Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications

In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism.


Introduction
Let X be a smooth projective geometrically connected curve over a finite field F q . We denote by F its function field, A the ring of adèles of F and O the ring of integral adèles.
Let G be a connected split reductive group over F q .
Let Ξ be a cocompact subgroup in Z G (F)\Z G (A), where Z G is the center of G. Then the quotient Z G (F)\Z G (A)/Z G (O)Ξ is finite. Let N ⊂ X be a finite subscheme. We denote by O N the ring of functions on N and K N := Ker(G(O) → G(O N )).
Let be a prime number not dividing q. Let E be a finite extension of Q containing a square root of q. Let O E be the ring of integers of E. We denote by C c (G(F)\G(A)/K N Ξ, E) the vector space of automophic forms with compact support.
Let u be a place in X N . Let O u be the complete local ring at u and let F u be its field of fractions. Let H G,u := C c (G(O u )\G(F u )/G(O u ), E) be the Hecke algebra of G at the place u. The algebra H G,u acts on Since H 0 G,N ,∅,1,x = C c (G(F)\G(A)/K N Ξ, E), Theorem 2 is a generalization of Proposition 1. The proof of Theorem 2 is given in Section 2. The idea is similar to the case of automorphic forms explained in Section 1. In addition, we use the constant term morphisms of the cohomology groups of stacks of shtukas and the contractibility of deep enough Harder-Narasimhan strata established in [Xue20].
As an application, in Section 3, we extend the excursion operators, defined in [Laf18] on the vector space of cuspidal automorphic forms C 1.0.3. Let l := deg(u). Let g be the genus of X. Let d 0 = max{0, g − 1} + l. Lemma 1.0.4. We have Proof. Applying successively Lemma 1.0.8 below to d 0 + 1, d 0 + 2, d 0 + 3, · · · , we deduce that for any d > d 0 , 1.0.6. The action of h G ω on F • G is defined in the following way: let Γ (G(O u ) ω G(O u )) be the groupoid classifying pairs (G G ), where G, G are rank 2 vector bundles of trivial determinant, G G is an isomorphism outside u such that when restricted to the formal disc on u, the relative position of G and G is equal to ω . We have a Hecke correspondence (where the arrows are morphisms of groupoids): The proof of Lemma 1.0.7 will be given in the remaining part of this section.
Lemma 1.0.8. For any d ∈ Z ≥0 such that d > d 0 , we have Proof. This is a direct consequence of Lemma 1.0.7.
We have a constant term morphism When d ≥ 1, we have a commutative diagram: Proof. (i) For any d ∈ Z, the action of h T ω induces an isomorphism: As a consequence, the Hecke operator Applying (1.5) to all d ≤ d and taking the product, we get a morphism: Recall that gr d Proof of Lemma 1.0.7. Let d > l. Since the constant term morphism commutes with the action of the Hecke algebra H G,u , we have a commutative diagram then the two vertical morphisms in (1.6) are isomorphisms. Moreover, by Lemma 1.0.13, the lower line in (1.6) is an isomorphism. So the upper line in (1.6) is an isomorphism.
Remark 1.0.14. For a general group G, to prove Proposition 1, we need [DG15, Proposition 9.2.2] to show that an analogue of (1.7) is bijective. We do not give details here because we will prove some more general statements in Section 2.

Proof of Theorem 2
In this section, we consider the general case.

Reminder of cohomologies of stacks of shtukas
The stacks of shtukas and their cohomologies are defined in [Var04], recalled in [Laf18, Section 2] and in [Xue20, Section 1 and 2].
As in the introduction, let N ⊂ X be a finite subscheme. Let I be a finite set and W be a representation of G I (in this paper, this always means a finite dimensional E-linear representation of G I ).
where pr 1 and pr 2 are finite étale morphisms. For any µ ∈ 1 r R + G ad , the projection pr 1 sends pr −1 2 (Cht

2.1.15.
For i ∈ I, let pr i : X I → X be the projection to the i-th factor. From now on, let x be a geometric point of (X N ) I such that for every i, j ∈ I, the image of the composition is not included in the graph of any non-zero power of Frobenius morphism Frob : X N → X N . (This condition will be needed in 2.3.11 and Proposition 2.3.15.) In particular, when i = j ∈ I, the above condition is equivalent to the condition that the composition x → (X N ) I pr i − − → X N is over the generic point η of X N . One example of geometric point satisfying the above condition is η I , a geometric point over the generic point η I of X I . Another example of geometric point satisfying the above condition is ∆(η), where ∆ : X → X I is the diagonal inclusion and η is a geometric point over η.
However, for any i ∈ I, let Frob {i} : X I → X I be the morphism sending (x j ) j∈I to (x j ) j∈I , with x i = Frob(x i ) and x j = x j if j i. Then Frob {i} ∆(η) does not satisfy the above condition.
Since every F µ G has finite dimension, Theorem 2 is a direct consequence of the following proposition: Proposition 2.2.4. There exists µ 0 ∈ 1 r R + G ad large enough (a priori depending on u and x), such that Proposition 2.2.4 will follow from Lemma 2.2.8 below.

