Semisimplicity of certain Galois representations occurring in étale cohomology of unitary Shimura varieties

Karam Fayad, Jan Nekovář

Abstract: We show that a certain part of the middle degree ℓ-adic cohomology of an arbitrary simple unitary Shimura variety is a semisimple Galois representation.

Introduction

In [N], the second named author developed a new method for obtaining non-trivial information about the Galois action on certain parts of ℓ-adic étale cohomology of Shimura varieties, which was then applied to quaternionic Shimura varieties.

The goal of the general Langlands–Kottwitz programme is to describe (up to semisimplification) the action of the Hecke algebra and of the absolute Galois group Γ_{E} of the reflex field E on the full ℓ-adic cohomology of a given Shimura variety Sh(G, 𝒳 ), using a variety of sophisticated automorphic and geometric tools. The method of [N] combines more elementary techniques involving Sh(G, 𝒳 ) with deep automorphic results coming from different – usually simpler – Shimura varieties.

In more concrete terms, fix a cohomological automorphic representation π = π_{∞} ⊗ π^{∞} of G(A) and denote by ρ the Galois representation given by the Galois action on the subspace of ℓ-adic cohomology on which the Hecke algebra acts via the non-archimedean component π^{∞} of π. According to [N], the Galois representation ρ can be analysed using the following three ingredients.

(1) A very strong automorphic input: the existence of a certain “standard” ℓ-adic Galois representation ρ_{π} (satisfying a number of properties) attached to π.

(2) A non-trivial, but relatively modest geometric input from the Shimura variety Sh(G, 𝒳 ): the Eichler– Shimura relations (either for the full Frobenius or for all partial Frobenii) at all primes p of good reduction such that G splits over Q_{p}.

(3) New tools from representation theory of Lie algebras and profinite groups.

The main new result proved in [N] using these techniques was a proof of the semisimplicity of the Galois action on ℓ-adic cohomology of quaternionic Shimura varieties (with a minor restriction in the case of coefficients of non-motivic weight).

The principal abstract semisimplicity criterion proved in [N, Thm. 2.6] and recalled in 1.4 below is fairly strong, even though its Lie-algebraic nature means that it can be applied only in those situations when the Galois representation ρ_{π} is strongly irreducible. This criterion was applied in [N] only through its more concrete consequence [N, Thm. 3.3]. However, the latter theorem is insufficient for applications to Shimura varieties more general than the quaternionic ones, for the following reason. Under appropriate assumptions inherent in the method of [N], the expected form ρ_{exp} of the semisimplification ρ^{ss} of ρ is given by a classical formula due to Langlands

L(ρ_{exp}, s) = L(π, r_{−μ}, s − dim/2)^{m},  (0.1)

where m ≥ 1 is a suitable integer, dim is the dimension of Sh(G, 𝒳 ) and μ is the cocharacter attached to the Shimura datum. The main drawback of [N, Thm. 3.3] is the fact that its assumption (B) amounts to saying that ρ^{ss} does not differ too much from ρ_{exp}; more precisely, that there exist an integer a ≥ 1 and an open subgroup U ⊂ Γ_{E} such that

ρ^{ss}|_{U} ⊂ (ρ_{exp}|_{U})^{⊕a} .  (0.2)

In other words, in order to apply [N, Thm. 3.3] one needs to know, in the given situation, a weak form of the conjecture of Langlands–Kottwitz

ρ^{ss} =^{?} ρ_{exp}.  (0.3)

For quaternionic Shimura varieties considered in [N] it was possible to verify (0.2) by two independent arguments: either by invoking deep results of Langlands [La] (and Brylinski–Labesse [BL]) that imply the strong statement (0.3), or by appealing to a combination of the Eichler–Shimura relations with [N, Prop. 3.10(1)]. Unfortunately, the latter argument works only in the cases when ρ_{π} has a very large image. In the quaternionic case, when ρ_{π} is the two-dimensional Galois representation attached to a cuspidal Hilbert modular eigenform f, the latter condition is automatic when ρ_{π} is strongly irreducible (i.e., when f does not have complex multiplication), since \overline{Q}_{ℓ} · Lie(Im(ρ_{π})) = gl(2) acts on \overline{Q}_{ℓ}^{2} by its standard representation, which is minuscule.

However, this minuscule property does not hold for general n-dimensional irreducible representations of Lie algebras if n > 2, nor for many n-dimensional Galois representations of the form ρ_{π}, which means that for such π one cannot deduce (0.2) from the Eichler–Shimura relations (cf. the discussion in [N, 0.15]).

This unsatisfactory state of affairs was a starting point of the first author’s thesis [F] at Université Pierre et Marie Curie (Paris 6). In the present article, which grew out of [F], we first deduce from [N, Thm. 2.6] a new semisimplicity criterion (Theorem 1.7 below) which replaces [N, Thm. 3.3] and which seems to be optimal for applications to Shimura varieties (when ρ_{π} is strongly irreducible; certain special cases of irreducible, but not strongly irreducible ρ_{π} were treated by ad hoc methods in [N, §4] and [F, ch. 2]).

Apart from strong irreducibility of ρ_{π}, the assumptions of Theorem 1.7 only require the Eichler–Shimura relations for partial Frobenii (at split primes) and a multiplicity one assumption, which ties in naturally with the multiplicity one property of the Hodge–Tate weights of the n-dimensional Galois representations ρ_{Π,ℓ} discussed below.

In particular, Theorem 1.7 avoids using (0.2) as an input, which means that it can be applied even in a situation when the Langlands–Kottwitz conjecture (0.3) is not yet proved and the Galois representation ρ_{π} does not have sufficiently large image (hence [N, Prop. 3.10(1)] cannot be used to deduce (0.2) from the Eichler–Shimura relations).

Once Theorem 1.7 is proved, one can apply the methods of [N] to Shimura varieties attached to unitary similitude groups G^{∗} defined by general simple PEL data of type (A), and to slightly larger groups G ⊃ G^{∗}, for which the similitude factor lies in the underlying totally real number field F, rather than in Q (see 2.1 and 2.2 for precise definitions). In this setting, the automorphic input (1) is provided by the existence and fundamental properties of the n-dimensional ℓ-adic representations ρ_{Π,ℓ} : Γ_{F_{c}} ⟶ GL_{n}(\overline{Q}_{ℓ}) attached to cohomological self-dual cuspidal automorphic representations Π of the group GL(n) over the underlying CM field F_{c}. The geometric input (2) comes from the Eichler–Shimura relations in the form proved by Wedhorn [W, §5-6] (see also [Mo, §4]), with the sign conventions corrected as in [N, Appendix]. Finally, the abstract semisimplicity criterion (3) is given by Theorem 1.7.

In order to state the main semisimplicity result obtained by combining (1), (2) and (3) we need to introduce some notation. Fix an isomorphism C ≃ \overline{Q}_{ℓ}. An irreducible algebraic representation ξ^{∗} of G_{C}^{∗} then defines an ℓ-adic lisse sheaf ℒ_{ξ^{∗},ℓ} on the Shimura variety Sh(G^{∗}, 𝒳 ^{∗}). The common dimension of this Shimura variety and the one attached to G is dim = ∑_{x∈X} a_{x}b_{x}, where X = {F ↪ R} and G(R) = ∏_{x∈X} GU(a_{x}, b_{x}). For any i ≥ 0, let

H_{∗}^{i} := H_{et}^{i}(Sh(G^{∗}, 𝒳 ^{∗}) ⊗_{E} \overline{Q}, ℒ_{ξ^{∗},ℓ}) = \underrightarrow{lim}_{K^{∗}} H_{et}^{i}(Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) ⊗_{E} \overline{Q}, ℒ_{ξ^{∗},ℓ}).

(2.20) Theorem. Let ξ^{∗} be an irreducible algebraic representation of G_{C}^{∗} and π^{∗} a cuspidal automorphic representation of G^{∗}(A) such that π_{∞}^{∗} is cohomological with respect to ξ^{∗} in degree dim = ∑_{x∈X} a_{x}b_{x}. Assume that π^{∗} is a quotient of the restriction to G^{∗}(A) of a cuspidal automorphic representation π of G(A) which admits a weak transfer (in the sense of 2.14) to a cuspidal automorphic representation (Π, ψ) of GL_{n}(A_{F_{c}}) × A_{F_{c}}^{×}. If the ℓ-adic Galois representation ρ_{Π,ℓ} : Γ_{F_{c}} ⟶ GL_{n}(\overline{Q}_{ℓ}) attached to Π satisfies the following two conditions

(1) ρ_{Π,ℓ} is strongly irreducible;

(2) either (2a) \overline{Q}_{ℓ} · Lie(ρ_{Π,ℓ}(Γ_{F_{c}})) ⊃ sl(n, \overline{Q}_{ℓ}), or (2b) G(R) = ∏_{x∈X} GU(a_{x}, b_{x}) with ∀x ∈ X (a_{x}, b_{x}) ∈ {(n, 0), (n − 1, 1), (1, n − 1), (0, n)}, then the \overline{Q}_{ℓ}[Γ_{E}]-module

Hom_{G^{∗}(_{Q})}((π^{∗})^{∞}, H_{∗}^{dim})

(which is finite-dimensional over \overline{Q}_{ℓ}) is semisimple.

At the time of writing, the Langlands–Kottwitz conjecture (0.3) has not yet been established under the assumptions (1) and (2) for the (π^{∗})^{∞}-component of H_{∗}^{dim}. On the other hand, under the assumptions (1) and (2a) one can deduce the weak form (0.2) of (0.3) from the Eichler–Shimura relations and [N, Prop. 3.10(1)] (and therefore appeal to [N, Thm. 3.3] rather than to Theorem 1.7 in order to prove Theorem 2.20 in this case). Theorem 1.7 is indispensable, though, for a proof of Theorem 2.20 under the assumptions (1) and (2b).

Similar results hold for Sh(G, 𝒳 ) (Theorem 2.25) and – assuming that an analogue of the main result of [KMSW, Thm. 1.7.1] is valid for unitary similitude groups – also for the ℓ-adic intersection cohomology of the Baily–Borel compactifications of Sh(G^{∗}, 𝒳 ^{∗}) (Corollary 2.23) and Sh(G, 𝒳 ) (Corollary 2.26).

The same method should apply to Shimura varieties attached to groups GSpin(n) over totally real number fields (with some restrictions if n is even), once the corresponding Eichler–Shimura relations are proved.

Acknowledgements. We would like to thank N. Bergeron and G. Chenevier for helpful discussions.

General conventions and notation. As in [N], all representations and characters are assumed to be continuous with respect to the natural topologies involved. By an automorphic representation we mean an irreducible automorphic representation.

The characteristic polynomial of an endomorphism u of a finite-dimensional vector space over a field k will be denoted by P_{u}(X) = det(X · id − u) ∈ k[X]. If k ⊂ K are fields and X is a k-vector subspace of a K-vector space Y, we denote by K · X the K-vector subspace of Y generated by X. We abbreviate ⊗_{Z} as ⊗. For an abelian group A we let Â = A ⊗ Ẑ. We denote by A and A_{k}, respectively, the ring of adeles of Q and of a number field k, and by _{k}T the algebraic torus _{k}T = R_{k/Q}(G_{m,k}). Let \overline{Q} be the algebraic closure of Q in C. The absolute Galois group of a subfield K ⊂ \overline{Q} will be denoted by Γ_{K} = Gal(\overline{Q}/K). The reciprocity map of class field theory will be normalised by letting uniformisers correspond to geometric Frobenius elements Fr(v).

