\Lambda}} \pi)=0, \end{align*}$$

since this is by proper base change the stalk of $\mathrm {CT}_{B,!}(j_{b,!}\pi )$ at a geometric point over $\mathcal {O}(d-m)\oplus \mathcal {O}(m)$ . But $\widetilde {Z_{{m}}^\circ }$ is isomorphic to $Z^\circ _{m}\times E^\times $ because giving an isomorphism of the kernel with $\mathcal {O}(d-m)$ is an $E^\times $ -torsor equivalent to the (split) $E^\times $ -torsor of isomorphisms $\mathrm {det}(\mathcal {E}_b)\cong \mathcal {O}(d)$ . In particular, $G_b(E)$ acts on the factor $E^\times $ via the determinant. One can therefore reduce to proving that

$$\begin{align*}R\Gamma_{\natural}(G_b(E),R\Gamma_c(Z_{{m}}^\circ,\Lambda)\otimes_{\Lambda} \chi \otimes_{{\Lambda}} \pi)=0, \end{align*}$$

for any character $\chi $ . For ${m}>d/2$ , this follows by the known cohomology of the punctured BC-space $Z_{{m}}^{\circ }$ (a copy of $\Lambda $ in degree $0$ and $2m-d$ ) and the fact that $R\mathrm {Hom}_{G_b(E)}(\pi ,\chi )=0$ for any character $\chi $ by supercuspidality of $\pi $ (cf. the proof of [Reference Drevon and Sécherre5, Proposition 6.1]).Footnote ⁵ For $m=d/2$ , this follows as supercuspidal representations are cuspidal (i.e., they have vanishing Jacquet functors).

Conversely, this argument shows that if $\pi $ is geometrically cuspidal, it must be cuspidal (case $m=d/2$ ). Moreover, if $\pi $ was not supercuspidal, which can happen only when $q=-1$ modulo $\ell $ , then $\pi $ would be a twist by a character $\chi $ of the representation $\pi _1$ from [Reference Vignéras26, Corollaire 5]. In particular, we would have

$$ \begin{align*}\mathrm{Ext}_{G_b(E)}^1(\pi, (-1)^{\mathrm{val}} \chi)\neq 0, \end{align*} $$

contradicting geometric cuspidality of $\pi $ (case $m>d/2$ ).

Remark 4.11. The argument from Proposition 4.10 can easily be modified to show the following: Assume $d=\kappa (b)$ odd, and let $\pi \in \mathrm {Rep}^\infty _\Lambda (G_b(E))$ be the inflation of a character along the reduced norm $G_b(E)\to E^\times $ . Then $\pi $ is not geometrically cuspidal. Indeed, $G_b(E)$ acts on $\mathrm {det}(\mathcal {E}_b)$ through the reduced norm, and it suffices to note that by the arguments in Proposition 4.10, we have

$$\begin{align*}R\Gamma_{\natural}(G_b(E),R\Gamma_c(\widetilde{Z_{{m}}^\circ},\Lambda)\otimes_{{\Lambda}} \pi)\cong R\Gamma_{\natural}(H,R\Gamma_c(Z_{{m}}^\circ,\Lambda)\otimes_{\Lambda}\pi)\neq 0, \end{align*}$$

where $H:=\mathrm {ker}(G_b(E)\to E^\times )$ .

5.1 The definition of the averaging functors

Let $d,n\geq 0$ and let

$$\begin{align*}\mathrm{Mod}^d_n \end{align*}$$

(standing for modifications) be the small v-stack on $\mathrm {Perf}_k$ sending $S\in \mathrm {Perf}_k$ to the groupoid of the data

$$\begin{align*}(\mathcal{E},\mathcal{E}^{\prime},x,\gamma) \end{align*}$$

with $\mathcal {E},\mathcal {E}^{\prime }\in \mathrm {Bun}_n(S)$ satisfying $\kappa (\mathcal {E}^{\prime })-\kappa (\mathcal {E})=d$ , $x\in \mathrm {Div}^1(S)$ , and

$$\begin{align*}\gamma\colon \mathcal{E}\hookrightarrow \mathcal{E}^{\prime} \end{align*}$$

a morphism of vector bundles, which is fiberwise injective with cokernel supported at x. We will actually only need the case that $d=1$ . Consider the diagram

(3)

with

$$\begin{align*}\begin{matrix} \overleftarrow{h}(\mathcal{E},\mathcal{E}^{\prime},x,\gamma)=\mathcal{E}, \\ \overrightarrow{h}(\mathcal{E},\mathcal{E}^{\prime},x,\gamma)=\mathcal{E}^{\prime}, \\ \end{matrix} \end{align*}$$

and let

$$\begin{align*}\mathbb{L}\in D_{\mathrm{{\acute{e}t}}}(\mathrm{Div}^1,\Lambda) \end{align*}$$

be any object, e.g., the sheaf associated to a $\Lambda $ -representation of the Weil group $W_E$  of E.

Note that $\overleftarrow {h}$ and $\overrightarrow {h}$ are representable in locally spatial diamonds, proper, of finite $\mathrm {dim.trg}$ and cohomologically smooth.

Definition 5.1. We define the averaging functor associated with $\mathbb {L}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Div}^1,\Lambda )$ to be

$$\begin{align*}\mathrm{Av}_{\mathbb{L},n}\colon D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda), ~ \mathcal{F}\mapsto \overrightarrow{h}_!(\overleftarrow{h}^\ast(\mathcal{F}){\otimes}_\Lambda \alpha^{\ast}\mathbb{L}(\frac{n-1}{2})[n-1]) \end{align*}$$

If n is clear from the context, then we write

$$\begin{align*}\mathrm{Av}_{\mathbb{L}}:=\mathrm{Av}_{\mathbb{L},n}. \end{align*}$$

Thus, the averaging functors provide examples of, potentially interesting, endofunctors of $D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ .

5.2 Averaging functors and constant terms

One of the most important properties of the averaging functors is their commutation with constant terms.

Proposition 5.3. Let $P\subseteq G$ be a maximal standard parabolic, and let $M=\mathrm {GL}_{n_1}\times \mathrm {GL}_{n_2}$ be its Levi quotient. Then there is a canonical exact triangle

$$\begin{align*}(\mathrm{Av}_{\mathbb{L}(\frac{n-n_1}{2})[n-n_1],n_1}\times \mathrm{Id}_{\mathrm{Bun}_{n_2}})\circ \mathrm{CT}_{P,!} \to \mathrm{CT}_{P,!}\circ \mathrm{Av}_{\mathbb{L},n}\to ((\mathcal{L}_1\otimes(-)\times \mathrm{Av}_{\mathbb{L}\otimes_{\Lambda} \mathcal{L}_2,n_2})\circ \mathrm{CT}_{P,!} \end{align*}$$

of functors $D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )\to D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_M,\Lambda )=D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_{n_1}\times \mathrm {Bun}_{n_2},\Lambda )$ . Here, $\mathcal {L}_1\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_{n_1},\Lambda ), \mathcal {L}_2\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Div}^1,\Lambda )$ are shifts of $\Lambda $ -local systems of rank $1$ .Footnote ⁶

Here, we used the notation $\mathrm {Av}_{\mathbb {L},n_1}\times \mathrm {Id}_{\mathrm {Bun}_{n_2}}, ((\mathcal {L}_1\otimes (-)\times \mathrm {Av}_{\mathbb {L}\otimes _{\Lambda } \mathcal {L}_2,n_2})$ for the functors arising from the correspondence (with kernel the pullback of $\mathbb {L}$ , $\mathcal {L}_1$ or $\mathbb {L}\otimes _{\Lambda }\mathcal {L}_2$ ) obtained by taking the product of (3) (for $n=n_1$ resp. $n=n_2$ ) with $\mathrm {Bun}_{n_2}$ resp. $\mathrm {Bun}_{n_1}$ . For an analog in the classical case, we refer to [Reference Frenkel, Gaitsgory and Vilonen12, Lemma 9.8].

Proof. Let

$$\begin{align*}Y:=\mathrm{Mod}^1_n\times_{\overrightarrow{h},\mathrm{Bun}_n}\mathrm{Bun}_P \end{align*}$$

be the stack parametrizing injections $\gamma \colon \mathcal {E}\to \mathcal {E}^{\prime }$ of $\mathcal {E},\mathcal {E}^{\prime }\in \mathrm {Bun}_n$ with $\kappa (\mathcal {E}^{\prime })=\kappa (\mathcal {E})+1$ and filtrations

$$\begin{align*}0\to \mathcal{E}^{\prime}_1\to \mathcal{E}^{\prime}\to \mathcal{E}^{\prime}_2\to 0 \end{align*}$$

with $\mathcal {E}^{\prime }_i\in \mathrm {Bun}_{n_i}, i=1,2$ . Let

$$\begin{align*}j\colon U\to Y \end{align*}$$

be the open substack defined by the condition that the composition $\mathcal {E}\to \mathcal {E}^{\prime }\to \mathcal {E}^{\prime }_2$ is surjective. Let

$$\begin{align*}i\colon Z\to Y \end{align*}$$

be the closed complement of $U\subseteq Y$ . Let $\mathcal {K}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n\times \mathrm {Bun}_M,\Lambda )$ be the kernel representing the composition

$$\begin{align*}\mathrm{CT}_{P,!}\circ \mathrm{Av}_{\mathbb{L},n}, \end{align*}$$

that is, $\mathcal {K}$ is the $!$ -pushforward for the morphism

$$\begin{align*}Y\to \mathrm{Bun}_n\times \mathrm{Bun}_M \end{align*}$$

of the pullback $\mathbb {L}_Y\in D_{\mathrm {{\acute {e}t}}}(Y,\Lambda )$ of $\mathbb {L}(\frac {n-1}{2})[n-1]$ along

$$\begin{align*}Y\to \mathrm{Mod}^1_n\to \mathrm{Div}^1. \end{align*}$$

Our aim is to identify the functors

$$\begin{align*}&(\mathrm{Av}_{\mathbb{L}(\frac{n-n_1}{2})[n-n_1],n_1}\times \mathrm{Id}_{\mathrm{Bun}_{n_2}})\\&\quad\circ \mathrm{CT}_{P,!}, (\mathrm{Id}_{\mathrm{Bun}_{n_2}}\times \mathrm{Av}_{\mathbb{L}(\frac{n-n_2}{2})[n-n_2],n_1})\circ \mathrm{CT}_{P,!}\colon D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_M,\Lambda) \end{align*}$$

as the functors with kernels the two outer terms in the triangle

$$\begin{align*}j_!j^\ast \mathcal{K}\to \mathcal{K}\to i_\ast i^\ast \mathcal{K}. \end{align*}$$