2.2.5.
We denote by Γ G the set of vertices of the Dynkin diagram of G. For any i ∈ Γ G , we denote by α i (resp.α i ) the simple root (resp. simple coroot) of G associated to i. We denote by , the natural pairing between coweights and weights. Let P i be the standard maximal parabolic with Levi quotient where Γ M i is the set of vertices of the Dynkin diagram of M i . By a quasi-fundamental coweight of G we mean the smallest positive multiple of a fundamental coweight of G ad , which is a coweight of G. Let ω i be the quasi-fundamental coweight of G such that ω i , α j = 0 for any j ∈ Γ M i .
Notation 2.2.6. Let C G ∈ Q ≤0 be the constant in Proposition 5.1.5 (c) of [Xue20]. Let Remark 2.2.7. The above definition implies that for any µ ∈ 1 r R + G ad and any Lemma 2.2.8. Let µ ∈ 1 r R + G ad such that µ, α j > C(G, u) for all j ∈ Γ G , then for any i ∈ Γ G , we have The proof of Lemma 2.2.8 will be given after Lemma 2.2.11 below.
Proof of Proposition 2.2.4 admitting Lemma 2.2.8. We use the same argument as in the proof of Lemma 5.3.6 in [Xue20]. Let Z(C) be the set of µ ∈ 1 r R + G ad such that µ, α j > C(G, u) for all j ∈ Γ G . Let Ω(C) be the set of µ ∈ Z(C) such that µ − 1 Applying successively Lemma 2.2.8 to λ (0) , λ (1) , · · · , until λ (m) , we deduce that Notation 2.2.9. For any quasi-fundamental coweight

Suppose that
They induce a morphism on the quotient spaces The proof of this lemma will be given in 2.5. The following figure shows an illustration of Lemma 2.2.11 for G = GL 3 , and i = 2, where the set S M 2 (µ) is defined in 2.3.9 below.
Proof of Lemma 2.2.8 admitting Lemma 2.2.11. We deduce from Lemma 2.2.11 that As a consequence, The goal of the remaining part of Section 2 is to prove Lemma 2.2.11. We need to use the cohomology group H M i of the stack of M i -shtukas, the constant term morphism from H G to H M i and the contractibility of deep enough Harder-Narasimhan strata of Cht G . We recall these in Section 2.3. Then we study the action of h G ω i on H M i in Section 2.4. Finally in Section 2.5, we use Section 2.3 and Section 2.4 to prove Lemma 2.2.11. C. Xue

Cohomology of stacks of M-shtukas and constant term morphism
For the convenience of the reader, we extract some results from [Xue20]. The goal of this subsection is to state Corollary 2.3.17.
As before, let I be a finite set and W be a representation of G I . Let P be a standard parabolic subgroup of G. Let M be the Levi quotient of P .

As in
be the projection. In [Xue20, § 1.5.13], we defined a partial order "≤" on Λ where ≤ is the partial order in Λ Q G ad .

2.3.10.
In [Xue20, Definition 4.1.10], we defined a locally closed substack Cht of Cht G and an open and closed substack Cht

2.3.11.
In [Xue20, § 3.5.8 and 4.6.3] for any µ ∈ 1 r R + G ad , we defined the constant term morphism: We have a long exact sequence of compact support cohomology groups: In particular, f j and f j+1 in (2.5) are injective. So we have a short exact sequence: Thus the left vertical map C P , j, ≤µ G is an isomorphism.

Action of Hecke operators
Let M as in Notation 2.3.12. The goal of this section is to prove Lemma 2.4.3.

Let
To prove this lemma we need some preparations.

We define
Since ω is dominant for G, for any θ ∈ Ω, we have θ ≤ ω for the order in Λ G .