1. An abstract semisimplicity criterion

(1.1) We need to recall several general facts about Lie algebras, as stated in [N, 1.7-1.9, 2.6]. Let V be a finite-dimensional vector space over a field k ⊃ Q and 𝔤 ⊂ End_{k}(V) a k-Lie subalgebra. As in [LIE, Ch. VII, §5, no. 3], denote by 𝔫_{V} (𝔤) the set of all elements of the radical of 𝔤 that are nilpotent in End_{k}(V). It is a nilpotent ideal of 𝔤 containing the intersection of the radical with 𝒟𝔤.

Recall that 𝔤 is called a decomposable linear Lie algebra [LIE, Ch. VII, §5, Def. 1] if both the semisimple and the nilpotent part of every element of 𝔤 belong to 𝔤.

(1.2) Proposition. (1) [LIE, Ch. VII, §5, Thm. 2] The Lie algebra 𝔤 ⊂ End_{k}(V) is decomposable if and only if some (equivalently, each) Cartan subalgebra of 𝔤 is decomposable, which is, in turn, equivalent to the radical of 𝔤 being decomposable.

(2) [LIE, Ch. VII, §5, Thm. 1] If 𝔤 is generated as a k-Lie algebra by a subset S ⊂ 𝔤 such that every X ∈ S is either semisimple or nilpotent in End_{k}(V), then 𝔤 is decomposable.

(3) [LIE, Ch. VII, §5, Prop. 7] If 𝔤 is decomposable, then there exists a Lie subalgebra 𝔪 ⊂ 𝔤, reductive in End_{k}(V) (in particular, 𝔪 is a reductive Lie algebra acting semisimply on V), such that 𝔤 = 𝔪 ⋉ 𝔫_{V} (𝔤).

(4) [LIE, Ch. VII, §3, Ex. 16] The set of elements of 𝔤 that are semisimple in End_{k}(V) is Zariski dense in 𝔤 if and only if some (equivalently, each) Cartan subalgebra of 𝔤 is commutative and consists of elements that are semisimple in End_{k}(V).

(1.3) Corollary. If the set of elements of 𝔤 that are semisimple in End_{k}(V) is Zariski dense in 𝔤, then:

(1) 𝔤 is decomposable;

(2) 𝔤 = 𝔪 ⋉ 𝔫_{V} (𝔤) for a suitable reductive Lie subalgebra 𝔪 ⊂ 𝔤 acting semisimply on V;

(3) There exists a flag {0} = V_{0} ⊊ V_{1} ⊊ ⋯ ⊊ V_{s} = V of 𝔤-submodules such that 𝔫_{V} (𝔤) acts trivially (and 𝔪 semisimply) on gr(V) = ⨁_{i=1}^{s} V_{i}/V_{i−1} (one such flag is V_{0} = {0} and V_{i+1} = {v ∈ V | 𝔫_{V} (𝔤) v ∈ V_{i}}). The isomorphism class of the semisimple 𝔤/𝔫_{V} (𝔤)-module gr(V) does not depend on the choice of {V_{i}}.

(1.4) Theorem [N, Thm. 2.6]. Let V be a non-zero vector space of finite dimension over an algebraically closed field k ⊃ Q. If \overline{𝔤} ⊂ End_{k}(V) is a k-Lie subalgebra such that (H1-ZAR) \overline{𝔤} contains a Zariski dense set of elements that are semisimple in End_{k}(V), then:

(1) \overline{𝔤} ⊂ End_{k}(V) is a decomposable linear Lie algebra.

(2) Let \overline{𝔫} = 𝔫_{V} (\overline{𝔤}) and fix \overline{𝔪} ⊂ \overline{𝔤} (\overline{𝔪} ≃ \overline{𝔤}/\overline{𝔫}), a Cartan subalgebra \overline{𝔥} ⊂ \overline{𝔪} and \{0\} = V_{0} ⊊ V_{1} ⊊ ⋯ ⊊ V_{s} = V as in Corollary 1.3. Assume that the following condition holds: (H2) all weights of \overline{𝔥} occurring in gr(V) lie in one coset of the root lattice of (\overline{𝔪}, \overline{𝔥}). Then \overline{𝔫} = 0, \overline{𝔤} = \overline{𝔪} is a reductive Lie algebra and V is a semisimple \overline{𝔤}-module.

(1.5) Let Γ be a profinite group, V ≠ 0 a finite-dimensional vector space over \overline{Q}_{ℓ} and ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) ≃ GL_{n}(\overline{Q}_{ℓ}) a (continuous) representation. The image ρ(Γ) is a compact Lie group of finite dimension over Q_{ℓ}. Its Lie algebra 𝔤 = Lie(ρ(Γ)) ⊂ End_{\overline{Q}_{ℓ}}(V) is a Q_{ℓ}-Lie algebra of finite dimension. The corresponding \overline{Q}_{ℓ}-Lie algebra \overline{Q}_{ℓ} · 𝔤 ⊂ End_{\overline{Q}_{ℓ}}(V) is a quotient of \overline{𝔤} = 𝔤 ⊗_{Q_{ℓ}} \overline{Q}_{ℓ}.

It is well-known that the representation ρ is semisimple if and only if the restriction ρ|_{Γ′} to each (equivalently, to some) open subgroup Γ′ ⊂ Γ is semisimple, which is, in turn, equivalent to V being a semisimple \overline{Q}_{ℓ} · 𝔤-module.

Recall that ρ is called strongly irreducible if ρ|_{Γ′} is irreducible for every open subgroup Γ′ ⊂ Γ, which is equivalent to V being a simple \overline{Q}_{ℓ} · 𝔤-module. If this is the case, then \overline{Q}_{ℓ} · 𝔤 is a reductive Lie algebra over \overline{Q}_{ℓ}. This implies that 𝔤 (resp. \overline{𝔤}) is a reductive Lie algebra over Q_{ℓ} (resp. over \overline{Q}_{ℓ}) and that 𝔷(\overline{𝔤}) acts on V through a map 𝔷(\overline{𝔤}) ⟶ \overline{Q}_{ℓ} · id_{V} ⊂ End_{\overline{Q}_{ℓ}}(V).

(1.6) In [N, Thm. 3.3] it was deduced from Theorem 1.4 above that ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) is semisimple, provided there exists an open subgroup Γ′ ⊂ Γ satisfying the following two conditions:

(1.6.1) the semisimplification ρ|_{Γ′}^{ss} = ρ^{ss}|_{Γ′} is a subrepresentation of (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})^{⊕m} (for some m ≥ 1), where

each ρ_{i} is a strongly irreducible representation of Γ′; (1.6.2) ρ(g) is a semisimple element of Aut_{\overline{Q}_{ℓ}}(V), for all g lying in a dense subset Σ ⊂ Γ′.

In the geometric context considered in [N, §5], the representations ρ_{i} were suitable twists of the Galois representation attached to a cuspidal Hilbert eigenform without complex multiplication and V was the corresponding Hecke eigenspace of the middle degree étale cohomology of a quaternionic Shimura variety. As recalled in the Introduction, the condition (1.6.1) can be deduced in this situation either from deep results of Langlands [La] and Brylinski–Labesse [BL], or from the Eichler–Shimura relation for the usual Frobenius at split primes and [N, Prop. 3.10(1)]. On the other hand, (1.6.2) was deduced in [N] from the Eichler–Shimura relation for partial Frobenii at split primes.

As explained in more detail in the Introduction, under the current state of the art (when the Langlands– Kottwitz conjecture (0.3) has not yet been established in sufficient generality), the condition (1.6.1) is too restrictive for applications to more general Shimura varieties.

This is the principal motivation behind the following abstract result, in which (1.6.1) is replaced by a weaker assumption and (1.6.2) by a stronger one.

(1.7) Theorem. Let Γ be a profinite group, V, W_{1}, …, W_{r} non-zero vector spaces of finite dimension over \overline{Q}_{ℓ}, ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) and ρ_{i} : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(W_{i}) representations with Lie algebras 𝔤_{i} = Lie(ρ_{i}(Γ)), 𝔤 = Lie(ρ(Γ)), \overline{𝔤}_{i} = 𝔤_{i} ⊗_{Q_{ℓ}} \overline{Q}_{ℓ}, \overline{𝔤} = 𝔤 ⊗_{Q_{ℓ}} \overline{Q}_{ℓ}. If the following three conditions hold, then the representation ρ = ρ^{ss} is semisimple. (A) Each ρ_{i} is strongly irreducible (which implies that each \overline{𝔤}_{i} is a reductive \overline{Q}_{ℓ}-Lie algebra and each element of its centre acts on W_{i} by a scalar). (B) For each i = 1, …, r, every (equivalently, some) Cartan subalgebra \overline{𝔥}_{i} of \overline{𝔤}_{i} acts on W_{i} without multiplicities (i.e., all weight spaces of \overline{𝔥}_{i} on W_{i} are one-dimensional). (C) There exists an open subgroup Γ′ ⊂ Γ and a dense subset Σ ⊂ Γ′ such that for each g ∈ Σ there exists a finite dimensional vector space over \overline{Q}_{ℓ} (depending on g) V (g) ⊃ V and elements u_{1}, …, u_{r} ∈ Aut_{\overline{Q}_{ℓ}}(V (g)) such that u_{i}u_{j} = u_{j}u_{i}, P_{ρ_{i}(g)}(u_{i}) = 0 for all i, j = 1, …, r, V is stable under u_{1} ⋯ u_{r} and u_{1} ⋯ u_{r}|_{V} = ρ(g).

Proof. We can replace Γ by Γ′ and assume that Γ′ = Γ. We are going to apply Theorem 1.4 to the \overline{Q}_{ℓ}- Lie algebra \overline{Q}_{ℓ} · 𝔤 ⊂ End_{\overline{Q}_{ℓ}}(V). According to [N, Prop. 3.6], assumptions (A) and (B) imply that there exists an open subgroup Γ_{0} ⊂ Γ and an open dense subset U_{0} ⊂ Γ_{0} such that the polynomials P_{ρ_{i}(g)}(X) do not have multiple roots, for any g ∈ U_{0} and i = 1, …, r. The intersection Σ ∩ U_{0} is then dense in Γ_{0} and for each g ∈ Σ ∩ U_{0} the mutually commuting elements u_{i} ∈ End_{\overline{Q}_{ℓ}}(V (g)) are semisimple, hence so is ρ(g) = u_{1} ⋯ u_{r}|_{V}. Therefore (H1-ZAR) is satisfied for \overline{Q}_{ℓ} · 𝔤.