This will imply the proposition. We start with identifying the functor with kernel $j_!j^\ast \mathcal {K}$ . It is calculated via the correspondence

with kernel the pullback $\mathbb {L}_U\in D_{\mathrm {{\acute {e}t}}}(U,\Lambda )$ of $\mathbb {L}$ . Over U, the intersection

$$\begin{align*}\mathcal{E}_1:=\mathcal{E}\cap \mathcal{E}_1^{\prime} \end{align*}$$

(i.e., the kernel of the surjection $\mathcal {E}\twoheadrightarrow \mathcal {E}^{\prime }_2$ ) is again a vector bundle. Note that the injection $\mathcal {E}_1\hookrightarrow \mathcal {E}_1^{\prime }$ lies in $\mathrm {Mod}_{n_1}^1$ . The morphism

$$\begin{align*}U\to \mathrm{Mod}^1_{n_1}\times_{\overleftarrow{h},\mathrm{Bun}_{n_1}}\mathrm{Bun}_P \end{align*}$$

sending the above data to $\mathcal {E}_1\hookrightarrow \mathcal {E}_1^{\prime }$ and the filtration

$$\begin{align*}0\to \mathcal{E}_1\to \mathcal{E}\to \mathcal{E}_2^{\prime}\to 0 \end{align*}$$

is an equivalence as necessarily $\mathcal {E}^{\prime }$ is the pushout of the diagram of

$$\begin{align*}\mathcal{E}\leftarrow \mathcal{E}_1\rightarrow \mathcal{E}_1^{\prime}. \end{align*}$$

In particular, we obtain the diagram

with cartesian square. Now, $\mathbb {L}_U\in D_{\mathrm {{\acute {e}t}}}(U,\Lambda )$ is the pullback of $\mathbb {L}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Div}^1,\Lambda )$ along the morphism

$$\begin{align*}U\to \mathrm{Mod}^1_{n_1}\times \mathrm{Bun}_{n_2}\to \mathrm{Div}^1. \end{align*}$$

This implies that the functor induced by $j_!j^\ast \mathcal {K}$ is $(\mathrm {Av}_{\mathbb {L}(\frac {n-n_1}{2})[n-n_1],n_1}\times \mathrm {Id}_{\mathrm {Bun}_{n_2}})\circ \mathrm {CT}_{P,!} $ , as desired. Let us now analyze Z (i.e., the locus of Y where the composition

$$\begin{align*}\mathcal{E}\to \mathcal{E}^{\prime}\to \mathcal{E}^{\prime}_2 \end{align*}$$

is not surjective (over each geometric point)). Over each geometric point the cokernel of $\mathcal {E}\to \mathcal {E}^{\prime }$ is of length $1$ , and supported on one point. Thus,

$$\begin{align*}\mathcal{E}^{\prime}/\mathcal{E} \cong \mathrm{coker}(\mathcal{E}\to \mathcal{E}^{\prime}_2). \end{align*}$$

In particular, $\mathcal {E}^{\prime }_1$ lies in $\mathcal {E}$ as its image in $\mathcal {E}^{\prime }/\mathcal {E}$ is trivial. Moreover, the quotient

$$\begin{align*}\mathcal{E}_2:=\mathcal{E}/\mathcal{E}^{\prime}_1 \end{align*}$$

is again a vector bundle, and the injection $\mathcal {E}_2\to \mathcal {E}_2^{\prime }$ lies in $\mathrm {Mod}^1_{n_2}$ . The map

$$\begin{align*}Z\to \mathrm{Mod}^1_{n_2}\times_{\overrightarrow{h},\mathrm{Bun}_{n_2}}\mathrm{Bun}_P \end{align*}$$

sending the above data to $\mathcal {E}_2\hookrightarrow \mathcal {E}^{\prime }_2$ and $0\to \mathcal {E}_1^{\prime }\to \mathcal {E}^{\prime }\to \mathcal {E}_2^{\prime }\to 0$ is an isomorphism as $\mathcal {E}$ is necessarily the pullback of the diagram

$$\begin{align*}\mathcal{E}^{\prime}\to \mathcal{E}^{\prime}_2\leftarrow \mathcal{E}_2. \end{align*}$$

However, in order to calculate the functor with kernel $i_\ast i^\ast \mathcal {K}$ , we need to understand the $!$ -pushforward of $i^\ast \mathcal {K}$ along the morphism

$$ \begin{align*}\Phi\colon Z\to \mathrm{Bun}_{P}\times_{\mathrm{Bun}_{n_1}\times \mathrm{Bun}_{n_2}} (\mathrm{Bun}_{n_1}\times \mathrm{Mod}^1_{n_2}) \end{align*} $$

with $\mathrm {Mod}^1_{n_2}\to \mathrm {Bun}_{n_2}$ given by $\overleftarrow {h}$ , where the morphism $Z\to \mathrm {Bun}_P$ sends the data in Z to the filtration $0\to \mathcal {E}_1^{\prime }\to \mathcal {E}\to \mathcal {E}_2\to 0$ , and the morphism $Z\to \mathrm {Mod}^1_{n_2}$ sends the data in Z to $\mathcal {E}_2\to \mathcal {E}^{\prime }_2$ . Let us note that $i^\ast \mathcal {K}$ is pulled back from $\mathrm {Div}^1$ , and that the morphism $Z\to \mathrm {Div}^1$ factors through $\Phi $ composed with the projection to $\mathrm {Mod}^1_{n_2}$ . By the projection formula, we can therefore reduce to the case that $\mathbb {L}$ is trivial. As in [Reference Frenkel, Gaitsgory and Vilonen12, Lemma 9.8], the morphism $\Phi $ is a principal homogeneous space under the relative affine space ${H:=\mathrm {Ext}^1(\mathrm {coker}(\mathcal {E}_2\to \mathcal {E}_2^{\prime }), \mathcal {E}_1^{\prime })}$ . It suffices (again by the projection formula) to show that $\Phi _!(\Lambda )$ is the pullback of the exterior tensor product of a $\Lambda $ -local system $\mathcal {L}_1$ on $\mathrm {Bun}_{n_1}$ of rank $1$ with the pullback to $\mathrm {Mod}^1_{n_2}$ of a $\Lambda $ -local system $\mathcal {L}_2$ on $\mathrm {Div}^1$ of rank $1$ . To see this, it suffices to note that for any H-torsor $g\colon T'\to T$ of small v-sheaves $T',T$ over $\mathrm {Bun}_{n_1}\times \mathrm {Div}^1$ the object $g_!(\Lambda )$ is canonically isomorphic to the pullback to T of the $!$ -pushforward of $\Lambda $ along $H\to \mathrm {Bun}_{n_1}\times \mathrm {Div}^1$ (this uses [Reference Fargues and Scholze11, Proposition V.2.1]). In particular, we can conclude that $g_!(\Lambda )$ is a $\Lambda $ -local system of rank $1$ , which is pulled back from $\mathrm {Bun}_{n_1}\times \mathrm {Div}^1$ (because H is a relative affine space). By [Reference Fargues and Scholze11, Proposition IV.7.3], we can conclude that this $\Lambda $ -local system is the exterior tensor product of local systems $\mathcal {L}_1, \mathcal {L}_2$ of rank $1$ on $\mathrm {Bun}_{n_1}$ and $\mathrm {Div}^1$ . This finishes the proof.

Remark 5.4. For $m\geq 1$ , let

$$\begin{align*}V_{\mathrm{std},m}\in \mathrm{Rep}_\Lambda (\mathrm{GL}_{m,\Lambda}) \end{align*}$$

be the standard representation, with associated Hecke functor

$$\begin{align*}T_{\mathrm{std},m}\colon D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_m,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_m\times \mathrm{Div}^1,\Lambda). \end{align*}$$

With the same strategy as in the proof of Proposition 5.3, one can prove that there exists a canonical exact triangle

$$\begin{align*}(T_{V_{\mathrm{std},n_1}}(\frac{n-n_1}{2})[n-n_1]\times \mathrm{Id}_{\mathrm{Bun}_{n_2}})\circ {\mathrm{CT}_{P,!}} \to {\mathrm{CT}_{P,!}}\circ T_{V_{\mathrm{std},n}} \to (\mathcal{L}_1\otimes (-)\times \mathcal{L}_2\otimes_{\Lambda}T_{V_{\mathrm{std},n_2}}\circ {\mathrm{CT}_{P,!}} \end{align*}$$

of functors

$$\begin{align*}D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_{G},\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_M\times \mathrm{Div}^1,\Lambda). \end{align*}$$

As a first consequence, we get that averaging functors preserve the cuspidal subcategory.