For any dominant coweight
. We can view h G ω as an element in H M,u by the Satake transformation H G,u → H M,u . We have the following equality (up to multiplication by a power of q) in H M,u : Proof of Lemma 2.4.3. Recall that in 2.3.4 we defined a projection pr ad We will write pr := pr ad P to simplify the notations.
where pr 1 and pr 2 are finite étale morphisms. Combining (2.14), (2.15) and (2.16), we deduce that the RHS of (2.13) can be sent in is an isomorphism. Finally, we deduce that (2.9) is an isomorphism.

Proof of Lemma 2.2.11
By [Xue20, Lemma 6.2.12], the action of H G,u commutes with C P G . The following diagram is commutative, where the horizontal lines are the exact sequences in 2.3.13 and 2.3.14, the vertical morphisms are constant term morphisms: Consider the right face (c): where the isomorphism of the lower line follows from Lemma 2.4.3. Moreover, by Corollary 2.3.17, the vertical morphisms are isomorphisms. We deduce that the upper line is an isomorphism.  . Let y be a geometric point of X I such that for every i, the composition y → (X N ) I pr i − − → X N is over the generic point η of X N . Then there exists x satisfying the condition in 2.1.15 and (n i ) i∈I ∈ N I such that y = Frob  (Frob(∆(η))), where Frob is the total Frobenius so Frob(∆(η)) = ∆(η), which satisfies 2.1.15.) We deduce from (2.18) that Proposition 2.2.4 is also true for y.

Global excursion operators
Recall that in Definition 2.1.11, we defined an inductive limit of E-constructible sheaves H It induces the homomorphism of specialization: ( {i} )(∆(η)), (m i ) i∈I ∈ N I } is Zariski dense in X I . Thus there exists (n i ) i∈I ∈ N I such that ( i∈I Frob n i {i} )(∆(η)) ∈ Ω 0 .
Step 2. The image by i∈I Frob n i {i} of the specialization map (3.1) gives a specialization map sp : We have a commutative diagram: where Frob is a shortcut of i∈I Frob  . This proves the surjectivity of (b). Step

Drinfeld's lemma
where (F I ) perf is the perfection of F I , Frob {i} is the partial Frobenius defined in Notation 3.1.3. C. Xue 20 C. Xue
We have π 1 (η I , η I ) = Gal(F I /F I ). Let Weil(η I , η I ) be the Weil group in π 1 (η I , η I ) (also denoted by Weil(F I /F I )). That is to say, As in [Laf18,rem. 8.18], the specialization map sp induces an inclusion (depending on the choice of sp) and where Z I has the discrete topology.

3.2.3.
We denote by Frob : F I → F I the absolute Frobenius morphism over F q . We have a morphism  [ (η I ,η I ) ) contains Q. Taking the limit on n we get the lemma.
Since π geom 1 (η I , η I ) is profinite and H is a closed and normal subgroup, the quotient group π Proof. For any maximal ideal m of A, since A is finitely generated over E (in particular A is Noetherian), for any n ∈ N, the quotient A/m n is of finite dimension over E. Since M is an A-module of finite type, M/m n M is an A/m n -module of finite type. Thus M/m n M is an E-vector space of finite dimension.
Applying Lemma 3.2.10 to M/m n M, we deduce that the action of Q on M/m n M is trivial. Since A is Noetherian, for any q ∈ Q and x ∈ M, we have where M m is the localization of M on A − m. (a) follows from [Mat89, Theorem 8.9] and (b) follows from [Mat89, Theorem 4.6]. We deduce that q · x = x. Thus the action of Q on M is trivial.
3.2.14. By the discussion after remarque 8.18 of [Laf18], we have a continuous action of FWeil(η I , η I ) on H j I,W η I (depending on the choice of η I and sp) which combines the action of π 1 (η I , η I ) and the action of the partial Frobenius morphisms.
Concretely, let θ ∈ FWeil(η I , η I ) such that θ ( It induces a specialization map (which is in fact a morphism): The action of θ on H j I,W η I is defined to be the composition: where the second map is the partial Frobenius morphism on H