We are now going to check (H2). After replacing Γ by an open subgroup we can assume that the filtration \{V_{i}\} is Γ-stable; then gr(V) = ρ^{ss} and 𝔪 = \overline{Q}_{ℓ} · Lie(ρ^{ss}(Γ)). Step 1: It follows from (C) that

P_{(ρ_{1}⊗···⊗ρ_{r})(g)}(ρ(g)) = 0 ∈ End_{\overline{Q}_{ℓ}}(V)  (1.7.1)

holds for all g ∈ Σ (cf. [N, Thm. 3.12(1)]), hence for all g ∈ Γ, by continuity. Consider the representation ρ_{0} = ρ_{1} ⊕⋯⊕ρ_{r} : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(W_{0}), where W_{0} = W_{1} ⊕⋯⊕W_{r}. The formula (1.7.1) implies that (ρ(g)−1)^{N} = 0 for all g ∈ Ker(ρ_{0}), where N = dim(W_{1} ⊗⋯⊗W_{r}). It follows that the Lie ideal 𝔞 = \overline{Q}_{ℓ} ·Lie(ρ^{ss}(Ker(ρ_{0}))) in 𝔪 = \overline{Q}_{ℓ} · 𝔤/𝔫_{V} (\overline{Q}_{ℓ} · 𝔤) = \overline{Q}_{ℓ} · Lie(ρ^{ss}(Γ)) ⊂ End_{\overline{Q}_{ℓ}}(gr(V)) consists of nilpotent elements, hence is a nilpotent Lie ideal [LIE, Ch. I, §4, no. 2, Cor. 3], and so 𝔞 = 0, since 𝔪 is reductive. Therefore ρ^{ss}(Ker(ρ_{0})) is a finite subgroup of Aut_{\overline{Q}_{ℓ}}(gr(V)), hence Ker(ρ_{0}) ∩ Ker(ρ^{ss}) is an open subgroup of Ker(ρ_{0}). The surjection Γ/(Ker(ρ_{0}) ∩ Ker(ρ^{ss})) ↠ Γ/Ker(ρ^{ss}) yields canonical surjections f : 𝔤_{0} = Lie(ρ_{0}(Γ)) ↠ Lie(ρ^{ss}(Γ)) and \overline{f} : \overline{𝔤}_{0} = 𝔤_{0} ⊗_{Q_{ℓ}} \overline{Q}_{ℓ} ↠ 𝔪. In particular, 𝔤_{0} acts on gr(V) via \overline{f}. Step 2: After replacing Γ by an open subgroup we can assume that we can apply the ℓ-adic logarithm to ρ_{i}(g) and ρ(g), for all g ∈ Γ. The formula (1.7.1) then implies that

∀X = (X_{1}, …, X_{r}) ∈ \overline{𝔤}_{0} ⊂ \overline{𝔤}_{1} × ⋯ × \overline{𝔤}_{r} P_{(X_{1},...,X_{r})|W_{1}⊗···⊗W_{r}}(\overline{f}(X)) = 0 ∈ End_{\overline{Q}_{ℓ}}(gr(V))

(above, X acts on W_{1} ⊗ ⋯ ⊗ W_{r} by ∑_{i} 1 ⊗ ⋯ ⊗ 1 ⊗ X_{i} ⊗ 1 ⊗ ⋯ ⊗ 1). Restricting this formula to a Cartan subalgebra \overline{𝔥}_{0} of the reductive Lie algebra \overline{𝔤}_{0} we see that each weight of \overline{𝔥}_{0} occurring in gr(V) must occur in W_{1} ⊗ ⋯ ⊗ W_{r}. Step 3: Recall that \overline{𝔤}_{0} projects surjectively on each \overline{𝔤}_{1}, …, \overline{𝔤}_{r} and that each W_{i} is a simple \overline{𝔤}_{i}-module. This implies, by [N, Prop. 2.2], that all weights of \overline{𝔥}_{0} occurring in W_{1} ⊗ ⋯ ⊗ W_{r} lie in one coset modulo the root lattice of (\overline{𝔤}_{0}, \overline{𝔥}_{0}). By Step 2, the same holds for the weights of \overline{𝔥}_{0} occurring in gr(V). As in [N, Cor. 2.3], writing \overline{𝔤}_{0} = 𝔪 × Ker(\overline{f}) and choosing \overline{𝔥}_{0} of the form 𝔥 × 𝔥′, where 𝔥 ⊂ 𝔪 and 𝔥′ ⊂ Ker(\overline{f}) are Cartan subalgebras, we deduce that all weights of 𝔥 occurring in gr(V) lie in one coset modulo the root lattice of (𝔪, 𝔥), which is precisely (H2). Having checked (H1-ZAR) and (H2) for the linear Lie algebra \overline{Q}_{ℓ} ·𝔤 ⊂ End_{\overline{Q}_{ℓ}}(V), we can apply Theorem 1.4, which tells us that \overline{Q}_{ℓ} · 𝔤 = 𝔪 is reductive and V is a semisimple \overline{Q}_{ℓ} · 𝔤-module, which is equivalent to ρ being semisimple. The Theorem is proved.

2. Unitary Shimura varieties

(2.1) Let (B, ∗, V, ⟨ , ⟩) be a simple PEL datum of type (A): B is a finite-dimensional simple Q-algebra, ∗ is a positive involution of second kind on B, V is a non-zero left B-module of finite type and ⟨ , ⟩ : V ×V ⟶ Q is a non-degenerate alternating Q-bilinear form such that ⟨bv, v′⟩ = ⟨v, b^{∗}v′⟩ for all b ∈ B and v, v′ ∈ V.

The centre Z(B) = F_{c} is then a CM field, with maximal real subfield F = F_{c}^{∗=id} of degree r = [F : Q]. There is a unique (non-degenerate) alternating F -bilinear form ⟨ , ⟩_{F} : V × V ⟶ F such that ⟨ , ⟩ = Tr_{F/Q} ∘ ⟨ , ⟩_{F}. The centraliser C = End_{B}(V) is again a simple Q-algebra with centre F_{c}; let # : C ⟶ C be the F -linear involution given by ⟨cv, v′⟩ = ⟨v, c^{#}v′⟩. (2.2) These data define the following algebraic groups and morphisms: H = GSp_{B}(V, ⟨ , ⟩_{F}) ⟶^{ν} G_{m,F} (over F), its restriction of scalars G = R_{F/Q}(H) and the closed subgroup G^{∗} ⊂ G defined by a cartesian diagram of connected reductive groups

\xymatrix{G^{∗} \ar[r] \ar[d]^{ν} & G \ar[d]^{ν} \\ G_{m,Q} \ar[r] & R_{F/Q}(G_{m,F}).}

In concrete terms

H(S) = {h ∈ GL_{B}(V ⊗_{F} S) | ∃ ν(h) ∈ S^{×} ∀v, v′ ∈ V ⟨hv, hv′⟩_{F} = ν(h)⟨v, v′⟩_{F} }
= {h ∈ (C ⊗_{F} S)^{×} | hh^{#} = ν(h) ∈ S^{×}},
G^{∗}(R) = {g ∈ (C ⊗ R)^{×} | gg^{#} = ν(h) ∈ R^{×}},

for all F -algebras S and Q-algebras R. (2.3) If S is an F_{c}-algebra, then C ⊗_{F} S ⟶^{∼} (C ⊗_{F_{c}} S) × (C ⊗_{F_{c},∗} S) and # ⊗ id interchanges the two factors, which gives an isomorphism [HT, p. 47]

α : H(S) = {h = (u, λ(u^{#})^{−1}) | u ∈ (C ⊗_{F_{c}} S)^{×}, λ = ν(h) ∈ S^{×}} ⟶^{∼} (C ⊗_{F_{c}} S)^{×} × S^{×}, α(h) = (u, λ).

If S = S^{+} ⊗_{F} F_{c} for an F -algebra S^{+}, then the complex conjugation ∗ = id_{S^{+}} ⊗ ∗ : S ⟶ S acts on (C ⊗_{F} S^{+})^{×} × (S^{+} ⊗_{F} F_{c})^{×} by α ∘ (id_{S^{+}} ⊗ ∗) ∘ α^{−1} : (A, b) ↦ ((b^{∗})^{−1}A^{#}, b^{∗}). (2.4) There exists a morphism of R-algebras h : C ⟶ C ⊗ R, unique up to conjugation by an element c ∈ (C ⊗ R)^{×} with cc^{#} = 1 [K, Lemma 4.3], such that h(\overline{z}) = h(z)^{#} and for which the symmetric R-bilinear form ⟨v, h(i)v′⟩ : V_{R} × V_{R} ⟶ R is positive definite. This gives rise to a Shimura datum (G^{∗}, 𝒳 ^{∗}) (resp. (G, 𝒳 )), where 𝒳 ^{∗} (resp. 𝒳 ) is the conjugacy class of h : S ⟶ G_{R}^{∗} under G^{∗}(R) (resp. under G(R)). The corresponding Shimura varieties Sh(G^{∗}, 𝒳 ^{∗}) and Sh(G, 𝒳 ) have a common reflex field E ⊂ F_{c}^{gal}, where F_{c}^{gal} is the Galois closure of F_{c} in \overline{Q}.

The action of h(i) defines a complex structure on V_{R}, hence a Hodge decomposition V_{C} = V^{−1,0} ⊕V^{0,−1} of weight -1, with h(z) ⊗ id acting as z (resp. \overline{z}) on V^{−1,0} (resp. on V^{0,−1}). The cocharacter μ_{h} : G_{m,C} ⟶ G_{C} attached to h acts on V_{C} as follows: μ_{h}(z) acts as z · id (resp. as id) on V^{−1,0} (resp. on V^{0,−1}). (2.5) Let X = {x : F ↪ R} be the set of infinite primes of F. Fix a CM type Φ = {σ_{x} : F_{c} ↪ C}_{x∈X} (σ_{x}|_{F} = x) of F_{c}. The choice of Φ induces an isomorphism

B ⊗ R = ∏_{x∈X} B ⊗_{F,x} R = ∏_{x∈X} B ⊗_{F_{c},σ_{x}} C ⟶^{∼} ∏_{x∈X} M_{N}(C)

under which V^{−1,0} ⟶^{∼} ⨁_{x∈X} ((C^{N})^{a_{x}} ⊕ (\overline{C}^{N})^{b_{x}}), where a_{x} + b_{x} = n and C ⊗ R ⟶^{∼} ∏_{x∈X} M_{n}(C). The

common dimension of the Shimura varieties Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) and Sh_{K}(G, 𝒳 ) is equal to dim := ∑_{x∈X} a_{x}b_{x}. The isomorphism α from 2.3 is functorial in S; it defines an isomorphism of algebraic groups over F_{c}

H ⊗_{F} F_{c} ⟶^{∼} C^{×} × G_{m,F_{c}},

hence an isomorphism H ⊗_{F} k ⟶^{∼} (C^{×})_{k} × G_{m,k}, for any field embedding F_{c} ↪ k. In particular, Φ gives rise to an isomorphism

G_{C} = ∏_{x∈X}(H ⊗_{F} F_{c}) ⊗_{F_{c},σ_{x}} C ⟶^{∼} ∏_{x∈X}(GL(n)_{C} × G_{m,C}) = (GL(n)_{C} × G_{m,C})^{X}  (2.5.1)

under which G_{C}^{∗} corresponds to the subgroup GL(n)_{C}^{X} ×G_{m,C} (with G_{m,C} embedded diagonally into G_{m,C}^{X}) and μ_{h} is given by μ_{h}(z) = ((zI_{a_{x}}, I_{b_{x}})_{x∈X}, z). (2.6) Assume that ξ (resp. ξ^{∗}) is an irreducible algebraic representation of G_{C} (resp. of G_{C}^{∗}). Under (2.5.1),

ξ = ⨂_{x∈X} ξ_{x}, ξ_{x} = η_{x} ⊗ ξ_{0,x}, ξ^{∗} = (⨂_{x∈X} η_{x}^{∗}) ⊗ ξ_{0}^{∗},

where η_{x} (resp. η_{x}^{∗}) is an irreducible algebraic representation of GL(n)_{C} with highest weight (with respect to the upper Borel subgroup) equal to diag(t_{1}, …, t_{n}) ↦ ∏ t_{i}^{m_{i,x}}, m_{1,x} ≥ ⋯ ≥ m_{n,x} (resp. ∏ t_{i}^{m_{i,x}^{∗}}, m_{1,x}^{∗} ≥ ⋯ ≥ m_{n,x}^{∗}) and ξ_{0,x}, ξ_{0}^{∗} : G_{m,C} ⟶ G_{m,C} given by t ↦ t^{m_{0,x}} and t ↦ t^{m_{0}^{∗}}, respectively.