Lemma 5.5. If $\mathcal {F}\in D_{\mathrm {{\acute {e}t}},\mathrm {cusp}}(\mathrm {Bun}_n,\Lambda )$ is a cuspidal object, then

$$\begin{align*}\mathrm{Av}_{\mathbb{L},n}(\mathcal{F}) \end{align*}$$

is again cuspidal.

5.3 Averaging functors and twisting by characters

We want to prove a compatibility of averaging functors with twists by characters. In spirit, a similar statement is [Reference Fargues and Scholze11, Theorem IX.6.1].

Let

$$\begin{align*}\mathrm{det}\colon \mathrm{Bun}_n\to \mathrm{Bun}_1,\ \mathcal{E}\mapsto \mathrm{det}(\mathcal{E})=\wedge^n\mathcal{E} \end{align*}$$

be the determinant morphism.

Proposition 5.6. Let $\chi \colon W_E\to \Lambda ^\times $ be a character, with corresponding $\Lambda $ -local system also denoted $\mathbb {L}_{\chi }\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Div}^1,\Lambda )$ . Let $\mathcal {G}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_1,\Lambda )$ be a Hecke eigensheaf with eigenvalue $\mathbb {L}_{\chi }$ . Then for any $\mathcal {F}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ , there exists a natural isomorphism

$$\begin{align*}\mathrm{Av}_{\mathbb{L}}(\mathcal{F}\otimes_\Lambda \mathrm{det}^\ast \mathcal{G})\cong {\mathrm{Av}_{\mathbb{L}\otimes_\Lambda \mathbb{L}_\chi^\vee}}(\mathcal{F})\otimes_\Lambda \mathrm{det}^\ast (\mathcal{G}). \end{align*}$$

Proof. Consider the map

$$\begin{align*}f\colon \mathrm{Bun}_n\to \mathrm{Bun}_n\times \mathrm{Bun}_1,\ \mathcal{E}\mapsto (\mathcal{E},\mathrm{det}(\mathcal{E})). \end{align*}$$

Then

$$\begin{align*}\mathcal{F}\otimes_\Lambda \mathrm{det}^\ast \mathcal{G}=f^\ast(\mathcal{F}\boxtimes \mathcal{G}). \end{align*}$$

For $r\geq 0$ , let

$$\begin{align*}\tilde{f}\colon \mathrm{Mod}_n^1\to \mathrm{Mod}_n^1\times \mathrm{Mod}_1^1 \end{align*}$$

be induced by the identity in the first component, and by taking the determinant in the second. In the diagram

the right square is cartesian.Footnote ⁷ Indeed, given a modification $\mathcal {E}\hookrightarrow \mathcal {E}^{\prime }\in \mathrm {Mod}^1_n$ with cokernel of length $1$ supported at a point $x\in X$ , and a modification $\mathcal {L}\hookrightarrow \mathrm {det}(\mathcal {E}^{\prime })$ of length $1$ supported on the same point, then there exists a unique isomorphism

$$\begin{align*}\mathcal{L}\cong \mathrm{det}(\mathcal{E}) \end{align*}$$

compatible with the morphisms to $\mathrm {det}(\mathcal {E}^{\prime })$ as both are canonically isomorphic to $\mathrm {det}(\mathcal {E}^{\prime })(-x)$ . Let $V_{\mathrm {std},r}$ be the standard representation of $\mathrm {GL}_{r,\Lambda }$ . Recall that

$$\begin{align*}T_{V_{\mathrm{std},n}}(\mathcal{F})=\overrightarrow{h}_!(\overleftarrow{h}^\ast(\mathcal{F})((n-1)/2)[n-1]) \end{align*}$$

(i.e., $T_{V_{\mathrm {std},n}}$ is defined as the correspondence given by the top horizontal line (with kernel $\Lambda ((n-1)/2)[n-1]$ )). Using the proven cartesianess, proper base change and the projection formula, we can deduce that

$$\begin{align*}\begin{matrix} & T_{V_{\mathrm{std},n}}(\mathcal{F}\otimes \mathrm{det}^\ast \mathcal{G}) \\ \cong & T_{V_{\mathrm{std},n}}(f^\ast(\mathcal{F}\boxtimes \mathcal{G})) \\ \cong & (f\times \Delta)^\ast(\overrightarrow{k}_!(\overleftarrow{k}^\ast(\mathcal{F}\boxtimes \mathcal{G})((n-1)/2)[n-1]))) \\ \cong & (f\times \Delta)^\ast(T_{V_{\mathrm{std},n}}(\mathcal{F})\boxtimes T_{V_{\mathrm{std},1}}(\mathcal{G})) \\ \end{matrix}. \end{align*}$$

(in these formulas, the Tate twist is only on the factor $\mathrm {Mod}_n^1$ ). Since $\mathcal {G}$ is a Hecke eigensheaf with eigenvalue $\mathbb {L}_{\chi }$ , the last object can be rewritten as

$$\begin{align*}(f\times \Delta)^\ast(T_{V_{\mathrm{std},n}}(\mathcal{F})\boxtimes \mathbb{L}_{\chi}^\vee \boxtimes \mathcal{G}). \end{align*}$$

All this isomorphisms are $W_{E}$ -equivariant if $W_E$ acts diagonally on the exterior powers. Let $p_2\colon \mathrm {Bun}_n \times \mathrm {Div}^1 \to \mathrm {Bun}_n$ be the first projection. From here, we can conclude

$$\begin{align*}\begin{matrix} \mathrm{Av}_{\mathbb{L}}(\mathcal{F}\otimes \mathrm{det}^\ast \mathcal{G}) & \cong & p_{2,!} (\mathbb{L} \otimes T_{V_{\mathrm{std},n}}(\mathcal{F}\otimes \mathrm{det}^\ast \mathcal{G})) \\ & \cong & p_{2,!}(\mathbb{L}\otimes_{\Lambda} \mathbb{L}_{\chi}^\vee \otimes_\Lambda T_{V_{\mathrm{std},n}}(\mathcal{F}))\otimes _{\Lambda} \mathrm{det}^\ast \mathcal{G} \\ & \cong & \mathrm{Av}_{\mathbb{L}\otimes_\Lambda \mathbb{L}_\chi^\vee}(\mathcal{F})\otimes_\Lambda \mathrm{det}^\ast (\mathcal{G}), \end{matrix} \end{align*}$$

as desired.

5.4 The averaging functors of rank $1$ local systems

It is expected that if $\mathbb {L}$ is a $\Lambda $ -local system of rank $<n$ , then $\mathrm {Av}_{\mathbb {L}}$ sends each object in

$$\begin{align*}D_{\mathrm{{\acute{e}t}},\mathrm{cusp}}(\mathrm{Bun}_n,\Lambda) \end{align*}$$

to zero (indeed, it should follow from a suitably refined version of the categorical enhancement of Fargues’ conjecture mentioned below in Remark 7.5 that any cuspidal object on $\mathrm {Bun}_n$ corresponds to an object supported on the irreducible locus in the stack of L-parameters; we do not make this precise here). We want to prove this in the case $\mathbb {L}$ has rank $1$ , and $n=2$ .

Theorem 5.7. Let $\mathbb {L}_\chi \colon W_E\to \Lambda ^\times $ be a character, seen as a local system $\mathbb {L}_\chi $ in $\mathrm {Div}^1$ . For each $\mathcal {F}\in D_{\mathrm {{\acute {e}t}},\mathrm {cusp}}(\mathrm {Bun}_2,\Lambda )$ ,

$$\begin{align*}\mathrm{Av}_{\mathbb{L}_\chi}(\mathcal{F})=0. \end{align*}$$

If $n=1$ , then each object is cuspidal for our definition (as there are no proper parabolics), and the statement is wrong. The proof follows [Reference Gaitsgory15, Theorem A.4].

Proof. Most of the proof works for any $n\geq 2$ , so we write it in this generality, only specifying $n=2$ when needed (although we expect the same argument to extend to all $n\geq 2$ with more work). By Proposition 5.6 and Lemma 4.4, we can assume that $\mathbb {L}_\chi \cong \Lambda (\frac {1-n}{2})$ . Set

$$\begin{align*}\mathrm{Av}:=\mathrm{Av}_{\Lambda(\frac{1-n}{2})}. \end{align*}$$