More on Drinfeld's lemma
We need the following variant of [Laf18, lem. 8.2].
with an action of Weil(F q /F q ): φ : (Id Y × Frob) * F ∼ → F. The category of smooth E-sheaves over Y is a full subcategory of the category of smooth Weil E-sheaves over Y .
We have an equivalence of categories • the composition with the restriction functor of the representations of Weil(U , η) I to the representations of By Lemma 3.2.10, this morphism factors through Weil(η, η) I . We deduce a continuous morphism: Since E is unramified over U I , the above morphism factors through Weil(U , η) I . We deduce a continuous morphism: Since E is smooth over U I , the homomorphism of specialization sp * : E ∆(η) → E η I is an isomorphism. We deduce a continuous morphism (not depending on η I and sp): Weil(U , η) I → Aut(E ∆(η) ).
We will also need the following lemma, whose proof is the same as in [Laf18,lem. 8.12].
Lemma 3.3.4 (rational coefficients version of lemme 8.12 in [Laf18] is induced by sp * (defined in Construction 3.4.1); • the action on E ∆(η) is given by Lemma 3.3.2, hence is independent of the choice of η I and sp.
By 3.4.3, f is independent of the choice of η I and sp. We deduce that the action of Weil(η, η) I on is independent of the choice of η I and sp.
Step 2. Through the morphism (3.12), the action of Weil(η, η) I on H is compatible with the action ) ∧ m . Applying Step 1 to I = m n , n ∈ N, we deduce that the action of Weil(η, η) I on m (H .

3.4.5.
Let I be a finite set and W be a representation of G I . Let x ∈ W and ξ ∈ W * be invariant under the diagonal action of G. We denote by E (X N ) the constant sheaf over X N . In [Laf18, défi. 5.1-5.2] and the beginning of Section 9 of loc. cit., V. Lafforgue defined the creation operator , and the annihilation operator (3.14) We denote by C x η (resp. C ξ η ) the restriction of (3.13) (resp. (3.14)) on η.
Construction 3.4.6. Let I, W , x and ξ as in 3.4.5. Let (γ i ) i∈I ∈ Weil(η, η) I . We construct an excursion operator S I,W ,x,ξ,(γ i ) i∈I on C c (Bun G,N (F q )/Ξ, E) as the composition of morphisms: where the action of (γ i ) i∈I is defined in Construction 3.4.1. The action of S I,W ,x,ξ,(γ i ) i∈I commutes with the action of the global Hecke algebra Remark 3.4.8. In [Laf18,défi. 9.3], the notation H I,W is used for the Hecke-finite part of the cohomology group in degree 0, which we denote in this paper by (H 0 I,W ∆(η) ) Hf . In this paper, the notation H j I,W is used for the cohomology group in degree j, which is also denoted by H The proof is the same as in [Laf18, lemme 10.6].

3.4.11.
For any place v of X, fix an algebraic closure F v of F v and fix the embeddings such that the following diagram commutes: Let k v be the residue field of F v and let k v be the residue field of the maximal unramified extension of be the specialization map associated to F ⊂ F v . We denote still by sp v the image by ∆ of the above specialization map (3.16) Lemma 3.4.12 (cf. lemme 10.4 in [Laf18] for the cuspidal part). Let v be a place in X N . Let us consider is the partial Frobenius morphism, C x v (resp. C ξ v ) is the restriction of (3.13) (resp. (3.14)) on v.
The proof will be given in the next subsection.
C. Xue 28 C. Xue 3.4.13. Let V be a representation of G and V * be the dual of V . Let δ V : 1 → V ⊗V * and ev V : V ⊗V * → 1 be the canonical morphisms.
Let v ∈ |X N |. Let h V ,v ∈ H G,v be the spherical function associated to V by the Satake isomorphism. Let T (h V ,v ) be the Hecke operator on C c (Bun G,N (F q )/Ξ, E) associated to h V ,v .
Proof. This follows from Lemma 3.4.12 above and [Laf18, prop. 6.2] (where the statement is already for the whole cohomology, not only for the cuspidal part).
Proposition 3.4.15 (cf. lemme 10.10 in [Laf18] for the cuspidal part). For any I and f , S I,f ,(γ i ) i∈I depends only on the image of (γ i ) i∈I in Weil(X N , η) I .
Proof. (The argument is the same as in [Laf18,lem. 10.10].) Thus for (γ i ) i∈I ∈ Weil(F/F) I and (δ i ) i∈I ∈ (I v ) I , we have S I,f ,(γ i ) = S I,f ,(δ i γ i ) . We have this for any choice of inclusion F ⊂ F v .
Since Weil(X N , η) is the quotient of Weil(η, η) by the subgroup generated by the I v for v ∈ X N and their conjugates, we deduce that S I,f ,(γ i ) depends only on the image of (γ i ) by Weil(η, η) I → Weil(X N , η) I .
Remark 3.4.16. The statement [Laf18,prop. 8.10] can be generalized to the excursion operators acting on C c (Bun G,N (F q )/Ξ, E) and the proof is the same. So we will not state them here.