The weight morphism attached to h is defined as

wt : G_{m,R} ⟶ S ⟶ G_{R}^{∗}, t ↦ h(t^{−1}),

which means that

ξ^{∗}(wt(t)) = t^{w^{∗}}, w^{∗} = − ∑_{x∈X} ∑_{i=1}^{n} m_{i,x}^{∗} − 2m_{0}^{∗}, ξ(wt(t)) = t^{∑ w_{x}}, w_{x} = − ∑_{i=1}^{n} m_{i,x} − 2m_{0,x}.

For j ∈ Z, the Tate twist ξ^{∗}(j) is defined as follows: the components η_{x}^{∗} do not change, but m_{0}^{∗} is replaced by m_{0}^{∗} + j.

(2.7) Proposition. (1) The respective centres of H and G are equal to Z_{H} = R_{F_{c}/F} (G_{m,F_{c}}) and Z_{G} = F_{c}T ⊃ Z_{G}^{+} = _{F} T, where the map (Z_{G}^{+})_{C} ↪ G_{C} is given by

(G_{m,C} ↪ GL(n)_{C} × G_{m,C})^{X}, ∀x ∈ X t ↦ (t · I_{n}, t^{2}).

(2) Given ξ as in 2.6, the following two conditions are equivalent: ξ(Z_{G}(Q) ∩ K) = ξ(F_{c}^{×} ∩ K) = {1} for every neat open compact subgroup K ⊂ G(Q̂) = H(F̂) ⟺ the integers w_{x} = w_{x}(ξ) do not depend on x ∈ X.

(3) Given ξ^{∗} as in 2.6, the following two conditions are equivalent: there exist j ∈ Z and ξ satisfying (2) such that ξ^{∗}(j) = ξ|_{G_{C}^{∗}} ⟺ ∑_{i=1}^{n} m_{i,x}^{∗} (mod 2) ∈ Z/2Z does not depend on x ∈ X.

Proof. (1) This follows from the definitions. (2) If K is neat, then Δ = F_{c}^{×} ∩ K is a subgroup of finite index in O_{F,+}^{×} and ξ^{∗}(β) = ∏_{x∈X} x(β)^{w_{x}} for β ∈ Δ. This product identically vanishes on Δ if and only if all w_{x} are equal to each other. (3) If j and ξ exist, then m_{i,x} = m_{i,x}^{∗} for all x ∈ X and 1 ≤ i ≤ n. We need to choose j ∈ Z and m_{0,x} so that ∑_{i=1}^{n} m_{i,x}^{∗} + 2m_{0,x} do not depend on x ∈ X and ∑_{i=1}^{n} m_{0,x} = m_{0}^{∗} + j, which is possible if and only if ∑_{i=1}^{n} m_{i,x}^{∗} (mod 2) ∈ Z/2Z does not depend on x ∈ X. (2.8) Any ξ^{∗} : G_{C}^{∗} ⟶ GL(N^{∗})_{C} from 2.6 (resp. any ξ : G_{C} ⟶ GL(N)_{C} satisfying the equivalent conditions from Proposition 2.7(2)) gives rise, for small enough open compact subgroups K^{∗} ⊂ G^{∗}(Q̂) (resp. K ⊂ G(Q̂)), to a locally constant sheaf of complex vector spaces ℒ_{ξ^{∗}} (resp. ℒ_{ξ}) on Sh_{K^{∗}}^{an} = G^{∗}(Q)\(𝒳 ^{∗} × G^{∗}(Q̂)/K) (resp. on Sh_{K}^{an} = G(Q)\(𝒳 × G(Q̂)/K)). Denote by j : Sh_{K^{∗}} ↪ Sh_{K^{∗},BB} (resp. j : Sh_{K} ↪ Sh_{K,BB}) the Baily–Borel compactification. The respective analytic intersection cohomology groups

H^{i}(Sh_{∗,BB}^{an}, j_{!∗}ℒ_{ξ^{∗}}) = \underrightarrow{lim}_{K^{∗}} H^{i}(Sh_{K^{∗},BB}^{an}, j_{!∗}ℒ_{ξ^{∗}}), H^{i}(Sh_{BB}^{an}, j_{!∗}ℒ_{ξ}) = \underrightarrow{lim}_{K} H^{i}(Sh_{K,BB}^{an}, j_{!∗}ℒ_{ξ})

have an automorphic description which generalises Matsushima’s formula [BW, Thm. VII.5.2] in the compact case, thanks to [BC, Th. A] combined with [BG, Prop. 5.6] and the proof of Zucker’s conjecture [Lo], [SS]:

H^{i}(Sh_{BB}^{an}, j_{!∗}ℒ_{ξ}) = ⨁_{π∈L_{disc}^{2}(G)} m_{disc}(π) H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ⊗ π^{∞}  (2.8.1)

(and similarly for G^{∗}), where π = π_{∞} ⊗ π^{∞} runs through the discrete automorphic representations of G(A) = G(R) × G(Q̂) such that π_{∞} is cohomological in degree i with respect to ξ (H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ≠ 0, where 𝔤 = Lie(G(R)) and K_{∞} ⊂ G(R) is the stabiliser of h) and m_{disc}(π) denotes the multiplicity of π in the discrete part of the automorphic spectrum of G.

The restriction of the canonical map

H^{i}(Sh_{BB}^{an}, j_{!∗}ℒ_{ξ}) ⟶ H^{i}(Sh^{an}, ℒ_{ξ}) = \underrightarrow{lim}_{K} H^{i}(Sh_{K}^{an}, ℒ_{ξ})

to the cuspidal subspace

⨁_{π∈L_{cusp}^{2}(G)} m_{cusp}(π) H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ⊗ π^{∞}

of (2.8.1) is injective, by [B, Cor. 5.5]. (2.9) Fix a prime number ℓ and an isomorphism C ⟶^{∼} \overline{Q}_{ℓ}. For any algebraic object (−) defined over a subfield of C we denote by (−)_{ℓ} its base change to \overline{Q}_{ℓ}.

The algebraic representation ξ_{ℓ} : G_{\overline{Q}_{ℓ}} ⟶ GL(N)_{\overline{Q}_{ℓ}} defines, for small enough K, a lisse \overline{Q}_{ℓ}-sheaf ℒ_{ξ,ℓ} on Sh_{K}(G, 𝒳 )_{et}. The decomposition (2.8.1) yields, thanks to the comparison isomorphism between Betti and ℓ-adic cohomology, a G(Q̂)-equivariant isomorphism

H_{BB}^{i} := H_{et}^{i}(Sh_{BB} ⊗_{E} \overline{Q}, j_{!∗}ℒ_{ξ,ℓ}) = \underrightarrow{lim}_{K} H_{et}^{i}(Sh_{K,BB} ⊗_{E} \overline{Q}, j_{!∗}ℒ_{ξ,ℓ}) = ⨁_{π^{∞}} V^{i}(π^{∞}) ⊗ π^{∞},  (2.9.1)

where π^{∞} is an irreducible smooth representation of G(Q̂) over \overline{Q}_{ℓ} and V^{i}(π^{∞}) = Hom_{G(_{Q})}(π^{∞}, H_{BB}^{i}) is a

finite-dimensional \overline{Q}_{ℓ}-vector space. The action of the Galois group Γ_{E} on étale cohomology defines an ℓ-adic representation

Γ_{E} ⟶ Aut_{G(_{Q})}(V^{i}(π^{∞}) ⊗ π^{∞}) = Aut_{\overline{Q}_{ℓ}}(V^{i}(π^{∞})),

under which (2.9.1) becomes Γ_{E} ×G(Q̂)-equivariant. The only π^{∞} contributing to (2.9.1) are those for which there exists π_{∞} cohomological in degree i with respect to ξ such that π = π_{∞} ⊗π^{∞} is a discrete automorphic representation of G(A), since

dim V^{i}(π^{∞}) = ∑_{π=π_{∞}⊗π^{∞}} m_{disc}(π) dim H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ).

Similarly, there is a decomposition

H_{∗,BB}^{i} := H_{et}^{i}(Sh_{∗,BB} ⊗_{E} \overline{Q}, j_{!∗}ℒ_{ξ^{∗},ℓ}) = \underrightarrow{lim}_{K^{∗}} H_{et}^{i}(Sh_{K^{∗},BB} ⊗_{E} \overline{Q}, j_{!∗}ℒ_{ξ^{∗},ℓ}) = ⨁_{(π^{∗})^{∞}} V^{i}((π^{∗})^{∞}) ⊗ (π^{∗})^{∞},

(2.9.2)

π^{∗} = π_{∞}^{∗} ⊗ (π^{∗})^{∞} ∈ L_{disc}^{2}(G^{∗}). (2.10) Chevalley’s theorem on units [C] implies [D, Cor. 2.0.12] (see also [TX, Lemma 2.5] and [N, Prop. 5.7]) that the canonical map i : Sh(G^{∗}, 𝒳 ^{∗}) ⟶ Sh(G, 𝒳 ) induces an isomorphism between the topological connected components containing (h, 1):

Sh(G^{∗}, 𝒳 ^{∗})^{an,+} = Sh(G, 𝒳 )^{an,+}.  (2.10.1)

It follows that there is a G(Q̂)-equivariant isomorphism Sh(G^{∗}, 𝒳 ^{∗})×^{U} G(Q̂) ≃ Sh(G, 𝒳 ), where U ⊂ G(Q̂) is the stabiliser of the subset Sh(G^{∗}, 𝒳 ^{∗}) ⊂ Sh(G, 𝒳 ), which yields an isomorphism of Γ_{E} × G(Q̂)-modules

H_{BB}^{i} ≃ Ind_{U}^{G(_{Q})} H_{∗,BB}^{i}  (2.10.2)

(with smooth induction on the right hand side), provided that ξ^{∗} = ξ|_{G_{C}^{∗}}.

(2.11) From now on until the end of §2 we assume that ξ^{∗} is as in 2.6 and π^{∗} is an automorphic representation of G^{∗}(A) for which π_{∞}^{∗} is cohomological in degree i with respect to ξ^{∗}.

Recall that the fixed CM type Φ of F_{c} gives rise to identifications F_{c} ⊗ R ≃ C^{X} and

Z(R) = (F_{c} ⊗ R)^{×} ≃ (C^{×})^{X}, Z^{∗}(R) ≃ \{z = (z_{x}) ∈ (C^{×})^{X} | λ = z_{x}\overline{z_{x}} ∈ R is independent of x ∈ X\}.

The infinitesimal characters of π_{∞}^{∗} and (ξ^{∗})^{∨} coincide. In particular, the infinity type of the central character ω_{π^{∗}} of π^{∗} is equal to

ω_{π_{∞}^{∗}} = ω_{ξ^{∗}}^{−1}|_{Z^{∗}(R)} : z = (z_{x}) ↦ λ^{u} ∏_{x∈X} z_{x}^{u_{x}}  (λ = z_{x}\overline{z_{x}}),  u = −m_{0}^{∗} ∈ Z,  u_{x} = − ∑_{i=1}^{n} m_{i,x}^{∗} ∈ Z.
(2.11.1)

There is a unique character Z(R) ≃ (C^{×})^{X} ⟶ C^{×} trivial on O_{F,+}^{×} and extending ω_{π_{∞}^{∗}}, namely,

z = (z_{x}) ↦ ∏_{x∈X} z_{x}^{u_{x}}|z_{x}|^{t−u_{x}}, t = (2u + ∑_{x∈X} u_{x})/r.

This character is “algebraic up to a twist”, i.e., is of the form

z = (z_{x}) ↦ ∏_{x∈X} z_{x}^{c_{x}} \overline{z_{x}}^{d_{x}}|z_{x}|^{s} (c_{x}, d_{x} ∈ Z, s ∈ R)

if and only if u_{x} (mod 2) ∈ Z/2Z does not depend on x ∈ X, which is equivalent to ξ^{∗} being as in Proposition 2.7(3). (2.12) Assume, in addition, that π^{∗} is cuspidal. Every (cuspidal) automorphic form on G^{∗}(A) extends to a (cuspidal) automorphic form on G(A) (cf. [LSc, Prop. 3.5]), which implies that there exists a cuspidal automorphic representation π of G(A) such that π^{∗} is a quotient of π|_{G^{∗}(A)}. (2.13) Automorphic representations of G(A) = H(A_{F}) are closely related to suitably self-dual automorphic representations (Π, ψ) of H_{0}(A_{F_{c}}) = G_{0}(A), where H_{0} = GL(n)_{F_{c}} × GL(1)_{F_{c}} and G_{0} = R_{F_{c}/Q}(H_{0}). For unitary groups of a special kind, very precise results in this direction were proved in [Cl1], [Cl2], [ClL], [ClHL], [HL], [HT], [L1], [L2], [Mr] and [Sh]. Recently, endoscopic transfer of discrete automorphic representations of the unitary group Ker(ν)(A_{F}) to automorphic representations of GL(n, A_{F_{c}}) has been completely described in the quasi-split case [Mk] and in full generality in [KMSW] (with some of the proofs relegated to subsequent articles). As in [HT, Thm. VI.2.1], this yields a transfer of discrete automorphic representations of G(A) to G_{0}(A). However, our main result in the form 2.20 and 2.25 below requires only a fairly weak form of this transfer, as formulated in Definition 2.14 below.

The discussion in 2.3 together with (2.5.1) give an isomorphism

(G_{0})_{C} ⟶^{∼} ∏_{x∈X}((GL(n)_{F_{c}} × G_{m,F_{c}}) ⊗_{F_{c},σ_{x}} C × (GL(n)_{F_{c}} × G_{m,F_{c}}) ⊗_{F_{c},cσ_{x}} C),

where c denotes the complex conjugation.

(2.14) Definition (cf. [HT, Thm. VI.2.1]). Let π be as in 2.12. We say that an automorphic representation (Π, ψ) of GL_{n}(A_{F_{c}}) × A_{F_{c}}^{×} is a weak transfer of π if the following conditions are satisfied.

(1) Π^{∨} ≃ Π^{c};

(2) the restriction of Π_{∞} to SL_{n}(C)^{X} ⊂ (GL_{n}(C) × C^{×})^{X} = G_{0}(R) is cohomological with respect to the restriction of ξ^{∗} ⊗ (ξ^{∗})^{c} to (SL(n)_{C} × SL(n)_{C})^{X} ⊂ (G_{0})_{C};

(3) the central characters are related as follows: ω_{π} = ψ^{c}, ω_{Π} = ψ^{c}/ψ;

(4) for all but finitely many finite primes v of F at which F_{c}/F and C are split (v = ww^{c} in F_{c}), the local representation π_{v} of H(F_{v}) = H((F_{c})_{w}) is isomorphic, via α : H((F_{c})_{w}) ⟶^{∼} (C ⊗_{F_{c}} (F_{c})_{w})^{×} × ((F_{c})_{w})^{×} = H_{0}((F_{c})_{w}), to the local representation (Π, ψ)_{w}.

(2.15) Remarks. Firstly, we are going to use 2.14(4) only for primes w of degree one, i.e., such that (F_{c})_{w} = Q_{p(w)}, where p(w) is the residue characteristic of w.

Secondly, the condition 2.14(2) implies that Π_{∞} is cohomological with respect to a certain algebraic representation of (GL(n)_{C} × GL(n)_{C})^{X}, since the infinity type

ω_{Π_{∞}} : (F_{c} ⊗ R)^{×} ≃ (C^{×})^{X} ⟶ C^{×}, z = (z_{x}) ↦ ω_{π}(z)/ω(\overline{z}) = ∏_{x∈X}(z_{x}/\overline{z_{x}})^{u_{x}}

of the central character of Π is algebraic, thanks to 2.14(3) combined with (2.11.1).

According to the discussion in 2.11, the Hecke character ψ∥ · ∥_{F_{c}}^{−s} : A_{F_{c}}^{×}/F_{c}^{×} ⟶ C^{×} is algebraic for some s ∈ R if and only if ξ^{∗} is as in Proposition 2.7(3). (2.16) The key assumption. From now on until the end of §2 we assume that the representation π from 2.12 admits a weak transfer (Π, ψ) as in 2.14, with Π a cuspidal automorphic representation of GL_{n}(A_{F_{c}}).

This cuspidality condition together with 2.14(1) and 2.15 imply that there exists an ℓ-adic Galois representation

ρ_{Π,ℓ} : Γ_{F_{c}} ⟶ GL_{n}(\overline{Q}_{ℓ}),

which is suitably self-dual (ρ_{Π,ℓ}^{∨} ≃ ρ_{Π,ℓ}^{c}(n − 1), but we are not going to use this fact) and is compatible with Π in a very strong sense [CH, Thm. 3.2.3], [BLGGT1, Thm. 2.1.1], [BLGGT2, Thm. A]. These results are an outcome of a collective effort of many mathematicians (Clozel, Harris, Taylor, Labesse, Shin,...). We need only the following properties of ρ_{Π,ℓ}.

(2.16.1) ρ_{Π,ℓ} is semisimple. (In fact, ρ_{Π,ℓ} is expected to be irreducible. Hopefully, this will be established in a foreseeable future. At the time of writing, the irreducibility is known only if ℓ belongs to a set of primes (depending on Π) of positive density [PT, Thm. D]. See also [CG] for results on irreducibility over F.)

(2.16.2) For all but finitely many finite primes w ∤ ℓ of F_{c} there is an equality in C[(Nw)^{−s}] ≃ \overline{Q}_{ℓ}[(Nw)^{−s}]

L_{w}(Π, Std_{n}, s)^{−1} = L_{w}(ρ_{Π,ℓ}, s + (n − 1)/2)^{−1},

where Std_{n} denotes the standard representation of GL(n). In particular, ρ_{Π,ℓ} is unramified outside a finite set of primes of F_{c}.

(2.16.3) For each embedding τ : F_{c} ↪ \overline{Q}_{ℓ}, the restriction of ρ_{Π,ℓ} to Γ_{(F_{c})_{τ}} = Gal(\overline{Q}_{ℓ}/(F_{c})_{τ}) is a Hodge-Tate

representation with n distinct Hodge-Tate weights. (2.16.4) For all but finitely many finite primes w ∤ ℓ of F_{c} all eigenvalues of ρ_{Π,ℓ}(Fr(w)) in \overline{Q}_{ℓ} ≃ C have absolute

value (Nw)^{(n−1)/2}.

(2.17) Proposition. Let \overline{𝔤} = \overline{Q}_{ℓ} · Lie(ρ_{Π,ℓ}(Γ_{F_{c}})) ⊂ gl(n, \overline{Q}_{ℓ}) be the \overline{Q}_{ℓ}-Lie algebra generated by the image of ρ_{Π,ℓ}.

(1) \overline{𝔤} is reductive and its centre acts semisimply on \overline{Q}_{ℓ}^{n}.

(2) \overline{Q}_{ℓ}^{n} is a simple \overline{𝔤}-module if and only if ρ_{Π,ℓ} is strongly irreducible.

(3) Any Cartan subalgebra \overline{𝔥} ⊂ \overline{𝔤} acts on \overline{Q}_{ℓ}^{n} by n distinct weights (each with multiplicity one).

Proof. The statement (1) follows from (2.16.1), while (2) is standard. It is enough to prove (3) for a single Cartan subalgebra after extending the scalars from \overline{Q}_{ℓ} to C_{ℓ}. According to Sen’s theory [Se, Thm. 1], \overline{𝔤} ⊗_{\overline{Q}_{ℓ}} C_{ℓ} contains a semisimple element whose eigenvalues are the n distinct Hodge-Tate weights of the restriction of ρ_{Π,ℓ} to Γ_{(F_{c})_{τ}} (for any τ : F_{c} ↪ \overline{Q}_{ℓ}). This element, being semisimple, lies in some Cartan subalgebra of \overline{𝔤} ⊗_{\overline{Q}_{ℓ}} C_{ℓ}, which then has the required property.

(2.18) Corollary. Let \overline{𝔥} ⊂ \overline{𝔤} be any Cartan subalgebra.

(1) \overline{𝔥} acts without multiplicities on ⋀^{i} \overline{Q}_{ℓ}^{n} for i = 1, n − 1.

(2) If \overline{𝔤} ⊃ sl(n, \overline{Q}_{ℓ}) (which is equivalent to \overline{𝔤} = gl(n, \overline{Q}_{ℓ}), since det(ρ_{Π,ℓ}) : Γ_{F_{c}} ⟶ Q_{ℓ}^{×} is a character of infinite order), then \overline{𝔥} acts without multiplicities on ⋀^{i} \overline{Q}_{ℓ}^{n}, for all i = 1, …, n − 1.

(2.19) We are now ready to state the main result of this article. Let

H_{∗}^{i} := H_{et}^{i}(Sh(G^{∗}, 𝒳 ^{∗}) ⊗_{E} \overline{Q}, ℒ_{ξ^{∗},ℓ}) = \underrightarrow{lim}_{K^{∗}} H_{et}^{i}(Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) ⊗_{E} \overline{Q}, ℒ_{ξ^{∗},ℓ}).

(2.20) Theorem. Let ξ^{∗} be an irreducible algebraic representation of G_{C}^{∗} and π^{∗} a cuspidal automorphic representation of G^{∗}(A) such that π_{∞}^{∗} is cohomological with respect to ξ^{∗} in degree dim = ∑_{x∈X} a_{x}b_{x}. Assume that π^{∗} is a quotient of the restriction to G^{∗}(A) of a cuspidal automorphic representation π of G(A) which admits a weak transfer (in the sense of 2.14) to a cuspidal automorphic representation (Π, ψ) of GL_{n}(A_{F_{c}}) × A_{F_{c}}^{×}. If the ℓ-adic Galois representation ρ_{Π,ℓ} : Γ_{F_{c}} ⟶ GL_{n}(\overline{Q}_{ℓ}) attached to Π satisfies the following two conditions

(1) ρ_{Π,ℓ} is strongly irreducible;

(2) either (2a) \overline{Q}_{ℓ} · Lie(ρ_{Π,ℓ}(Γ_{F_{c}})) ⊃ sl(n, \overline{Q}_{ℓ}), or (2b) G(R) = ∏_{x∈X} GU(a_{x}, b_{x}) with ∀x ∈ X (a_{x}, b_{x}) ∈ {(n, 0), (n − 1, 1), (1, n − 1), (0, n)}, then the \overline{Q}_{ℓ}[Γ_{E}]-module

Hom_{G^{∗}(_{Q})}((π^{∗})^{∞}, H_{∗}^{dim})

(which is finite-dimensional over \overline{Q}_{ℓ}) is semisimple.

(2.21) If we admit the validity of an analogue of the main global result of [KMSW] for the unitary similitude group H, then the following Proposition will allow us to deduce from Theorem 2.20 a semisimplicity result for the ℓ-adic intersection cohomology of the Baily–Borel compactification of Sh(G^{∗}, 𝒳 ^{∗}).

(2.22) Proposition. Assume that (an expected variant of) [KMSW, Thm. 1.7.1] holds for H. If ξ^{∗}, π^{∗}, π and (Π, ψ) are as in 2.6, 2.11, 2.12 and 2.16, respectively, then:

(1) If π^{′∗} is a discrete automorphic representation of G^{∗}(A) such that (π^{′∗})^{∞} = (π^{∗})^{∞}, then π^{′∗} is cuspidal.

(2) The map V^{i}((π^{∗})^{∞}) ⊗ (π^{∗})^{∞} ↪ H_{∗,BB}^{i} ⟶ H_{∗}^{i} is injective.

(3) If V^{i}((π^{∗})^{∞}) ≠ 0, then i = dim = ∑_{x∈X} a_{x}b_{x}.

Proof. (1) A variant of [KMSW, Thm. 1.7.1] for H (rather than for Ker(ν)) would imply that, under the assumption (π^{′∗})^{∞} = (π^{∗})^{∞}, the representations π and π′ have the same transfer (Π, ψ) to (GL(n) × GL(1))_{F_{c}}. The cuspidality of π′ then follows from the cuspidality of Π.

(2) This is a consequence of (1) and [B, Cor. 5.5].

(3) Let K^{∗} ⊂ G^{∗}(Q̂) be an open compact subgroup such that ((π^{∗})^{∞})^{K^{∗}} ≠ 0. The sheaf ℒ_{ξ^{∗},ℓ} can be constructed using the universal abelian variety on the PEL Shimura variety Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}), as in [HT, III.2]. It is pure of weight w^{∗} = 2u + ∑_{x∈X} u_{x}, in the notation of 2.6 and 2.11. This implies that, for all but finitely many finite primes P̃ of F_{c}^{gal}, all eigenvalues in \overline{Q}_{ℓ} ≃ C of Fr(P̃) ∈ Γ_{F_{c}^{gal}} ⊂ Γ_{E} acting on V^{i}((π^{∗})^{∞}) ⊗ (π^{∗})^{∞} have absolute value (NP̃)^{(i+w^{∗})/2}. On the other hand, if P̃ is the restriction of a prime P_{S} as in (3.3.1) below, then NP̃ = p and i + w^{∗} = dim +w^{∗}, by (2) and Proposition 3.12(3) below (the reader is invited to check that our arguments are not circular).

(2.23) Corollary. Under the assumptions of 2.20(1), 2.20(2) and 2.22, the \overline{Q}_{ℓ}[Γ_{E}]-module V^{dim}((π^{∗})^{∞}) (which is isomorphic to its image in Hom_{G^{∗}(_{Q})}((π^{∗})^{∞}, H_{∗}^{dim})) is semisimple.

(2.24) Similar results hold for the Shimura variety Sh(G, 𝒳 ). In the compact case (when the derived group G^{der} is anisotropic) the following variant of Theorem 2.20 was proved in [F].

(2.25) Theorem. Let ξ be an irreducible algebraic representation of G_{C} satisfying the conditions from Proposition 2.7(2) and π a cuspidal automorphic representation of G(A) such that π_{∞} is cohomological in degree dim with respect to ξ. If π admits a weak transfer to a cuspidal automorphic representation (Π, ψ) of GL_{n}(A_{F_{c}}) × A_{F_{c}}^{×} and if the ℓ-adic Galois representation ρ_{Π,ℓ} : Γ_{F_{c}} ⟶ GL_{n}(\overline{Q}_{ℓ}) attached to Π satisfies 2.20(1) and 2.20(2), then the \overline{Q}_{ℓ}[Γ_{E}]-module

Hom_{G(_{Q})}(π^{∞}, H_{et}^{dim}(Sh(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ})) = Hom_{G(_{Q})}(π^{∞}, H^{dim})

is semisimple.

(2.26) Corollary. Under the assumptions of Theorem 2.25 and Proposition 2.22, the \overline{Q}_{ℓ}[Γ_{E}]-module V^{dim}(π^{∞}) (which is isomorphic to its image in Hom_{G(_{Q})}(π^{∞}, H^{dim})) is semisimple.

3. Proof of Theorems 2.20 and 2.25

(3.1) The proof will combine Theorem 1.7 with arguments along the lines of [N, §6]. Let ξ^{∗}, π^{∗}, π, (Π, ψ) and ρ_{Π,ℓ} be as in Theorem 2.20.

(3.2) Assume that V^{dim}((π^{∗})^{∞}) ≠ 0. Fix open compact subgroups K ⊂ G(Q̂) and K^{∗} ⊂ G^{∗}(Q̂) ∩ K such that (π^{∞})^{K} ≠ 0 ≠ ((π^{∗})^{∞})^{K^{∗}}. Let S be a sufficiently large finite set of places of Q containing {2, ℓ, ∞}, all primes where 2.14(4), (2.16.2) or (2.16.4) fails, all primes at which F_{c}/Q and B ramify, and such that K = K_{S}K^{S} and K^{∗} = K_{S}^{∗}K^{∗S}, where K^{S} = ∏_{p∉S} K_{p} and K^{∗S} = ∏_{p∉S} K_{p}^{∗} with K_{p} ⊂ G(Q_{p}) and K_{p}^{∗} ⊂ G^{∗}(Q_{p}) maximal compact subgroups.

(3.3) Denote by Q_{S}/Q the maximal subextension of \overline{Q}/Q unramified outside S. The Galois representation ρ_{Π,ℓ} then factors through Gal(Q_{S}/F_{c}).

We are going to consider primes P_{S} of Q_{S} which are unramified in Q_{S}/Q and which satisfy

Fr_{Q_{S}/Q}(P_{S}) ∈ Gal(Q_{S}/F_{c}^{gal}).  (3.3.1)

The rational prime p = P_{S} ∩ Z ∉ S then splits completely in F_{c}. Extend each element σ_{x} : F_{c} ↪ \overline{Q} ⊂ C (x ∈ X) of the CM type Φ of F_{c} to an element s(x) ∈ Γ_{Q}. For each x ∈ X, the prime P_{x} = s(x)^{−1}P_{S} ∩ O_{F} of F (resp. the prime P′_{x} = s(x)^{−1}P_{S} ∩ O_{F_{c}} of F_{c}) depends only on x (resp. on σ_{x}). These primes satisfy

pO_{F} = ∏_{x∈X} P_{x},  P_{x}O_{F_{c}} = P′_{x}P′′_{x},  P′′_{x} = c(P′_{x}),  F_{P_{x}} = (F_{c})_{P′_{x}} = (F_{c})_{P′′_{x}} = Q_{p}.

(3.4) Let p be as in 3.3. Fix an embedding \overline{Q} ↪ \overline{Q}_{p}; this defines a prime v | p of the reflex field E ⊂ F_{c}^{gal} ⊂ \overline{Q} such that E_{v} = Q_{p}.

The map α from 2.3 yields an isomorphism

G_{Q_{p}} ≃ ∏_{x∈X}(GL(n) × GL(1))_{(F_{c})_{P′_{x}}} = (GL(n) × GL(1))_{Q_{p}}^{X}.

Therefore the pair (G(Q_{p}), K_{p} = ∏_{x∈X} K_{x}) is isomorphic to ((GL_{n}(Q_{p}) × Q_{p}^{×})^{X}, (GL_{n}(Z_{p}) × Z_{p}^{×})^{X}) and (G^{∗}(Q_{p}), K_{p}^{∗}) to (GL_{n}(Q_{p})^{X} × Q_{p}^{×}, GL_{n}(Z_{p})^{X} × Z_{p}^{×}), with Q_{p}^{×} embedded diagonally into (Q_{p}^{×})^{X}.

By assumption, there exists a surjection π|_{G^{∗}(A)} ⟶ π^{∗}. As G(Q_{p}) ≃ G^{∗}(Q_{p}) × (Q_{p}^{×})^{r−1}, the induced surjection π_{p}|_{G^{∗}(Q_{p})} ⟶ π_{p}^{∗} is an isomorphism, since π_{p} and π_{p}^{∗} are irreducible and Q_{p}^{×} is abelian. In

particular, it induces an isomorphism π_{p}^{K_{p}^{∗}} ⟶^{∼} (π_{p}^{∗})^{K_{p}^{∗}}, hence an isomorphism between the spherical lines

π_{p}^{K_{p}} = ⨂_{x∈X} π_{P_{x}}^{K_{x}} ⊆ π_{p}^{K_{p}^{∗}}

and (π_{p}^{∗})^{K_{p}^{∗}}. Note that

π_{p} = ⨂_{x∈X} π_{P_{x}} ≃ ⨂_{x∈X}(Π, ψ)_{P′_{x}},

hence

π_{p}^{∗}|_{GL_{n}(Q_{p})^{X}} ≃ ⨂_{x∈X} Π_{P′_{x}}.

(3.5) There exists an O_{F_{c}} ⊗Z_{p}-lattice Λ ⊂ V_{Q_{p}} which is stable under K_{p}^{∗}. Together with the O_{F_{c}} ⊗Z_{(p)}-order O_{B} = {b ∈ B | bΛ ⊂ Λ} ⊂ B, the pair (O_{B}, Λ) gives rise to an unramified PEL datum in the sense of [K, §5]. For K_{S} small enough, the construction in [K, §5] defines a smooth quasi-projective model of Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) over O_{E} ⊗Z_{(p)}. We denote by S_{K^{∗}} its base change to O_{E_{v}} = Z_{p} and by S_{K^{∗}}^{∘} the special fibre of S_{K^{∗}} (defined over the residue field k(v) = F_{p}).

(3.6) The sets of conjugacy classes of cocharacters G_{m} ⟶ G^{∗} defined over the fields C ⊃ \overline{Q} ↪ \overline{Q}_{p} coincide. Viewed over \overline{Q}_{p}, the conjugacy class [μ_{h}] of the cocharacter μ_{h} attached to the Shimura datum (G^{∗}, 𝒳 ^{∗}) contains a character defined over Q_{p}, namely,

μ = (μ_{x})_{x∈X} : G_{m,Q_{p}} ⟶ G_{Q_{p}}^{∗} = GL(n)_{Q_{p}}^{X} × G_{m,Q_{p}} ⊂ G_{Q_{p}}, μ(t) = ((tI_{a_{x}}, I_{b_{x}})_{x∈X}, t).

The parabolic subgroups P_{μ}^{∗} ⊂ G_{Q_{p}}^{∗} and P_{μ} ⊂ G_{Q_{p}} attached to μ [N, A1.2] are equal to

P_{μ}^{∗} = P × G_{m,Q_{p}} ⊂ P_{μ} = P × G_{m,Q_{p}}^{X}, P = ∏_{x∈X} ( \begin{matrix}GL(a_{x}) & ∗ \\ 0 & GL(b_{x}) \end{matrix} )_{Q_{p}} .

We identify their Levi subgroups M^{∗} ⊂ M with their respective sets of Q_{p}-points, namely,

M^{∗} = ( ∏_{x∈X} GL_{a_{x}}(Q_{p}) × GL_{b_{x}}(Q_{p})) × Q_{p}^{×} ⊂ M = ∏_{x∈X} M_{x} = ∏_{x∈X}(GL_{a_{x}}(Q_{p}) × GL_{b_{x}}(Q_{p}) × Q_{p}^{×}).

The subgroups L^{∗} = M^{∗} ∩ K_{p}^{∗} and L = M ∩ K_{p} = ∏_{x∈X} L_{x} are given by the same formulas, with Q_{p} replaced by Z_{p}. (3.7) We use the notation H(G^{∗}(Q_{p})//K_{p}^{∗}, Q) for the (commutative) Hecke algebra of locally constant functions with compact support f : K_{p}^{∗}\G^{∗}(Q_{p})/K_{p}^{∗} ⟶ Q (and similarly for other Hecke algebras).

The maps K_{p}^{∗}gK_{p}^{∗} ↦ K_{p}gK_{p} and L^{∗}mL^{∗} ↦ LmL define embeddings of Hecke algebras

H^{∗} = H(G^{∗}(Q_{p})//K_{p}^{∗}, Q) ↪ H(G(Q_{p})//K_{p}, Q) ≃ ⨂_{x∈X} H_{x}, H_{x} = H(GL_{n}(Q_{p}) × Q_{p}^{×}//GL_{n}(Z_{p}) × Z_{p}^{×}, Q)

and

H_{M}^{∗} = H(M^{∗}//L^{∗}, Q) ↪ H(M//L, Q) ≃ ⨂_{x∈X} H(M_{x}//L_{x}, Q),

H(M_{x}//L_{x}, Q) = H(GL_{a_{x}}(Q_{p}) × GL_{b_{x}}(Q_{p}) × Q_{p}^{×}//GL_{a_{x}}(Z_{p}) × GL_{b_{x}}(Z_{p}) × Z_{p}^{×}, Q)

that are compatible with the twisted Satake transforms ([N, A1.4])

\overline{S}_{μ} : H^{∗} ⟶ H_{M}^{∗}, \overline{S}_{μ_{x}} : H_{x} ⟶ H(M_{x}//L_{x}, Q)

given by

(\overline{S}_{μ}f)(m) = ∫_{N_{μ}} f(mu) du,

where N_{μ} is the common unipotent radical of P_{μ}^{∗} and P_{μ} and du is the Haar measure on N_{μ} giving N_{μ} ∩ K_{p} volume 1. (3.8) These Hecke algebras contain the following important elements.

• φ_{x} = L_{x}μ_{x}(p)^{−1}L_{x} = L_{x}((p^{−1}I_{a_{x}}, I_{b_{x}}), p^{−1})L_{x} ∈ H(M_{x}//L_{x}, Q) (the partial Frobenius at P′_{x})

• φ = ∏_{x∈X} φ_{x} = L^{∗}μ(p)^{−1}L^{∗} ∈ H_{M}^{∗} (the total Frobenius)

• U′_{x} = K_{x}(I_{n}, p^{−1})K_{x} ∈ H_{x}

• U_{x} = \overline{S}_{μ_{x}}(U′_{x}) = L_{x}(I_{n}, p^{−1})L_{x} ∈ H(M_{x}//L_{x}, Q)

• U′ = ∏_{x∈X} U′_{x} ∈ H^{∗}

• U = \overline{S}_{μ}(U′) = ∏_{x∈X} U_{x} ∈ H_{M}^{∗} • φ̃_{x} = φ_{x}U_{x}^{−1} = φ_{x}U_{x}^{−1} ⊗ ⨂_{y≠x} 1_{y} ∈ H_{M}^{∗} (twisted partial Frobenius) By definition,

U ∏_{x∈X} φ̃_{x} = ∏_{x∈X} U_{x}φ̃_{x} = φ.

(3.9) According to [Bu, Prop. 3.4] (see also [W, Prop. 2.9]), the element φ_{x} ∈ H(M_{x}//L_{x}, Q) satisfies an abstract Eichler–Shimura relation of the form

(\overline{S}_{μ_{x}}(Q_{x})) (φ_{x}) = 0 ∈ H(M_{x}//L_{x}, Q),  (3.9.1)

where Q_{x} ∈ H_{x}[Y] is an explicit polynomial of degree deg(Q_{x}) = \binom{n}{a_{x}} (this uses the fact that the cocharacter μ_{x} is minuscule). The image of Q_{x} under the morphism

H_{x} ⟶ C, T ↦ T |_{(Π,ψ)_{P′_{x}}^{Kx}}

given by the action of H_{x} on the K_{x}-spherical line of (Π, ψ)_{P′_{x}} is given by the following formula ([N, A5.10]; our notation and conventions are those of [N, §A1]), in which (R_{λ}Q)(Y) = λ^{deg(Q)}Q(λ^{−1}Y):

(R_{p^{−(axbx/2)}} Q_{x})|_{(Π,ψ)_{P′_{x}}^{Kx}} (Y) = ∏_{|I|=a_{x}}(Y − t_{I}^{−1}ψ_{x}(p)^{−1}) (I ⊂ {1, …, n}, t_{I} = ∏_{i∈I} t_{i}),

where

L_{P′_{v}}(ρ_{Π,ℓ}, s + (n − 1)/2) = L_{P′_{v}}(Π, Std_{n}, s) = ∏_{i=1}^{n}(1 − t_{i}p^{−s})^{−1},  ψ_{x}(p) = ψ(P′_{x}).  (3.9.2)

It follows from (3.9.1) that

(R_{U_{x}^{−1}}(\overline{S}_{μ_{x}}(Q_{x}))) (φ̃_{x}) = (\overline{S}_{μ_{x}}(R_{U_{x}^{′−1}}(Q_{x}))) (φ̃_{x}) = 0 ∈ H_{M}^{∗}.  (3.9.3)

The polynomial R_{U_{x}^{′−1}}(Q_{x}) lies in H^{∗}[Y] and satisfies

(R_{p^{−(axbx/2)}}R_{U_{x}^{′−1}} Q_{x})|_{(π_{p}^{∗})^{K^{∗}_{p}}} (Y) = (R_{p^{−(axbx/2)}ψ(P′_{x})}(Q_{x}|_{(Π,ψ)_{P′_{x}}^{Kx}}))(Y) = ∏_{|I|=a_{x}}(Y − t_{I}^{−1}).  (3.9.4)

For any representation ρ of Gal(Q_{S}/F_{c}^{gal}) and x ∈ X, denote by ^{s(x)}ρ the representation (^{s(x)}ρ)(g) = ρ(s(x)^{−1}gs(x)) of the same group; then

(^{s(x)}ρ)(Fr(P_{S})) = ρ(s(x)^{−1}Fr(P_{S})s(x)) = ρ(Fr(P′_{x})).

In particular, the representation

ρ′_{x} := ^{s(x)}((⋀^{a_{x}}ρ_{Π,ℓ}^{∨}) (a_{x}(a_{x} + 1)/2 − a_{x}n)) : Gal(Q_{S}/F_{c}^{gal}) ⟶ GL_{n_{x}}(\overline{Q}_{ℓ}),  n_{x} = \binom{n}{a_{x}} (3.9.5)

satisfies

ρ′_{x}(Fr(P_{S})) = (⋀^{a_{x}}ρ_{Π,ℓ}^{∨}) (Fr(P′_{x})) p^{a_{x}b_{x}/2} p^{a_{x}(n−1)/2},

which implies that

P_{ρ′_{x}(Fr(P_{S}))}(Y) = R_{U_{x}^{′−1}}(Q_{x})|_{(π_{p}^{∗})^{K^{∗}_{p}}} (Y),  (3.9.6)

thanks to (3.9.2) and (3.9.4).

(3.10) Following [FC, VII.3], Wedhorn [W, 3.3] defined a scheme of p-isogenies p − Isog_{K^{p}} ⟶ S_{K^{∗}} × S_{K^{∗}} over O_{E_{v}} = Z_{p} (Wedhorn worked with stacks, but if one imposes (as we do) a level structure K^{∗p} outside p with small enough K_{S}, then one obtains a scheme). The main result of [W] (see also [Mo, Thm. 4.2.13]) is the existence of a commutative diagram of Q-algebras (with the sign conventions corrected according to [N, A4.2, A4.4, A5.2])

\xymatrix{H(G^{∗}(Q_{p})_{−}//K_{p}^{∗}, Q) \ar[r]^{h} \ar[d]^{_{\overline{S}_{μ}}} & Q[p − Isog_{K^{p}} ⊗ E_{v}] \ar[d]^{σ} \\ H(M_{−}^{∗}//L^{∗}, Q) \ar[r]^{\overline{h}} & Q[p − Isog_{K^{p}} ⊗ F_{p}]}  (3.10.1)

(E_{v} = Q_{p}, k(v) = F_{p}). Above, Q[p − Isog_{K^{p}}/S] denotes the Q-vector space on the set of connected components of p − Isog_{K^{p}}(S) (with the ring structure defined as in [FC, p. 252]; see [N, A4.1] for the sign conventions in our context), the map σ is given by specialisation of cycles and

G^{∗}(Q_{p})_{−} = {g ∈ G^{∗}(Q_{p}) | g^{−1}(Λ) ⊂ Λ}, M_{−}^{∗} = G^{∗}(Q_{p})_{−} ∩ M^{∗}.

The rings H(G^{∗}(Q_{p})_{−}//K_{p}^{∗}, Q) and Q[p−Isog_{K^{p}} ⊗E_{v}] naturally act on H_{et}^{i}(Sh_{K^{∗}} ⊗_{E} \overline{Q}, ℒ_{ξ^{∗},ℓ}) ([FC, VII.2], [N, A5.5, A5.11]). The \overline{Q}_{ℓ}-sheaf ℒ_{ξ^{∗},ℓ} extends to a \overline{Q}_{ℓ}-sheaf on S_{K^{∗}}, whose restriction to the special fibre of S_{K^{∗}} will be denoted by ℒ_{ξ^{∗},ℓ}^{∘}. There is a canonical isomorphism

H_{et}^{i}(Sh_{K^{∗}} ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) ≃ H_{et}^{i}(S_{K^{∗}}^{∘} ⊗ \overline{F}_{p}, ℒ_{ξ,ℓ}^{∘})  (3.10.2)

([FC, Thm. VI.6.1], [LSt, Thm. 6.1]). The ring Q[p−Isog_{K^{p}} ⊗F_{p}] naturally acts on H_{et}^{i}(S_{K^{∗}}^{∘} ⊗\overline{F}_{p}, ℒ_{ξ,ℓ}^{∘}) and all of the above actions are compatible via (3.10.1). Under the isomorphism (3.10.2), the action of Fr(P_{S}) on the left hand side corresponds to the action of φ ∈ H(M_{−}//L, Q) on the right hand side.

(3.11) Let i ≥ 0 be arbitrary. According to [N, A5.5, A5.12],

H_{∗}^{i}[(π^{∗})^{∞}] := Hom_{G^{∗}(_{Q})}((π^{∗})^{∞}, H_{∗}^{i}) ⊗ (π^{∗})^{∞}

is a \overline{Q}_{ℓ}[Γ_{E} ×G^{∗}(Q̂^{(p)})]×(\overline{Q}_{ℓ}⊗_{Q}H^{∗})-submodule of H_{∗}^{i}, stable under the action of φ̃_{x}, for all x ∈ X. Therefore

W^{i} := H_{∗}^{i}[(π^{∗})^{∞}]^{K^{∗}} = Hom_{G^{∗}(_{Q})}((π^{∗})^{∞}, H_{∗}^{i}) ⊗ ((π^{∗})^{∞})^{K^{∗}} ⊂ H^{i}(Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) ⊗_{E} \overline{Q}, ℒ_{ξ^{∗},ℓ}) = (H_{∗}^{i})^{K^{∗}}

(3.11.1)

is a \overline{Q}_{ℓ}[Γ_{E}] × (\overline{Q}_{ℓ} ⊗_{Q} H^{∗})-submodule of (H_{∗}^{i})^{K^{∗}}, stable under all φ̃_{x}. In addition, ∏_{x∈X} φ̃_{x} = Uφ, with φ acting on (H_{∗}^{i})^{K^{∗}} as Fr(P_{S}).

The abstract Eichler–Shimura relation (3.9.3) together with the commutative diagram (3.10.1) and the compatibilities discussed in 3.10 imply that

R_{U_{x}^{′−1}}(Q_{x})|_{(π_{p}^{∗})^{K^{∗}_{p}}} (φ̃_{x}|_{W^{i}}) = 0 ∈ End_{\overline{Q}_{ℓ}}(W^{i}),

hence

∀x ∈ X  P_{ρ′_{x}(Fr(P_{S}))} (φ̃_{x}|_{W^{i}}) = 0 ∈ End_{\overline{Q}_{ℓ}}(W^{i}),  (3.11.2)

by (3.9.6) (above, ρ′_{x} is the Galois representation defined in (3.9.5)). Moreover,

( ∏_{x∈X} φ̃_{x})|_{W^{i}} = U′|_{W^{i}} · φ|_{W^{i}} = c(P_{S}) · Fr(P_{S})|_{W^{i}},  c(P_{S}) = ∏_{x∈X} ψ(P′_{x})^{−1}.  (3.11.3)

(3.12) Proposition. (1) There exists a character χ : Γ_{F_{c}^{gal}} ⟶ \overline{Q}_{ℓ}^{×} such that χ(Fr(P_{S})) = ∏_{x∈X} ψ(P′_{x}), for every P_{S} satisfying (3.3.1).

(2) If P_{S} satisfies (3.3.1), then

P_{(⊗_{x}ρ′_{x})(Fr(P_{S}))}(Fr(P_{S})|_{W^{i}⊗χ}) = 0 ∈ End_{\overline{Q}_{ℓ}}(W^{i} ⊗ χ).

(3) If P_{S} satisfies (3.3.1), then all eigenvalues of Fr(P_{S})|_{W^{i}} in \overline{Q}_{ℓ} ≃ C have absolute value p^{m/2}, where

m = 2u + ∑_{x∈X} u_{x} + ∑_{x∈X} a_{x}b_{x} = w^{∗} + dim,

in the notation of 2.6 and 2.11.

Proof. (1) The character

χ̃ : ∏_{x∈X} (ψ ∘ N_{F_{c}^{gal}/F_{c}} ∘ s(x)^{−1}) : A_{F_{c}^{gal}}^{×}/(F_{c}^{gal})^{×} ⟶ C^{×}

sends a uniformiser at P_{S} ∩ O_{F_{c}^{gal}} to ∏_{x∈X} ψ(P′_{x}). The infinity type of χ̃ is algebraic: if τ : F_{c}^{gal} ↪ C extends some σ_{x} : F_{c} ↪ C, then

χ̃_{τ} (z) = ∏_{x∈X} (\overline{z}^{u_{x}}|z|^{t−u_{x}}) = z^{a}\overline{z}^{b},  a = ∑_{x∈X}(t − u_{x})/2 = u ∈ Z,  b = ∑_{x∈X}(t + u_{x})/2 = u + ∑_{x∈X} u_{x} ∈ Z,

in the notation of (2.11.1). The ℓ-adic character χ : Γ_{F_{c}^{gal}} ⟶ \overline{Q}_{ℓ}^{×} attached to χ̃ ([Sc, ch. 0, §5], [HT, p. 20]) then satisfies χ(Fr(P_{S})) = ∏_{x∈X} ψ(P′_{x}), as claimed.

(2) This is a consequence of (3.11.2), (3.11.3) and (1).

(3) It follows from (2.16.4) that all eigenvalues of ρ′_{x}(Fr(P_{S})) in \overline{Q}_{ℓ} ≃ C have absolute value p^{m_{x}/2}, where m_{x} = a_{x}(1 − n) − a_{x}(a_{x} + 1) + 2a_{x}n = a_{x}(n − a_{x}) = a_{x}b_{x}. By construction, χ(Fr(P_{S})) has absolute value p^{−(a+b)/2}, with a and b as in the proof of (1). We deduce from (2) that m = a + b + ∑_{x∈X} m_{x}, as claimed. (3.13) We are now ready to conclude the proof of Theorem 2.20, by applying Theorem 1.7 to the following objects: Γ = Γ′ = Gal(Q_{S}/F_{c}^{gal}), Σ = {g = Fr(P_{S}) | P_{S} as in (3.3.1)}, V = V (g) = W^{dim}|_{Γ} ⊗ χ, {ρ_{1}, …, ρ_{r}} = {ρ′_{x}}_{x∈X}, {u_{1}, …, u_{r}} = {φ̃_{x}}_{x∈X}. We can (and will) disregard the elements x ∈ X for which (a_{x}, b_{x}) = (n, 0) or (0, n), since dim(ρ′_{x}) = 1 then.

For the remaining x ∈ X, assumption (A) in Theorem 1.7 is a consequence of 2.20(1) in the case 2.20(2b) (resp. of the fact that ⋀^{a}\overline{Q}_{ℓ}^{n} is an irreducible sl(n, \overline{Q}_{ℓ})-module, for all a, in the case 2.20(2a)). Assumption (B) follows from Corollary 2.18. Finally, assumption (C) is satisfied by (3.11.2), (3.11.3) and Proposition 3.12(1). Theorem 1.7 tells us that V is a semisimple \overline{Q}_{ℓ}[Gal(Q_{S}/F_{c}^{gal})]-module, hence so is Hom_{G^{∗}(_{Q})}((π^{∗})^{∞}, H_{∗}^{dim})|_{Γ}, since ((π^{∗})^{∞})^{K^{∗}} ≠ 0. Theorem 2.20 is proved.

(3.14) In the case 2.20(2a) one can avoid the use of Theorem 1.17 and combine instead Proposition 3.12(2) with [N, Prop. 3.10]. Indeed, all exterior powers ⋀^{a}\overline{Q}_{ℓ}^{n} are one-dimensional or minuscule sl(n, \overline{Q}_{ℓ})-modules, which means that [N, Prop. 3.10(1)] applies: there exists an open subgroup U ⊂ Γ such that V^{ss}|_{U} = (V |_{U})^{ss} is isomorphic to a subrepresentation of (⊗_{x}ρ′_{x})^{⊕n}|_{U}, for some n ≥ 1. Semisimplicity of V then follows from [N, Thm. 3.3, 3.4, Thm. 3.6]. (3.15) Theorem 2.25 is proved along the same lines as Theorem 2.20. One works directly with the Shimura variety attached to (G, 𝒳 ) and the contorsions involving φ̃_{x} are unnecessary.

One considers K ⊂ G(Q̂) such that (π^{∞})^{K} ≠ 0 and primes P_{S} as in (3.3.1). The Kottwitz model S_{K^{∗}} is replaced by its variant S_{K} described in [N, A3.7] and the diagram (3.10.1) by [N, (A5.1.1)].

As ξ satisfies the condition in Proposition 2.7(2), the Hecke character ψ is algebraic. Denote by ρ_{ψ} : Gal(Q_{S}/F_{c}) ⟶ Q_{ℓ}^{×} the corresponding ℓ-adic Galois representation ([Sc, ch. 0, §5], [HT, p. 20]): ρ_{ψ}(Fr(P′_{x})) = ψ_{x}(p) = ψ(P′_{x}) for all x ∈ X.

The \overline{Q}_{ℓ}[Γ_{E}] × (\overline{Q}_{ℓ} ⊗_{Q} H)-module

W^{i} := H^{i}[π^{∞}]^{K} = Hom_{G(_{Q})}(π^{∞}, H^{i}) ⊗ (π^{∞})^{K} ⊂ H^{i}(Sh_{K}(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) = (H^{i})^{K}  (3.15.1)

is stable under the action of all φ_{x}. The Eichler–Shimura relation [N, Thm. A5.3, (A5.5.2)] for φ_{x}

Q_{x}|_{π_{p}^{Kp}} (φ_{x}|_{W^{i}}) = 0 ∈ End_{\overline{Q}_{ℓ}}(W^{i})

becomes [N, (A5.10.2)]

∀x ∈ X  P_{ρ_{x}(Fr(P_{S}))} (φ_{x}|_{W^{i}}) = 0 ∈ End_{\overline{Q}_{ℓ}}(W^{i}),  (3.15.2)

where

ρ_{x} := ^{s(x)}((⋀^{a_{x}}ρ_{Π,ℓ}^{∨}) ⊗ ρ_{ψ}^{∨}(a_{x}(a_{x} + 1)/2 − a_{x}n)) : Gal(Q_{S}/F_{c}^{gal}) ⟶ GL_{n_{x}}(\overline{Q}_{ℓ}), n_{x} = \binom{n}{a_{x}}.

By definition,

( ∏_{x∈X} φ_{x})|_{W^{i}} = φ|_{W^{i}} = Fr(P_{S})|_{W} .  (3.15.3)

As in 3.13, we apply Theorem 1.17 to Γ = Γ′ = Gal(Q_{S}/F_{c}^{gal}), Σ = {g = Fr(P_{S}) | P_{S} as in (3.3.1)}, V = V (g) = W^{dim}|_{Γ} (with W^{dim} from (3.15.1), not (3.11.1)), {ρ_{1}, …, ρ_{r}} = {ρ_{x}}_{x∈X}, {u_{1}, …, u_{r}} = {φ_{x}}_{x∈X}. Again, we disregard the elements x ∈ X for which (a_{x}, b_{x}) = (n, 0) or (0, n).

Assumptions (A) and (B) of Theorem 1.17 are checked as in 3.13; assumption (C) is a consequence of (3.15.2) and (3.15.3). We obtain that V is a semisimple \overline{Q}_{ℓ}[Gal(Q_{S}/F_{c}^{gal})]-module, as claimed. (3.16) If ρ_{Π,ℓ} in Theorem 2.25 satisfies 2.20(2a), one can again avoid the use of Theorem 1.17, as in 3.14. The formulas (3.15.2-3) imply that

P_{(⊗_{x}ρ_{x})(Fr(P_{S}))}(Fr(P_{S})|_{W^{i}}) = 0 ∈ End_{\overline{Q}_{ℓ}}(W^{i})

and one deduces from [N, Prop. 3.10(1)] that there exists an open subgroup U ⊂ Γ such that V^{ss}|_{U} = (V |_{U})^{ss} is isomorphic to a subrepresentation of (⊗_{x}ρ_{x})^{⊕m}|_{U}, for some m ≥ 1. Semisimplicity of V then follows from [N, Thm. 3.3, 3.4, Thm. 3.6], and we obtain as a bonus that V^{dim}(π^{∞})|_{U} ⊂ (⊗_{x}ρ_{x})^{⊕m}|_{U} (if a_{x}b_{x} ≠ 0 only for one x ∈ X, then we can even take U = Γ, by [N, Prop. 3.10(2)]).

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Jan Nekovář Institut de Mathématiques de Jussieu Sorbonne Université Campus Pierre et Marie Curie Théorie des Nombres, Case 247 4, place Jussieu F-75252 Paris cedex 05 FRANCE

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