Recall that each cuspidal $\mathcal {F}$ is automatically supported on the semistable locus of $\mathrm {Bun}_n$ (cf. Lemma 4.6) and that $\mathrm {Av}(\mathcal {F})\in D_{\mathrm {{\acute {e}t}},\mathrm {cusp}}(\mathrm {Bun}_n,\Lambda )$ is again cuspidal (cf. Lemma 5.5). Thus, it suffices to prove that for each $b,c\in B(G)$ basic with $\kappa (c)=\kappa (b)+1$ , and each $\mathcal {F}\in D_{\mathrm {{\acute {e}t}},\mathrm {cusp}}(\mathrm {Bun}_n,\Lambda )$ which is supported on $\mathrm {Bun}_n^b$ , the stalk

$$\begin{align*}j_c^\ast\mathrm{Av}(\mathcal{F}) \end{align*}$$

at c of

$$\begin{align*}\mathrm{Av}(\mathcal{F}) \end{align*}$$

vanishes. To prove this, let

$$\begin{align*}Z \end{align*}$$

be the small v-stack over k parametrizing morphisms

$$\begin{align*}\mathcal{E}\to \mathcal{E}^{\prime} \end{align*}$$

with $\mathcal {E}$ locally isomorphic to $\mathcal {E}_b$ and $\mathcal {E}^{\prime }$ locally isomorphic to $\mathcal {E}_c$ . For each $r=0,\ldots , n$ and each $d\in B(\mathrm {GL}_r)$ , let

$$\begin{align*}Z_{r,d} \end{align*}$$

be the small v-stack parametrizing factorizations into a surjection followed by an injection

$$\begin{align*}\mathcal{E}\twoheadrightarrow \mathcal{F}\hookrightarrow \mathcal{E}^{\prime} \end{align*}$$

with $\mathcal {E}, \mathcal {E}^{\prime }$ locally isomorphic to $\mathcal {E}_b, \mathcal {E}_c$ , respectively, and $\mathcal {F}$ locally isomorphic to $\mathcal {E}_d$ . The morphism

$$\begin{align*}(\mathcal{E}\twoheadrightarrow \mathcal{F}\hookrightarrow \mathcal{E}^{\prime})\mapsto (\mathcal{E}\to \mathcal{E}^{\prime}) \end{align*}$$

defines an injection $j_{r,d}\colon Z_{r,d}\to Z$ , which is a locally closed embedding. Moreover,

$$\begin{align*}\coprod\limits_{r,d} |Z_{r,d}|=|Z| \end{align*}$$

as each geometric point of Z factors over a unique $Z_{r,d}$ . The space Z defines a correspondence

(with $q_1$ , resp. $q_2$ , factoring through $\mathrm {Bun}_n^b$ , resp. $\mathrm {Bun}_n^c$ ) and thus for each $r,d$ an endofunctor

$$\begin{align*}\mathrm{Av}_{r,d}\colon D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda),\ \mathcal{G}\mapsto q_{2,!}(q_1^{\ast}\mathcal{G}\otimes^{\mathbb{L}}_\Lambda j_{r,d,!}(\Lambda)) \end{align*}$$

as well as the endofunctor

$$\begin{align*}\overline{\mathrm{Av}}\colon D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda),\ \mathcal{G}\mapsto q_{2,!}(q_1^{\ast}\mathcal{G})). \end{align*}$$

Note that (up to a shift) $\mathrm {Av}_{n,b}=\mathrm {Av}$ , and $Z_{n,d}$ is empty for $d\neq b$ . We claim that for $\mathcal {F}\in D_{\mathrm {{\acute {e}t}},\mathrm {cusp}}(\mathrm {Bun}_n,\Lambda )$ , each of the objects

$$\begin{align*}\overline{\mathrm{Av}}(\mathcal{F}),\quad \mathrm{Av}_{r,d}(\mathcal{F}), \end{align*}$$

where $r= 0,\ldots , n-1$ and d is arbitrary, vanish at c. As the cone of the natural morphism

$$\begin{align*}\mathrm{Av}(\mathcal{F})\to \overline{\mathrm{Av}}(\mathcal{F}) \end{align*}$$

has (by excision) a filtration by the $\mathrm {Av}_{r,d}(\mathcal {F})$ for $r=0,\ldots ,n-1$ , this implies the desired vanishing of

$$\begin{align*}\mathrm{Av}_{n,b}(\mathcal{F})=\mathrm{Av}(\mathcal{F}). \end{align*}$$

For $\overline {\mathrm {Av}}$ , note that the base change along $k \to \mathrm {Bun}_n^c$ of the projection $Z\to \mathrm {Bun}_n^c$

$$\begin{align*}Z\times_{\mathrm{Bun}_n^c}k\cong [G_b({E})\setminus \mathrm{Hom}(\mathcal{E}_b, \mathcal{E}_c)], \end{align*}$$

where $\mathrm {Hom}(\mathcal {E}_b,\mathcal {E}_c)$ is a positive Banach-Colmez space (by our assumption that $b,c$ are basic with $\kappa (c)>\kappa (b)$ ). As the cohomology of positive Banach-Colmez spaces is $1$ -dimensional and concentrated in a single degree, the compactly supported cohomology of Z with coefficients in the pullback of a (geometrically) cuspidal object in

$$ \begin{align*}D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n^b,\Lambda)\cong D_{\mathrm{{\acute{e}t}}}([\ast/G_b({E})],\Lambda)\end{align*} $$

vanishes for $n=2$ : when $\kappa (b)$ is even (cf. the discussion of Proposition 4.10); when $\kappa (b)$ is odd, $G_b(E)$ is compact modulo center, so the claim follows from Remark 4.11. Indeed, already the cohomology of the kernel of the reduced norm will vanish in this case. Next, assume $r=0$ . Then

$$\begin{align*}Z_{r,d}\times_{\mathrm{Bun}_n^c}k=[\ast/G_b({E})], \end{align*}$$

and the claim follows as geometrically cuspidal objects do not have cohomology, exactly as above. For $r=1,\ldots , n-1$ and $d\in B(\mathrm {GL}_r)$ , we can write

$$\begin{align*}Z_{r,d}\times_{\mathrm{Bun}_n^c}k=[G_b({E})\setminus \mathrm{Hom}^{\mathrm{surj}}(\mathcal{E}_b,\mathcal{E}_d)\times^{\mathcal{G}_d} \mathrm{Hom}^{\mathrm{inj}}(\mathcal{E}_d, \mathcal{E}_c)] \end{align*}$$

with $\mathcal {G}_d$ acting diagonally on $\mathrm {Hom}^{\mathrm {surj}}(\mathcal {E}_b,\mathcal {E}_d)\times \mathrm {Hom}^{\mathrm {inj}}(\mathcal {E}_d, \mathcal {E}_c)$ . Let $\mathcal {F}\in D_{\mathrm {{\acute {e}t}}, \mathrm {cusp}}(\mathrm {Bun}_n,\Lambda )$ with support at b. The compactly supported cohomology

$$\begin{align*}R\Gamma_c([G_b({E})\setminus \mathrm{Hom}^{\mathrm{surj}}(\mathcal{E}_b,\mathcal{E}_d)],\mathcal{F}) \end{align*}$$

vanishes because it identifies with the fiber of $0=Rf_!(\mathrm {CT}_{P,!}(\mathcal {F}))$ at $\mathcal {E}_d$ , where f is the morphism

$$\begin{align*}\mathrm{Bun}_M\xrightarrow{f} \mathrm{Bun}_r, \end{align*}$$

and $P\subseteq G$ is the standard parabolic with Levi $M\cong \mathrm {GL}_{n-r}\times \mathrm {GL}_{r}$ . Using the projection

$$\begin{align*}[G_b({E})\setminus \mathrm{Hom}^{\mathrm{surj}}(\mathcal{E}_b,\mathcal{E}_d)\times^{\mathcal{G}_b} \mathrm{Hom}^{\mathrm{inj}}(\mathcal{E}_d, \mathcal{E}_c)]\to [\mathcal{G}_b\setminus \mathrm{Hom}^{\mathrm{inj}}(\mathcal{E}_d,\mathcal{E}_d)] \end{align*}$$

with fibers isomorphic to $[G_b({E})\setminus \mathrm {Hom}^{\mathrm {surj}}(\mathcal {E}_b,\mathcal {E}_d)]$ , this implies the desired vanishing of the stalk of

$$\begin{align*}\mathrm{Av}_{r,d}(\mathcal{F}) \end{align*}$$

at c (i.e., of the compactly supported cohomology of $Z_{r,d}\times _{\mathrm {Bun}_n^c}k$ with coefficients in  $\mathcal {F}$ ).

5.5 Averaging functors via the spectral action

Recall the stack $X_{\hat {G}}$ of n-dimensional continuous $\Lambda $ -representations of $W_E$ , with the natural projection

$$\begin{align*}f\colon X_{\hat{G}}\to [\mathrm{Spec}(\Lambda)/\hat{G}], \end{align*}$$

introduced in Section 3.4. If $V \in \mathrm {Rep}_{\hat {G}}$ , we also denote by V the associated vector bundle on $[\mathrm {Spec}(\Lambda )/\hat {G}]$ .

For a $W_E$ -representation $\mathbb {L}$ on a finite projective $\Lambda $ -module, we set

$$\begin{align*}\mathcal{A}v_{\mathbb{L}}:=R\Gamma(W_E,\mathbb{L}\otimes_\Lambda f^\ast V_{\mathrm{std}})\in \mathrm{Perf}(X_{\hat{G}}). \end{align*}$$

Note that $\mathcal {A}v_{\mathbb {L}}$ is indeed a perfect complex. Indeed, the wild inertia $P_E\subseteq W_E$ acts via a finite quotient and has no higher cohomology. As $p\in \Lambda ^\times $ , this implies that the invariants

$$\begin{align*}(\mathbb{L}\otimes_{\Lambda}f^\ast V_{\mathrm{std}})^{P_E} \end{align*}$$

are a direct summand of $\mathbb {L}\otimes _{\Lambda }f^\ast V_{\mathrm {std}}$ and hence a vector bundle. After fixing a generator $\tau \in I_E/P_E$ of the tame quotient of the inertia subgroup $I_E\subseteq W_E$ , and a Frobenius lift $\sigma \in W_{E}/P_E$ , the complex $\mathcal {A}v_{\mathbb {L}}$ is calculated as the homotopy limit of the diagram

(4)

where q is the cardinality of the residue field of E. Indeed, this follows because (after fixing a smooth representation of the wild inertia) the stack of $W_E$ -representations (with prescribed restriction to the wild inertia) is equivalent to the representations of a discretization. Moreover $\tau ,\sigma $ can be chosen such that the quotient modulo the wild inertia is isomorphic to $\mathbb {Z}\ltimes \mathbb {Z}[1/p]$ with $1\in \mathbb {Z}$ acting by multiplication by q on $\mathbb {Z}[1/p]$ , [Reference Fargues and Scholze11, Proof of theorem VIII.1.3]. Now, the cohomology of $\mathbb {Z}\ltimes \mathbb {Z}[1/p]$ on $\Lambda $ -modules is easily seen to be calculated by an analogous homotopy limit. This presentation as a finite homotopy limit implies that $\mathcal {A}v_{\mathbb {L}}$ is a perfect complex on $X_{\hat {G}}$ , as claimed.

Lemma 5.8. There exists a canonical isomorphism

$$\begin{align*}\mathrm{Av}_{\mathbb{L}}\cong \mathcal{A}v_{\mathbb{L}}\ast (-) \end{align*}$$

of functors

$$\begin{align*}D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda). \end{align*}$$

Proof. Note that the diagram (Equation (4)) is a diagram of perfect complexes in $X_{\hat {G}}$ . Applying it to some object $\mathcal {G}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ , we get the diagram

whose homotopy limit calculates $\mathrm {Av}_{\mathbb {L}}$ . But the spectral action

$$\begin{align*}\mathrm{Perf}(X_{\hat{G}})\times D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda) \end{align*}$$

commutes with finite homotopy limits in each variable. This implies the claim.

The following immediate consequence of the existence of the spectral action is the analog of the so-called vanishing conjecture of [Reference Frenkel, Gaitsgory and Vilonen12] (cf. also [Reference Gaitsgory14] for a generalized vanishing conjecture in the context of D-modules).

For clarity, let us add the subscript

$$\begin{align*}\mathrm{Av}_{\mathbb{L},n}, \end{align*}$$

when we consider the averaging functor on $D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ associated with $\mathbb {L}$ . Note that n need not be the rank of $\mathbb {L}$ .

Proposition 5.9. Assume that $\Lambda $ is a field and $\mathbb {L}$ is an irreducible $W_E$ -representation. Then

$$\begin{align*}\mathrm{Av}_{\mathbb{L},n}=0 \end{align*}$$

if $n<\mathrm {rk}(\mathbb {L})$ .

Proof. By Lemma 5.8, it suffices to see that the object

$$\begin{align*}\mathcal{A}v_{\mathbb{L}}\in \mathcal{X}_{\hat{G}} \end{align*}$$

vanishes. This can be checked at field valued points of $X_{\hat {G}}$ , where it follows from the fact that by irreducibility of $\mathbb {L}$

$$\begin{align*}R\Gamma(W_E,\mathbb{L}\otimes_\Lambda V)=0 \end{align*}$$

for any representation V of $W_E$ which is of rank $n<\text {rank } \mathbb {L}$ . Indeed, vanishing in degree $0$ is clear, which implies vanishing in degree $2$ by duality, and then vanishing of the $H^1$ as the Euler characteristic of $W_E$ on finite dimensional $\Lambda $ -representations is $0$ .

This vanishing has the following interesting consequence.

Corollary 5.10. Assume that $\Lambda $ is a field. Let $\mathcal {F}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ be a Hecke eigensheaf with eigenvalue $\mathbb {L}$ an irreducible $W_{E}$ -representation over $\Lambda $ of rank n. Then

$$\begin{align*}\mathcal{F}\in D_{\mathrm{{\acute{e}t}},\mathrm{cusp}}(\mathrm{Bun}_n,\Lambda) \end{align*}$$

(i.e., $\mathcal {F}$ is cuspidal).

Proof. From Proposition 5.9 and Proposition 5.3 (we note that $\mathbb {L}\otimes _{\Lambda } \mathcal {L}_2$ is again irreducible, so that Proposition 5.9 can be applied to it), we can conclude that

$$\begin{align*}\mathrm{CT}_{P,!}\circ \mathrm{Av}_{\mathbb{L}^\vee,n}(\mathcal{F})=0 \end{align*}$$

for any proper standard parabolic $P\subsetneq G$ . As $\mathcal {F}$ is a Hecke eigensheaf with eigenvalue $\mathbb {L}$ , we get that

$$\begin{align*}\mathrm{Av}_{\mathbb{L}^\vee,n}(\mathcal{F}) = R\Gamma(W_E,\mathbb{L}^\vee \otimes_\Lambda T_{V_{\mathrm{std}}}(\mathcal{F})) =R\Gamma(W_{E},\mathbb{L}^\vee{\otimes}_\Lambda \mathbb{L})\otimes_{\Lambda}^{L}\mathcal{F}. \end{align*}$$

Thus, $\mathrm {CT}_{P,!}(\mathcal {F})=0$ as

$$\begin{align*}0=\mathrm{CT}_{P,!}({\mathrm{Av}_{\mathbb{L}^\vee,n}}(\mathcal{F}))=R\Gamma(W_{E},\mathbb{L}^\vee{\otimes}_\Lambda \mathbb{L})\otimes_{\Lambda}^{L}\mathrm{CT}_{P,!}(\mathcal{F}) \end{align*}$$

and

$$\begin{align*}R\Gamma(W_{E},\mathbb{L}^\vee{\otimes}_\Lambda \mathbb{L})\neq 0. \end{align*}$$

This finishes the proof.

5.6 Averaging functors on $D_{\text {lis}}$

We explain how to extend the definition of the averaging functors

$$\begin{align*}\mathrm{Av}_{\mathbb{L}}\colon D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda)\to D_{\mathrm{{\acute{e}t}}}(\mathrm{Bun}_n,\Lambda) \end{align*}$$

for $\mathbb {L}\in D_{\mathrm {{\acute {e}t}}}(\mathrm {Div}^1,\Lambda )$ , to the case that the torsion ring $\Lambda $ is replaced by an arbitrary $\mathbb {Z}_\ell $ -algebra L (the most important case being $L=\overline {\mathbb {Q}}_\ell $ ).

Let $\mathbb {L}$ be a representation of $W_E$ on a finite, locally free L-module $\mathbb {L}$ , which is continuous when $\mathbb {L}$ is given the colimit topology in the presentation

$$\begin{align*}\mathbb{L}=\varinjlim\limits_{N\subseteq \mathbb{L} \text{ fin.gen. } \mathbb{Z}_\ell-\text{submodule}} N, \end{align*}$$

with each N equipped with the $\ell $ -adic topology. Then for each $\mathcal {F}\in D_{\text {lis}}(\mathrm {Bun}_n,L)$ , the object

$$\begin{align*}\mathbb{L} \overset{\blacksquare}\otimes_{L} \mathcal{F}\in D_{\blacksquare}(\mathrm{Bun}_n, L) \end{align*}$$

lies in $D_{\text {lis}}(\mathrm {Bun}_n,L)$ . Using the presentation (Equation (4)), we can therefore deduce that

$$\begin{align*}R\Gamma(W_E,\mathbb{L} \overset{\blacksquare}\otimes_L \mathcal{F})\in D_{\text{lis}}(\mathrm{Bun}_n,L) \end{align*}$$

for each $W_E$ -equivariant object $\mathcal {F}\in D_{\text {lis}}(\mathrm {Bun}_n,L)$ .

The functor

$$ \begin{align*}T_{V_{\mathrm{std}}}: D_\blacksquare(\mathrm{Bun}_n, L) \to D_\blacksquare(\mathrm{Bun}_n \times \mathrm{Div}^1, L), \end{align*} $$

defined by the formula

$$ \begin{align*}T_{V_{\mathrm{std}}} (\mathcal{F}) = \overrightarrow{h}_{\natural} \left( \overleftarrow{h}^\ast \mathcal{F} \overset{\blacksquare} \otimes_{L}^L \mathcal{S}_{V_{\mathrm{std}}}^{\prime} \right), \end{align*} $$

where $\mathcal {S}_{V_{\mathrm {std}}}^{\prime }$ denotes the object constructed in [Reference Fargues and Scholze11, §IX.2], induces, by [Reference Fargues and Scholze11, Corollary IX.2.3], a functor

$$ \begin{align*}T_{V_{\mathrm{std}},\text{lis}}^{\mathrm{equiv}}: D_{\text{lis}}(\mathrm{Bun}_n, L) \to D_{\text{lis}}(\mathrm{Bun}_n, L)^{BW_E}. \end{align*} $$

By the considerations above, if $\mathbb {L}$ is a representation of $W_E$ as before, for any $\mathcal {F} \in D_{\text {lis}}(\mathrm {Bun}_n, L)$ , we have

$$\begin{align*}\mathrm{Av}_{\mathbb{L}}(\mathcal{F}):=R\Gamma(W_E,\mathbb{L} \overset{\blacksquare}\otimes_L T_{V_{\mathrm{std}},\text{lis}}^{\mathrm{equiv}}(\mathcal{F}))\in D_{\text{lis}}(\mathrm{Bun}_n,L). \end{align*}$$

This recipe defines the averaging functor associated to $\mathbb {L}$ ,

$$\begin{align*}\mathrm{Av}_{\mathbb{L}}\colon D_{\text{lis}}(\mathrm{Bun}_n,L)\to D_{\text{lis}}(\mathrm{Bun}_n,L). \end{align*}$$

It recovers the previous definition when $L=\Lambda $ is torsion, through the identification $D_{\text {lis}}(\mathrm {Bun}_n,L)\cong D_{\mathrm {{\acute {e}t}}}(\mathrm {Bun}_n,\Lambda )$ , thanks to [Reference Fargues and Scholze11, Proposition VII.5.2].

The description

$$\begin{align*}\mathrm{Av}_{\mathbb{L}}(-)\cong \mathcal{A}v_{\mathbb{L}}\ast(-) \end{align*}$$

of it, via the spectral action for the analogously defined perfect complex

$$\begin{align*}\mathcal{A}v_{\mathbb{L}}:=R\Gamma(W_E,\mathbb{L}{\otimes}_L f^\ast V_{\mathrm{std}}) \end{align*}$$

on the algebraic stack $X_{\hat {G}}$ over L, remains valid.

7.1 Generalities on Hecke eigensheaves for irreducible representations

Let $\mathbb {L}$ be an irreducible L-representation of $W_E$ . Let

$$\begin{align*}\mathcal{F}\in D_{\text{lis}}(\mathrm{Bun}_n,L) \end{align*}$$

be a Hecke eigensheaf with eigenvalue $\mathbb {L}$ (i.e., we are given natural isomorphisms

$$\begin{align*}T_V(\mathcal{F})\cong r_{V,\ast}(\mathbb{L})\boxtimes_L \mathcal{F} \end{align*}$$

for each representation $V\in \mathrm {Rep}_L(\hat {G})$ ; cf. Section 3.3). We moreover assume that $\mathcal {F}$ is ULA.

Let

$$\begin{align*}\mathrm{LL}_{FS}\colon \mathcal{O}(X_{\hat{G},L})\to \mathcal{Z}(G,L) \end{align*}$$

be the morphism into the Bernstein center $\mathcal {Z}(G,L)$ for $D_{\text {lis}}(\mathrm {Bun}_n,L)$ constructed by Fargues-Scholze in [Reference Fargues and Scholze11], and let

$$\begin{align*}V_1,\ldots, V_m\in \mathrm{Rep}_L (\hat{G}), (\gamma_1,\ldots, \gamma_m)\in W_E^{m}, \alpha\colon L \to V_1\otimes_L\ldots, \otimes_L V_m, \beta\colon V_1\otimes_L\ldots, \otimes_L V_m\to L \end{align*}$$

be an excursion datum with associated function $f\in \mathcal {O}(X_{\hat {G},L})$ . By the Hecke eigensheaf property, we can then see that $\mathrm {LL}_{\mathrm {FS}}(f)$ acts on $\mathcal {F}$ via the same scalar $\in L$ , which gives the composition

$$\begin{align*}L\xrightarrow{\alpha} r_{V_1,\ast}(\mathbb{L})\otimes_L \ldots \otimes_L r_{V_m,\ast}(\mathbb{L}) \xrightarrow{\gamma_1\otimes\ldots \otimes \gamma_m} r_{V_1,\ast}(\mathbb{L})\otimes_L \ldots \otimes_L r_{V_m,\ast}(\mathbb{L}) \xrightarrow{\beta} L. \end{align*}$$

By compatibility with parabolic induction (cf. [Reference Fargues and Scholze11, Corollary IX.7.3]), this implies the following lemma.

In the case that L is replaced by a field $\Lambda $ of characteristic $\ell $ (and $\mathbb {L}$ an irreducible $W_E$ -representation over $\Lambda $ ), we proved the stronger statement that $\mathcal {F}$ is automatically cuspidal using averaging functors in Corollary 5.10. We do not know if this stronger statement is true in general.

Proof. Take $c\in B(G)$ basic with $\mathrm {deg}(\mathcal {E}_c)=\mathrm {deg}(\mathcal {E}_b)+1$ . We already know that $\mathcal {F}$ is supported on the semistable locus and that $\mathcal {F}_c$ is a complex of admissible supercuspidal representations. Then

$$\begin{align*}T_{V_{\mathrm{std}}^\ast}(\mathcal{F}_c)\cong \mathbb{L}^\vee\otimes_{L} \mathcal{F}_b \end{align*}$$

as $W_{E}$ -equivariant sheaves. The right-hand side is irreducible (as a $W_{E}$ -equivariant complex of sheaves). Thus, $T_{V_{\mathrm {std}}^\ast }$ kills all irreducible Jordan-Hölder factors of cohomology sheaves of $\mathcal {F}_c$ , except one. But using our assumption that L is of characteristic $0$ ,

$$\begin{align*}\mathcal{F}_c=T_{L}(\mathcal{F}_c)\to T_{V_{\mathrm{std}}}(T_{V_{\mathrm{std}}^\ast}(\mathcal{F}_c)) \end{align*}$$

is a split injection, and thus, $\mathcal {F}_c$ is an irreducible representation, placed in some degree. The case of $c\in B(G)$ basic such that $\mathrm {deg}(\mathcal {E}_c)=\mathrm {deg}(\mathcal {E}_b)-1$ is dealt with similarly using instead the Hecke operator for the standard representation. Repeating the argument, one gets the result for every $c\in B(G)$ basic by ascending or descending induction.

7.2 The spectral action and the definition of $\mathrm {Aut}_{{\mathbb {L}}}$

We continue to use the notation of the previous section. Let

$$\begin{align*}i_{\mathbb{L}}\colon \mathrm{Spec}(L)\to X_{\hat{G}} \end{align*}$$

be the natural morphism with image the point determined by $\mathbb {L}$ . As the $W_E$ -representation $\mathbb {L}$ is assumed to be irreducible, the image of $i_{\mathbb {L}}$ lands inside the smooth locus of $X_{\hat {G}}$ , and thus, the object

$$\begin{align*}k(\mathbb{L})_{\mathrm{reg}}:=i_{\mathbb{L},\ast}(L) \end{align*}$$

lies in $\mathrm {IndPerf}(X_{\hat {G}})$ . More concretely, the residual gerbe of $X_{\hat {G}}$ at the image of $i_{\mathbb {L}}$ is given by

$$\begin{align*}[\mathrm{Spec}(L)/\mathbb{G}_{m,L}] \end{align*}$$

and the (underived) stalk of $k(\mathbb {L})_{\mathrm {reg}}$ at x is the regular representation

$$\begin{align*}\mathcal{O}(\mathbb{G}_m) \end{align*}$$

of $\mathbb {G}_m$ . Let us write

$$\begin{align*}k(\mathbb{L})_{\mathrm{reg}}=\bigoplus\limits_{i\in \mathbb{Z}}k(\mathbb{L})_i, \end{align*}$$

with $k(\mathbb {L})_i\subseteq k(\mathbb {L})_{\mathrm {reg}}$ the summand where ${\mathbb {G}_m}$ acts via $z\mapsto z^i$ .

Let $\psi \colon E\to L^\times $ be a nontrivial character. This yields the standard Whittaker datum $(B,\psi )$ for $G=\mathrm {GL}_n$ , that is, B is the standard upper triangular Borel, and (using abuse of notation) $\psi $ denotes the regular character

$$\begin{align*}\psi\colon U(E)\twoheadrightarrow U/[U,U](E)\cong E^{n-1}\xrightarrow{\sum }E\xrightarrow{\psi}L^\times \end{align*}$$

with $U\subseteq B$ the unipotent radial of B, and $\sum $ the summation.

This yields the so-called Whittaker sheaf

$$\begin{align*}\mathcal{W}_\psi:=j_{1,!}(\mathcal{F}_{\mathrm{cInd}^{G_1(E)}_{U(E)} \psi})\in D_{\text{lis}}(\mathrm{Bun}_n,L). \end{align*}$$

We can now make the following definition. Recall the spectral action of Section 3.4, denoted with $\ast $ .

Definition 7.3. For $\mathbb {L}$ an irreducible $W_E$ -representation (as before), set

$$\begin{align*}\mathrm{Aut}_{\mathbb{L}}:=k(\mathbb{L})_{\mathrm{reg}}\ast \mathcal{W}_\psi \in D_{\text{lis}}(\mathrm{Bun}_n,L). \end{align*}$$

We note that the symbol $\mathrm {Aut}_{{\mathbb {L}}}$ is for now just a notation; the rest of this text is devoted to showing that this object of $D_{\text {lis}}(\mathrm {Bun}_n,L)$ indeed has the properties prescribed by Fargues’ conjecture and therefore deserves its name.

The Hecke eigensheaf property of $\mathrm {Aut}_{\mathbb {L}}$ is formal and holds more generally as recorded in the following proposition.

Proof. Let $V\in \mathrm {Rep}_L(\hat {G})$ . Then there is a natural isomorphism

$$\begin{align*}\begin{matrix} T_{V}(k(\mathbb{L})_{\mathrm{reg}}\ast \mathcal{F}) & \cong & f^\ast(V)\ast k(\mathbb{L})_{\mathrm{reg}}\ast \mathcal{F} \\ & \cong & (f^\ast(V)\otimes_{\mathcal{O}_{X_{\hat{G}}}} k(\mathbb{L})_{\mathrm{reg}})\ast \mathcal{F} \\ & \cong & (r_{V}(\mathbb{L})\otimes_{L} k(\mathbb{L})_{\mathrm{reg}})\ast \mathcal{F} \\ & \cong & r_V(\mathbb{L})\otimes_{L} k(\mathbb{L})_{\mathrm{reg}}\ast \mathcal{F} \end{matrix} \end{align*}$$

(compatible with the $W_{E}$ -action), and similarly for every finite set I, $V_i\in \mathrm {Rep}(\hat {G}), i\in I$ .

This proposition applies in particular to $\mathcal {G}=\mathcal {W}_{\psi }$ , and hence, the object $\mathrm {Aut}_{\mathbb {L}}$ is a Hecke eigensheaf, as desired. We also know that it is supported on the semistable locus and given by supercuspidal representations, by Lemma 7.1. However, it could potentially be zero. That this is not the case will be deduced in the next subsection from the local Langlands correspondence.

Remark 7.5. To motivate the definition of $\mathrm {Aut}_{\mathbb {L}}$ , let us recall that Fargues’ conjecture admits a strong conjectural enhancement: it is expected that one has an equivalence

$$ \begin{align*}\mathrm{IndCoh}(X_{\hat{G}}) \cong D_{\text{lis}}(\mathrm{Bun}_n,L) \end{align*} $$

(recall that here L is assumed to be algebraically closed of characteristic $0$ ), compatibly with the actions of $\mathrm {IndPerf}(X_{\hat {G}})$ on both sides (acting via tensor product on the left); cf. [Reference Fargues and Scholze11, Conjecture X.1.4.]. Via this equivalence, if $\mathbb {L}$ is as above, the Hecke eigensheaf whose existence is predicted by Fargues’ conjecture should correspond to $k(\mathbb {L})_{\mathrm {reg}}$ , and the Whittaker sheaf should correspond to the structure sheaf $\mathcal {O}_{X_{\hat {G}}}$ .Footnote ⁸ Since tautologically

$$ \begin{align*}{k(\mathbb{L})_{\mathrm{reg}} }\otimes_{\mathcal{O}_{X_{\hat{G}}}} \mathcal{O}_{X_{\hat{G}}} \cong {k(\mathbb{L})_{\mathrm{reg}}}, \end{align*} $$

since the conjectural categorical equivalence should be equivariant for the action of $\mathrm {IndPerf}(X_{\hat {G}})$ and since $k(\mathbb {L})_{\mathrm {reg}} \in \mathrm {IndPerf}(X_{\hat {G}})$ , we see that the definition of $\mathrm {Aut}_{\mathbb {L}}$ is reasonable (and meaningful independently of the conjecture!).

7.3 Generalities on the spectral action of $k(\mathbb {L})_i$

We use the notation from before. This section is devoted to the study of the spectral actions of the objects

$$\begin{align*}k(\mathbb{L})_i\in \mathrm{Perf}(X_{\hat{G}}) \end{align*}$$

for $i\in \mathbb {Z}$ . Let us set

$$\begin{align*}F_i(\mathcal{G}):=k(\mathbb{L})_i\ast \mathcal{G} \end{align*}$$

for $\mathcal {G}\in D_{\text {lis}}(\mathrm {Bun}_n,L)$ .

Recall the object

$$\begin{align*}\mathcal{A}v_{\mathbb{L}^{\prime}}=R\Gamma(W_E,\mathbb{L}^{\prime}\otimes_L f^\ast V_{\mathrm{std}})\in \mathrm{Perf}(X_{\hat{G}}) \end{align*}$$

associated with a $W_E$ -representation $\mathbb {L}^{\prime }$ .

Lemma 7.6. We have an equality

$$\begin{align*}\mathcal{A}v_{\mathbb{L}^\vee}\cong k(\mathbb{L})_1[-1]\oplus k(\mathbb{L}(\chi_{\mathrm{cyc}}))_1[-2]\in \mathrm{Perf}(X_{\hat{G}}), \end{align*}$$

where $\chi _{\mathrm {cyc}}$ denotes the cyclotomic character of $W_E$ . In particular,

$$\begin{align*}\mathrm{Av}_{\mathbb{L}^\vee}(\mathcal{G})=k(\mathbb{L})_1\ast \mathcal{G}[-1] \oplus k(\mathbb{L}(\chi_{\mathrm{cyc}}))_1\ast \mathcal{G}[-2] \end{align*}$$

for each $\mathcal {G}\in D_{\text {lis}}(\mathrm {Bun}_n,L)$ .

In particular, we see that $F_1(-)$ is a direct summand of $\mathrm {Av}_{\mathbb {L}^\vee }(-)[1]$ .

7.4 Computation of one stalk of $\mathrm {Aut}_{{\mathbb {L}}}$ and conclusion

Our aim is to identify the stalks of

$$\begin{align*}\mathrm{Aut}_{\mathbb{L}}\cong k(\mathbb{L})_{\mathrm{reg}}\ast \mathcal{W}_\psi\cong \bigoplus\limits_{i\in \mathbb{Z}} k(\mathbb{L})_i\ast \mathcal{W}_\psi \end{align*}$$

at points in the semistable locus.

We recall that we have fixed a completed algebraic closure C of E, giving rise to a (classical) point $\infty \in X_C$ on the Fargues-Fontaine curve.

Let $b\in B(G)$ be the basic element with $\kappa (b)=1$ . By [Reference Scholze and Weinstein25, Lecture 24], the space

$$\begin{align*}\mathcal{M}_{\mathrm{LT},\infty} \end{align*}$$

of injections of the trivial rank n vector bundle $\mathcal {E}_1$ into $\mathcal {E}_b$ with cokernel supported at $\infty $ is (the diamond associated with) the infinite level Lubin-Tate space.

For an irreducible $W_E$ -representation $\mathbb {L}^{\prime }$ of rank n over L we will denote by $\mathrm {LL}_1(\mathbb {L}^{\prime })$ the local Langlands correspondent of $\mathbb {L}^{\prime }$ and by $\mathrm {LL}_b(\mathbb {L}^{\prime })$ the local Jacquet-Langlands correspondent of $\mathrm {LL}_1(\mathbb {L}^{\prime })$ .

Theorem 7.8. Let $\mathbb {L}$ be an irreducible $W_E$ -representation of rank n over L and $\mathrm {LL}_1(\mathbb {L})\in \mathrm {Rep}^\infty _L G_1(E)$ its local Langlands correspondent. Then

$$\begin{align*}{j_b^\ast}T_{V_{\mathrm{std}}}({j_{1,!}}\mathcal{F}_{\mathrm{LL}_1(\mathbb{L})})\cong \mathbb{L}\otimes_{L} \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})} \end{align*}$$

as $W_E$ -equivariant objects in $D_{\text {lis}}(\mathrm {Bun}_n,L)$ .

This incarnation of the local Langlands correspondence allows us to calculate one stalk of $\mathrm {Aut}_{\mathbb {L}}$ .

Theorem 7.9. Let as before $b\in B(G)$ basic with $\kappa (b)=1$ . Then

$$\begin{align*}j_b^\ast \mathrm{Aut}_{\mathbb{L}}\cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})}. \end{align*}$$

Proof. We first analyze

$$\begin{align*}\mathrm{Av}_{\mathbb{L}^\vee}(\mathcal{W}_\psi)[1]\cong {k(\mathbb{L})_1}\ast \mathcal{W}_\psi \oplus k(\mathbb{L}(\chi_{\mathrm{cyc}}))_1[-1]\ast \mathcal{W}_{\psi}. \end{align*}$$

By Lemma 7.7, we can replace here $\mathcal {W}_\psi $ by its supercuspidal part. Let $\mathcal {B}:=\mathcal {B}_{\mathfrak {s}}\subseteq \mathrm {Rep}_L^\infty G_1(E)$ be a supercuspidal block, and let

$$\begin{align*}\mathcal{W}_{\psi, \mathfrak{s}} \end{align*}$$

be the Bernstein component of $\mathcal {W}_\psi $ in $\mathcal {B}$ . Let

$$\begin{align*}\mathfrak{z} \end{align*}$$

be the center of the block $\mathcal {B}$ . Then it is known (cf. [Reference Bernstein and Deligne1], [Reference Bushnell and Henniart2]) that

$$\begin{align*}\mathfrak{z}\cong \mathrm{End}_{\mathcal{B}}(\mathcal{W}_{\psi,\mathfrak{s}})\cong L[T^{\pm 1}], \end{align*}$$

that $\mathcal {W}_{\psi ,\mathfrak {s}}\in \mathcal {B}$ is projective, and that the functor

$$\begin{align*}\mathcal{B}\cong \mathrm{Mod}_{\mathfrak{z}},\ \pi\mapsto \mathrm{Hom}_{\mathcal{B}}(\mathcal{W}_{\psi,\mathfrak{s}},\pi). \end{align*}$$

is an equivalence. Implicit in these statements, we used the existence and uniqueness of Whittaker models. More precisely, for each irreducible, supercuspidal representation $\pi \in \mathcal {B}$ (such a $\pi $ is unique up to some unramified twist), we have

$$\begin{align*}\begin{matrix} & \mathrm{Hom}_{\mathcal{B}}(\mathcal{W}_{\psi,\mathfrak{s}},\pi) \\ \cong & \mathrm{Hom}_{\mathrm{Rep}^\infty_L G_1(E)}(W_{\psi},\pi) \\ \cong & \mathrm{Hom}_{\mathrm{Rep}^\infty_L G_1(E) } (\pi^\vee,\mathrm{Ind}_{N(E)}^{G_1(E)} \psi) \\ \cong & L, \end{matrix} \end{align*}$$

where $(-)^\vee $ denotes the smooth dual, and the last step uses the uniqueness and existence of Whittaker models for irreducible, supercuspidal representations (cf. [Reference Gel’fand and Kazhdan21]).

Let

$$\begin{align*}\mathcal{C}\subseteq \mathrm{Rep}^\infty_L G_b(E) \end{align*}$$

be a supercuspidal Bernstein block. By Lemma 7.7 (2), it suffices to describe the compositions

$$\begin{align*}F_{\mathcal{B},\mathcal{C}}\colon \mathcal{B}\subseteq D_{\text{lis}}(\mathrm{Bun}_n,L)\xrightarrow{\mathrm{Av}_{\mathbb{L}^\vee}[1]} D_{\text{lis}}(\mathrm{Bun}_n,L)\to D(\mathcal{C}) \end{align*}$$

for all such $\mathcal {B}, \mathcal {C}$ . The block $\mathcal {C}$ is again equivalent to modules over its center R, and thus (by commutation of $\mathcal {F}_{\mathcal {B},\mathcal {C}}$ with direct sums), we have

$$\begin{align*}F_{\mathcal{B},\mathcal{C}}(-)\cong M \otimes_{\mathfrak{z}}^{L}(-) \end{align*}$$

for some complex $M\in D(R)\cong D(\mathcal {C})$ with action by $\mathfrak {z}$ (i.e., an $R\otimes _{L}\mathfrak {z}$ -module). As $F_{\mathcal {B},\mathcal {C}}$ commutes with direct products and preserves compact objects, M is a perfect $\mathfrak {z}$ -complex. Let $x\in \mathrm {Spec}(\mathfrak {z})$ be a closed point, with corresponding irreducible, supercuspidal representation $\pi ^{\prime }$ . Let $\mathbb {L}^{\prime }$ the the Langlands correspondent of $\pi ^{\prime }$ . Then by Theorem 7.8

$$\begin{align*}M\otimes_{\mathfrak{z}}^L k(x)=F_{\mathcal{B},\mathcal{C}}(k(x))=0, \end{align*}$$

if $\mathbb {L}^{\prime }$ is not isomorphic to $\mathbb {L}$ or $\mathbb {L}(\chi _{\mathrm {cyc}})$ because if $\mathbb {L}^{\prime }$ is not isomorphic to $\mathbb {L}$ or $\mathbb {L}(\chi _{\mathrm {cyc}})$ the cohomology $R\Gamma (W_{E},\mathbb {L}^{\prime }\otimes _L \mathbb {L}^\vee )$ vanishes. Let $\pi :=\mathrm {LL}_1(\mathbb {L})$ with corresponding closed point $x_0\in \mathrm {Spec}(\mathfrak {z})$ , and let $x_1\in \mathrm {Spec}(\mathfrak {z})$ be the point corresponding to $\mathrm {LL}_1(\mathbb {L}(\chi _{\mathrm {cyc}}))$ . Let $y_0, y_1\in \mathrm {Spec}(R)$ be the closed points corresponding to $\mathrm {LL}_b(\mathbb {L})$ resp. $\mathrm {LL}_b(\mathbb {L}(\chi _{\mathrm {cyc}}))$ . Then we can deduce from Theorem 7.8 that

$$\begin{align*}M\otimes_{\mathfrak{z}}^L k(x_0)\cong k(y_0)[1]\oplus k(y_0), \end{align*}$$

while

$$\begin{align*}M\otimes_{\mathfrak{z}}^L k(x_1)\cong k(y_1)\oplus k(y_1)[-1]. \end{align*}$$

From here, we can deduce that $M\cong k(x_0)\oplus k(x_1)[-1]$ (as a $\mathfrak {z}$ -complex). This implies

$$\begin{align*}j^\ast_b\mathrm{Av}_{\mathbb{L}^{\vee}}(\mathcal{W}_{\psi})[1]\cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})}\oplus \mathcal{F}_{\mathrm{LL}_b(\mathbb{L}(\chi_{\mathrm{cyc}}))}[-1]. \end{align*}$$

This first implies that

$$ \begin{align*}k(\mathbb{L})_1\ast \mathcal{W}_{\psi} \neq 0. \end{align*} $$

Indeed, if it were zero, we would get

$$ \begin{align*}k(\mathbb{L}(\chi_{\mathrm{cyc}}))_1\ast \mathcal{W}_\psi[-1] \cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})}\oplus \mathcal{F}_{\mathrm{LL}_b(\mathbb{L}(\chi_{\mathrm{cyc}}))}[-1]. \end{align*} $$

But applying the same reasoning as above to $\mathbb {L}(\chi _{\mathrm {cyc}})$ instead of $\mathbb {L}$ , we have

$$ \begin{align*}{k(\mathbb{L}(\chi_{\mathrm{cyc}}))_1}\ast \mathcal{W}_\psi \oplus k(\mathbb{L}(\chi_{\mathrm{cyc}}^2))_1[-1]\ast \mathcal{W}_{\psi} \cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L}(\chi_{\mathrm{cyc}}))}\oplus \mathcal{F}_{\mathrm{LL}_b(\mathbb{L}(\chi_{\mathrm{cyc}}^2))}[-1], \end{align*} $$

contradicting this isomorphism (since $\mathcal {F}_{\mathrm {LL}_b(\mathbb {L})}[1]$ is not a direct summand of the right-hand side). In fact, these isomorphisms show that we cannot have

$$ \begin{align*}k(\mathbb{L}(\chi_{\mathrm{cyc}}))_1\ast \mathcal{W}_\psi[-1] \cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})} \end{align*} $$

and so

$$ \begin{align*}k(\mathbb{L})_1\ast \mathcal{W}_\psi \cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})}. \end{align*} $$

This finishes the proof.

By Lemma 7.2 and Theorem 7.9, we can conclude that each basic stalk of $\mathrm {Aut}_{\mathbb {L}}$ is irreducible. We also know it is a single supercuspidal representation, put in some degree. Let us identify this supercuspidal representation.

We will only explain how to identify the representation of $G_b(E)$ corresponding to $j_b^\ast \mathrm {Aut}_{\mathbb {L}}$ for b basic with $\kappa (b) \geq 1$ . The case where $\kappa (b) \leq 1$ is dealt with similarly, by replacing in what follows the standard representation $V_{\mathrm {std}}$ of $\hat {G}$ by its dual.

We argue inductively. Theorem 7.9 tells us that

$$ \begin{align*}j_{b}^\ast \mathrm{Aut}_{\mathbb{L}} \cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})}, \end{align*} $$

where b is the basic element with $\kappa (b)=1$ .

Let $b\in B(G)$ basic with $\kappa (b)\geq 1$ , and c basic with $\kappa (c)=\kappa (b)+1$ . Assume we know that $j_b^\ast \mathrm {Aut}_{\mathbb {L}}$ agrees up to shift with $\mathcal {F}_{\mathrm {LL}_b(\mathbb {L})}$ :

$$ \begin{align*}j_b^\ast \mathrm{Aut}_{\mathbb{L}}\cong \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})}[k_b], \end{align*} $$

$k_b\in \mathbb {Z}$ .

By the Hecke eigensheaf property of $\mathrm {Aut}_{\mathbb {L}}$ and our assumption, we have

$$ \begin{align*}j_c^*\mathrm{Aut}_{\mathbb{L}} \otimes_L \mathbb{L} \cong j_c^\ast T_{V_{\mathrm{std}}}(j_{b,!} \mathcal{F}_{\mathrm{LL}_b(\mathbb{L})}[k_b]). \end{align*} $$

Therefore, by [Reference Hansen, Kaletha and Weinstein19, Theorem 6.5.2], we deduce that

$$ \begin{align*}\mathrm{tr}(j_c^*\mathrm{Aut}_{\mathbb{L}}) = (-1)^{k_b}\mathrm{tr}(\mathrm{LL}_c(\mathbb{L})) \end{align*} $$

in $\mathrm {Dist}(G_c(E)_{\mathrm {ell}},L)^{G_c(E)}$ , where $G_c(E)_{\mathrm {ell}}$ denotes the elliptic regular locus in $G_c(E)$ . Since $j_c^*\mathrm {Aut}_{\mathbb {L}}$ corresponds to an irreducible supercuspidal representation, and since such representations are entirely determined by the restriction of their Harish-Chandra distribution character to the elliptic locus, this forces $j_c^*\mathrm {Aut}_{\mathbb {L}}$ to agree with $\mathcal {F}_{\mathrm {LL}_c(\mathbb {L})}$ , up to some shift $k_c \in \mathbb {Z}$ (having the same parity as $k_b$ ). This concludes the induction step.

Finally, it remains to show that all the shifts $k_b$ appearing above are zero – that is, that $\mathrm {Aut}_E$ is concentrated in degree $0$ (we so far only know that $k_b=0$ if b is such that $\kappa (b)=1$ ). But concentration in degree $0$ is implied by [Reference Hansen18, Theorem 1.1].

Altogether, this concludes the proof of Theorem 1.2.

Remark 7.10. Let $b, c\in B(G)$ basic with $\kappa (c)=\kappa (b)+1$ . We expect that

$$ \begin{align*}R\Gamma(\mathcal{M}_{b,c}, L) \end{align*} $$

is concentrated in degrees $[0,n-1]$ . When $b=1$ , this can be deduced from the fact that $\mathcal {M}_{b,c}$ is the infinite level Lubin-Tate space, which is an inverse limit of smooth Stein rigid-analytic varieties, but we do not know how to prove it for all b. If this were true, using commutation of $T_{V_{\mathrm {std}}}$ with Verdier duality, one could deduce concentration in degree $0$ of $j_b^\ast \mathrm {Aut}_{\mathbb {L}}$ for all b (we learnt this argument from a paper of Fargues, [Reference Fargues9]; it was recently reconsidered by Hansen, [Reference Hansen18]).

Note that $\mathcal {M}_{b,c}$ is Zariski-closed in the open perfectoid unit ball X of dimension n parametrizing maps from $\mathcal {E}_b$ to $\mathcal {E}_c$ , cut out by one equation. So, in fact, concentration in degree $0$ of $\mathrm {Aut}_{\mathbb {L}}$ would follow from the statement that if Z is a Zariski-closed subspace in the n-dimensional affinoid perfectoid ball, $R\Gamma (Z,L)$ vanishes in degrees bigger than the Krull dimension of Z.Footnote ⁹ We refer the interested reader to [Reference Hansen18] for more on this.