Proof of Lemma 3.4.12
Lemma 3.4.12 will follow from Lemma 3.5.4 below.
Lemma 3.5.1 (rational coefficients version of lemme 8.15 in [Laf18]). Let U be an open subscheme of X. We denote by j I : U I → X I the inclusion. Let E be a smooth E-sheaf over U I , equipped with the partial Frobenius morphisms. Let v be a place in X as in 3.4.11. Let (γ i ) i∈I and (d i ) i∈I as in Lemma 3.4.12. Then the following diagram is commutative where the vertical map on the right is the action of Weil(U , η) I on E ∆(η) given by Lemma 3.3.2.
Proof. (The argument is the same as in [Laf18,lem. 8.15].) It is enough to prove the lemma with E of the form i∈I E i (as in Lemma 3.3.2). Noting j : U → X the inclusion, we have ((j I ) * E) ∆(v) = ⊗ i∈I (j * E i v ). Thus it is enough to prove the lemma in the case where I is a singleton. In this case, the commutativity follows from the definition of deg : Definition 3.6.5. We denote by B I the sub-E-algebra of generated by all the excursion operators S I,f ,(γ i ) i∈I . It is finite dimensional. By Lemma 3.4.9, B I is commutative.
By Lemma 3.4.14, B I contains the Hecke algebras at all the places of X N .
Lemma 3.6.6 (cf. proposition 10.10 in [Laf18] for the cuspidal part). The morphism is continuous, where B I is endowed with the E-adic topology.
Proof. The proof is the same as [Laf18,prop. 10.10], except that we use Lemma 3.5.4 (of this paper) instead of [Laf18,lem. 10.4].
Then we use the same arguments as in [Laf18] Section 11, except that we replace C cusp c (Bun G,N (F q )/Ξ, Q ) by C c (Bun G,N (F q )/Ξ, Q )/I · C c (Bun G,N (F q )/Ξ, Q ) and replace π 1 (X N , η) by Weil (X N , η). We obtain: This decomposition is characterized by the following property: H σ is equal to the generalized eigenspace H ν associated to the character ν of B I defined by ν(S I,f ,(γ i ) i∈I ) = f ((σ (γ i )) i∈I ).
It is compatible with the Satake isomorphism at every place v of X N : for any irreducible representation V of G, we denote by T (h V ,v ) the Hecke operator at v associated to V . Then H σ is included in the generalized eigenspace of T (h V ,v ) for the eigenvalue ν(T (h V ,v )) = χ V (σ (Frob v )), where χ V is the character of V and Frob v is an arbitary lifting of the Frobenius element on v.
Remark 3.6.8. Contrary to the case of the cuspidal part C cusp c (Bun G,N (F q )/Ξ, Q ) in [Laf18], here the actions of the Hecke operators on C c (Bun G,N (F q )/Ξ, Q )/I · C c (Bun G,N (F q )/Ξ, Q ) are not always diagonalizable.

Excursion operators on cohomology groups
3.7.1. Let J be a finite set and V be a representation of G J . Let j ∈ Z. Applying Definition 2.1.11 to J and V , we define H j J,V , which is an inductive limit of constructible sheaves on (X N ) J . Let I be a finite set and W be a representation of G I . Let x ∈ W and ξ ∈ W * be invariant under the diagonal action of G. Let (γ i ) i∈I ∈ Weil(η, η) I . We will construct an excursion operator S I,W ,x,ξ,(γ i ) i∈I on , where ∆ J : X N → (X N ) J is the diagonal morphism.
Proposition 3.7.2 (cf. proposition 4.12 in [Laf18]). Let I 1 , I 2 be two finite sets and ζ : I 1 → I 2 be a map. Let ∆ ζ : X I 2 → X I 1 , (x j ) j∈I 2 → (x ζ(i) ) i∈I 1 be the morphism associated to ζ. Let W be a representation of Proof. The proof consists of 4 parts.
Step 1. The constant term morphisms commute with the creation operators: the following diagram is commutative where W ζ I is the representation of G via the diagonal inclusion G → G I , χ −1 ζ I and H(x) are defined in [Laf18, défi. 5.1] and C x = χ −1 ζ I • H(x). Indeed, the commutativity of (a) comes from the fact that C P , ν G is functorial on W . The commutativity of (b) follows from Lemma 4.1.5 below applied to J = {0}.
Step 2. The constant term morphisms commute with the specialization morphisms: the following diagram is commutative: Step 2. Similarly, we have a commutative diagram of stacks of shtukas, where all squares are cartesian: