SPRINGER CORRESPONDENCE IN NON CONNECTED REDUCTIVE GROUPS

KARINE SORLIN

Abstract:

Let G be an algebraic connected group and let σ be a quasi-semi-simple unipotent automorphism of G, i.e. an automorphism of G which stabilises a pair (T, B) where T is a maximal torus included in a Borel subgroup B of G. The purpose of this article is to show that there is a Springer correspondence in the component G.σ of the non connected reductive group G̃ = G.⟨σ⟩ between some pairs (C, E), where C is a unipotent conjugacy class of G.σ and E a G-equivariant irreducible local system on C, and the irreducible representations of the Weyl group of the group G^{σ} of σ-fixed points of G.

Introduction

In this article, we consider a non connected reductive algebraic group of the form G̃ = G.⟨σ⟩, where G is a connected reductive group on an algebraic closed field of characteristic p > 0 and σ is a unipotent quasi-semi-simple automorphism of G. An automorphism σ of G is said to be quasi-semi-simple if there is a σ-stable pair (T, B), where T is a maximal torus included in a Borel subgroup B of G. If G = GL(n), the subgroup B of upper triangular matrices is a Borel subgroup of G and the subgroup of diagonal matrices is a maximal torus of G included in B, moreover let σ be the automorphism of G defined by A ↦ J^{t}A^{−1}J^{−1}, where J is the matrix

J = ( \begin{matrix}& & 1 \\ & ... \\ 1 \end{matrix} )

Then σ stabilises both T and B, so it is quasi-semi-simple.

The purpose of this article is to define the analogous of the Springer correspondence for the connected component G^{1} = G.σ. In the connected case, the Springer correspondence was often studied. In the case of the connected reductive group G, it is a map from the set of irreducible representations of the Weyl group of G to the set of pairs (C, E), where C is a unipotent Gconjugacy class in G and E is a G-equivariant irreducible local system on C. For example, it is well-known that irreducible characters of the symmetric group of degree n, which is the Weyl group of GL(n), are in bijection with unipotent conjugacy classes of GL(n).

The Springer correspondence for G^{1} is a map from the set of irreducible representations of the Weyl group of the group G^{σ} of σ-fixed elements of G (which is then a connected reductive group) to the set of pairs (C, E), where C is a unipotent G-class in G^{1} and E is a G-equivariant irreducible local system on C. This correspondence induces an injective map from the set of unipotent conjugacy classes in G^{1} to the set of irreducible representations of the Weyl group of G^{σ}.

In section 1, we recall some results from [7] about quasi-semi-simple automorphisms of a connected reductive group. The subject of section 2 is the study of regular quasi-semi-simple elements of G^{1} which play the same role in the Springer correspondence for G^{1} as regular semisimple elements in the Springer correspondence for G. In section 3, we give some preliminary results on the variety B_{g.σ} of the Borel subgroups of G which are stable under conjugation by an element g.σ of G^{1}. In section 4, we construct representations of the group W^{σ} of σ-fixed elements of the Weyl group W of G on l-adic groups H^{i}(B_{g.σ}^{G}, \overline{ℚ}_{l}), where g.σ ∈ G^{1} (proposition 4.7), which we call Springer representations of W^{σ} from G^{1}.

1991 Mathematics Subject Classification. According to the 2000 classification: Primary 20G05; Secondary 20C33.

As σ is unipotent quasi-semi-simple, W^{σ} is also the Weyl group of G^{σ} and B_{σ}^{G} is isomorphic to the variety B^{G^{σ}} of Borel subgroups of G^{σ}. In section 5, we compare the Springer representation of W^{σ} from G^{1} on H^{∗}(B_{σ}^{G}, \overline{ℚ}_{l}) constructed in section 4 with the Springer representation of W^{σ} from the connected reductive group G^{σ} on H^{∗}(B^{G^{σ}}, \overline{ℚ}_{l}) (proposition 5.1), both of them coincide with the regular representation of W^{σ} (corollaire 5.4). This result is necessary for the proof of theorem 6.3 which defines the Springer correspondence for G^{1}.

The existence of a Springer correspondence is then proved in theorem 6.3 but the correspondence is not explicitely known. We have to use its properties to calculate it in some exemples. Two of these properties are proved at the end of this article. Section 6 ends with the proof of a property whose analogous is true in the connected case (proposition 6.7): for any unipotent element u.σ in G^{1}, let ρ be the irreducible representation of W^{σ} which corresponds to the pair (Cl_{G}(u.σ), \overline{ℚ}_{l}) by the Springer correspondence, then dim(B_{u.σ}) = a_{ρ}, where a_{ρ} is the valuation of the fake degree of ρ. The purpose of the last section is to prove a second property of the Springer correspondence for G^{1}. Let L be a σ-stable Levi subgroup of a σ-stable parabolic subgroup of G. Let C^{L} be a unipotent L-conjugacy class in L^{1} = L.σ and let C be the unipotent G-conjugacy class in G^{1} obtained by inducing C^{L} from L^{1} to G^{1} (definition 7.1). Then, the image ρ_{(C^{L},\overline{ℚ}_{l})} of (C^{L}, \overline{ℚ}_{l}) by the Springer correspondence for L^{1} is an irreducible component of the restriction to W (L)^{σ} of the image ρ_{(C,\overline{ℚ}_{l})} of (C, \overline{ℚ}_{l}) under the Springer correspondence for G^{1} (proposition 7.13).

1. Automorphims of connected reductive groups

Let G be a connected reductive group on an algebraic closed field K of characteristic p. According to [7] (théorème 1.8), when an automorphism is quasi-semi-simple, the group G^{σ} of fixed points of G under σ is a reductive group, and there is a one-to-one correspondence between σ-stable maximal tori of G included in a σ-stable Borel subgroup of G and maximal tori of G^{σ} defined by T ↦ T ∩ (G^{σ})^{0}.

A quasi-semi-simple automorphism σ is said to be quasi-central if the following equivalent properties are true:

Theorem 1.1. ([7] théorème 1.15) Let G be a connected reductive group and σ a quasi-semi-simple automorphism of G. The following properties are equivalent:

(i) dim(G^{σ}) > dim(G^{σ′}) for any quasi-semi-simple automorphism σ′ = σ ∘ ad(g), where ad(g) is the conjugation by g for all g ∈ G.

(ii) There is a bijection between σ-stable Borel subgroups of G and maximal Borel subgroups of G^{σ} defined by B ↦ B ∩ (G^{σ})^{0}.

(iii) There exists a σ-stable maximal torus T included in a σ-stable Borel subgroup B such that the Weyl group W (G^{σ}) of G^{σ} is isomorphic to the group W (G)^{σ} of fixed points by σ of the Weyl group of G.

(iii’) Statement (iii) is true for every pair (T, B) where T is a σ-stable maximal torus T included in a σ-stable Borel subgroup B.

Until the end of the article, we will consider a reductive group of the form G̃ = G.⟨σ⟩ where G is a connected reductive group and σ is a unipotent quasi-semi-simple automorphism of G. Such a σ is quasi-central by (i) of the following result:

Proposition 1.2. ([7] corollaire 1.33)

Let H be a connected reductive group and let σ be a unipotent quasi-semi-simple automorphism of H. Then,

(i) σ is quasi-central,

(ii) H^{σ} is connected,

(iii) For any σ-stable maximal torus T of H, we have T = T^{σ}L_{σ}(T) where L_{σ}(T) is the map from T to T defined by t ↦ t^{−1}σ(t).

We say that an element of G̃ is quasi-semi-simple if it induces a quasi-semi-simple automorphism of G. Like in [7], we will call quasi-borel of G̃ the normalizer in G̃ of a Borel subgroup of G and quasi-torus of G̃ the normalizer in G̃ of a pair (T, B) where T is a maximal torus included in a Borel subgroup B of G.

We will use the following notation: we first fix a pair (T, B) where T is a σ-stable maximal torus included in a σ-stable Borel subgroup B of G. Then, T^{σ} is a maximal torus and B^{σ} a Borel subgroup of the connected reductive group G^{σ}. Let U be the unipotent radical of B. We will denote by G^{1} the connected component G.σ and by U(G^{1}) the subvariety of unipotent elements of G^{1}. Let W be the Weyl group of G then W^{σ} is the Weyl group of G^{σ}. For all connected reductive group H, we will denote by rk(H) the rank of H and B^{H} the variety of Borel subgroups of H. Moreover, for all x ∈ G̃, B_{x}^{G} will be the variety of Borel subgroups of G which are stable by conjugation under x.

2. Regular quasi-semi-simple elements of G^{1}

Regular quasi-semi-simple elements of G^{1} play the same role for the Springer correspondence in G^{1} than that played by regular semi-simple elements for the Springer correspondence in G.

Definition 2.1. An element x of G̃ is said regular if it is regular in the connected component G.x of G̃ which it belongs to, that is to say, if its centralizer C_{G}(x) in G has minimal dimension among the C_{G}(y) for y ∈ G.x.

Proposition 2.1.

(i) An element g.σ of G^{1} is regular if and only if dim(C_{G}(g.σ)) = rk(G^{σ}).

(ii) An element g.σ of G^{1} is regular if and only if dim(B_{g.σ}^{G}) = 0.

(iii) Let t be a semi-simple element of G^{σ}. Then, t is regular in the connected reductive group G^{σ} if and only if t.σ is regular in G^{1}.

Proof: Let t be a semi-simple element of G^{σ}. Then, t (resp. σ) is the semi-simple (resp. unipotent) part of t.σ. So, for all x ∈ C_{G}(t.σ), we have (xtx^{−1})(xσx^{−1}) = t.σ, hence, by unicity of the Jordan decomposition, xtx^{−1} = x and xσx^{−1} = σ. So, we have C_{G}(t.σ) = C_{G}(t)∩C_{G}(σ) = C_{G^{σ}}(t). Then, we have dim(C_{G}(t.σ)) = dim(C_{G^{σ}}(t)).

But, t is semi-simple regular in G^{σ} if and only if dim(C_{G^{σ}}(t)) = rk(G^{σ}) ([6] proposition 14.2). So, if t is semi-simple regular in G^{σ} then dim(C_{G}(t.σ)) = rk(G^{σ}).

Moreover, according to [14] (théorème II.10.15), we have dim(CG̃(x)) = rk_{x}(G) + 2dim(B_{x}^{G}) for all x ∈ G̃, where rk_{x}(G) = max{rk(C_{G}(y))|y ∈ G.x}. But, as σ is quasi-central, we have rk_{g.σ}(G) = rk(G^{σ}) and dim(C_{G}(g.σ)) ≥ rk(G^{σ}) for all g.σ ∈ G^{1}. Hence, if t is semi-simple regular in G^{σ} then dim(C_{G}(t.σ)) = rk(G^{σ}) and that means that t.σ is regular in G^{1}. Whence, we have proved assertion (i) and the direct implication of (iii). Assertion (ii) is a direct consequence of (i) and of the above result of [14] (théorème II.10.15).

Let us prove the converse of (iii). Let t be a semi-simple element of G^{σ} such that t.σ is regular. Then, we have dim(C_{G}(t.σ)) = rk(G^{σ}) which implies that dim(C_{G^{σ}}(t)) = rk(G^{σ}), that is to say that t is regular in G^{σ}.

End of the proof.

Theorem 2.2. Regular quasi-semi-simple elements are dense in G^{1}.

Proof: We have shown in the proof of the previous proposition that if t ∈ T^{σ} is a regular element of G^{σ}, then t.σ is regular in G̃. Moreover t.σ is quasi-semi-simple since T and B are stable by conjugation under t.σ. As regular elements of G^{σ} are dense in T^{σ} ([6] proposition 14.7), so quasi-semi-simple regular elements of G^{1} are dense in T^{σ}.σ.

We will then need the following lemma:

Lemma 2.3. Let σ′ be a regular quasi-semi-simple element of G̃ stabilizing T and B; then the map f : U → U defined by u ↦ u^{σ′}u^{−1} is surjective.

Proof of lemma 2.3: Let σ′ = su be the Jordan decomposition of σ′. Then, we have dim(C_{G}(σ′)) = dim(C_{C_{G}(s)}(u)). But, as σ′ is regular, we have dim(C_{G}(σ′)) = rk_{σ′}(G), and according to [14] (lemma II.1.14), rk_{σ′}(G) = rk_{u}(C_{G}(s)). So, u is a regular element of C_{G}(s)^{0}.u. Moreover, according to [7] (lemma 1.14), u is quasi-semi-simple in CG̃(s)^{0}.u. We set H = CG̃(s)^{0}. Let W_{H} be the Weyl group of H, then the action of u on H induces an action of u on W_{H}. We set W_{H}^{u} = {w ∈ W_{H}|u.w = w}. Then, according to [14] (proposition II.2.20), as u is unipotent quasi-semi-simple in H.u, we have dim(B_{u}^{H}) = l_{u}(w^{H}) where l_{u}(w^{H}) is the length in W_{H}^{u} of the longest element of W^{H}. Moreover, by [14] (proposition II.10.2), as u is regular in H.u, we have |B_{u}^{H}| = 1. Thus, we have W_{H}^{u} = {1} and H is a torus.

So, we have proved that (G^{σ′})^{0} = ((G^{s})^{u})^{0} is a torus which leads to the fact that U^{σ′} = {1}. Thus, σ′ induces a surjective endomorphism of U with a finite number of fixed points. According to Lang-Steinberg’s theorem ([15] theorem 10), we can then conclude that f is surjective.

End of the proof of lemma 2.3.

Applying lemma 2.3, with σ′ = t.σ where t is any regular element of T^{σ}, we get that for all v ∈ U there exists u ∈ U such that v = u^{t.σ}u^{−1}, that is to say, such that tv.σ = ^{t}u(t.σ)(^{t}u)^{−1}. Hence, tv.σ is quasi-semi-simple (^{^{t}u}T and ^{^{t}u}B are stable under conjugation by tv.σ) and regular (C_{G}(tv.σ) = ^{^{t}u}C_{G}(t.σ)). So regular quasi-semi-simple elements of G^{1} are dense in T^{σ}U.σ.

By proposition 1.2, we have B = L_{σ}(T)T^{σ}U. Moreover, if b.σ ∈ T^{σ}U.σ is regular quasi-semi-simple, then, for all t ∈ T, we have L_{σ}(t)b.σ = ^{t^{−1}}b.σ, so L_{σ}(t)b.σ is also regular quasi-semi-simple. Hence, regular quasi-semi-simple elements of G^{1} are dense in B.σ.

Let B′ be a Borel subgroup of G, then there is x ∈ G such that B′ = ^{x}B and we have N_{G}(B′) ∩ G^{1} = ^{x}(B.σ). Hence, for any quasi-borel N_{G}(B′), quasi-semi-simple regular elements of G^{1} are dense in N_{G}(B′) ∩ G^{1}. Then, G̃ is the union of its quasi-borels ([15] proposition 7.2).

End of the proof of theorem 2.2.

Proposition 2.4.

(i) Let g.σ be a regular quasi-semi-simple element of G^{1}, then g.σ is conjugate under G to an element t.σ, where t is a regular element of G^{σ}.

(ii) Let g.σ be a regular quasi-semi-simple element of G^{1} then g.σ belongs to a unique quasitorus of G̃.

(iii) Let t.σ be a regular quasi-semi-simple element of T^{σ}.σ then B_{t.σ}^{G} = {^{w}B; w ∈ W^{σ}}.

Proof: (i) Let g.σ be a regular quasi-semi-simple element of G^{1}. Then, there is a pair (T′, B′) where T′ is a maximal torus included in a Borel subgroup B′ of G such that g.σ belongs to the quasi-torus NG̃(T′, B′). Let x be in G so that T′ = ^{x}T and B′ = ^{x}B, then NG̃(T′, B′) = ^{x}NG̃(T, B). So, ^{x^{−1}}(g.σ) is a regular element of NG̃(T, B) ∩ G.σ = T.σ. So, in order to prove (i), we can suppose that g.σ is a regular element of T.σ.

By proposition 1.2, every regular element of T.σ is conjugate to a regular element of T^{σ}.σ. Moreover, we have seen in 2.1 that t.σ ∈ T^{σ}.σ is regular in G^{1} if and only if t is regular in G^{σ}, whence the first assertion.

(ii) Let g.σ be a regular quasi-semi-simple element of G^{1}. According to (i), g.σ is then conjugate to t.σ ∈ T^{σ}.σ, where t is regular in G^{σ}. Moreover, t.σ belongs to a unique quasi-torus if and only if this is also true for g.σ. Let NG̃(T′, B′) be a quasi-torus containing t.σ. As t.σ is a Jordan decomposition, t and σ are also in NG̃(T′, B′). But, t is a regular semi-simple element of G^{σ}, so T^{σ} is the unique maximal torus of G^{σ} which contains t and B_{t}^{G^{σ}} = {^{w}B^{σ}, w ∈ W^{σ}} ([6] proposition 14.6).

So, T′^{σ} = T^{σ} and B′^{σ} = ^{w}B^{σ}. But, σ is quasi-central, hence B′ = ^{w}B and T′ = T, whence NG̃(T′, B′) = NG̃(T, B), which proves (ii).

Let us prove (iii). Let t.σ be a regular quasi-semi-simple element of T^{σ}.σ then t.σ is a Jordan decomposition. So, if t.σ is in the quasi-borel NG̃(B′) then both t and σ are in NG̃(B′). Hence, the fact that B_{t}^{G^{σ}} = {^{w}B^{σ}, w ∈ W^{σ}} is sufficient to prove (ii) because σ is quasi-central and because the map B′ ↦ (B′)^{σ} is a bijection between σ-stable Borel subgroups of G and Borel subgroups of G^{σ}.

End of the proof.

3. Borel subgroups stabilised by an element of G^{1}

The purpose of this section is to study the variety B_{g.σ}^{G} of the Borel subgroups stabilised by conjugation under g.σ ∈ G^{1}. We need the following varieties:

Z = {(g.σ, B_{1}, B_{2}) ∈ G^{1} × B^{G} × B^{G}|g.σ ∈ N_{1} ∩ N_{2}}
Z′ = {(g.σ, B_{1}, B_{2}) ∈ U(G^{1}) × B^{G} × B^{G}|g.σ ∈ N_{1} ∩ N_{2}},

where N_{i} is the normalizer of B_{i} in G̃.

Theorem 3.1.

(i) Let C be the conjugacy class of g.σ in G^{1}, then

2dim(B_{g.σ}^{G}) + dim(C) = dim(G) − rk(G^{σ})

(ii) Irreductible components of Z (resp. Z′) are parametrised by W^{σ}. Moreover, dim(Z) = dim(G) and dim(Z′) = dim(G) − rk(G^{σ}).

Proof:

(i) According to [14] (proposition II.10.15) we have dim(C_{G}(g.σ)) = rk(G^{σ}) + 2dim(B_{g.σ}), for all g.σ ∈ G^{1}, whence the result.

(ii) Let p : Z′ → B^{G} × B^{G} be the projection on the second and the third factors. Let O_{w} be the G-orbit of (B,^{w}B) in B × B. Then we set Z′_{w} = p^{−1}(O_{w}).

Let us show that Z′_{w} is empty when w is not σ-stable. Let w be in W such that Z′_{w} is not empty. This means that there is x ∈ G such that NG̃(^{x}B) ∩ NG̃(^{xw}B) ∩ U(G^{1}) is not empty, i.e. that E = NG̃(B) ∩ NG̃(^{w}B) ∩ U(G^{1}) is not empty. Let u.σ be in E then both u and w^{−1}uσ(w) belong to U, so that BwB ∩ Bσ(w)B is not empty and by the Bruhat decomposition we can conclude that σ(w) = w. So we have a partition of Z′ into subsets Z′_{w} such that w ∈ W^{σ}.

Let us show that when w is σ-stable then Z′_{w} is an irreducible variety. Let (B_{1}, B_{2}) be in O_{w}; then there is x ∈ G such that (B_{1}, B_{2}) = (^{xw_{0}}B,^{xw_{0}w}B), where we denote by w_{0} the longest element of W. Let V (B_{1}, B_{2}) be the subset of O_{w} defined by {(^{xuw_{0}}B,^{xuw_{0}w}B); u ∈ U}. Then (B_{1}, B_{2}) ∈ V (B_{1}, B_{2}) and V (B_{1}, B_{2}) is isomorphic to U and is dense open in O_{w}: {^{uw_{0}}B; u ∈ U} ≃ Uw_{0}B/B is indeed a dense open subset of B^{G} = G/B as, according to [5], Bw_{0}B is an open dense subset of G. Let L_{σ}(T) be the image of T by the map t ↦ t^{−1}σ(t).

Let ϕ : U × L_{σ}(U ∩ ^{w}U) → p^{−1}((B_{1}, B_{2})) be the morphism of varieties defined by (u, b) ↦ (^{xuw_{0}}(b.σ),^{xuw_{0}}B,^{xuw_{0}w}B). It is an isomorphism of varieties and the inverse morphism is defined by (g.σ, B′, B′′) ↦ (u, (xuw_{0})^{−1}(g.σ)(xuw_{0})) where u is the unique element of U such that B′ = ^{xuw_{0}}B. So Z′_{w} is a locally trivial fibration over O_{w} ≃ G/(B∩wBw^{−1}) with fiber isomorphic to L_{σ}(T)(U ∩^{w}U). Whence Z′_{w} is an irreducible variety and dim(Z′_{w}) = dim(G) − dim(T^{σ}). All the varieties Z′_{w} have the same dimension so dim(Z′) = dim(G) − dim(T^{σ}) and the varieties \overline{Z′_{w}} for w ∈ W^{σ} are the irreducible components of Z′.

In the case of Z, the proof is similar if we take, instead of Z′_{w}, the inverse image Z_{w} of O_{w} by the projection Z → B^{G} × B^{G}. The varieties Z_{w} for w ∈ W^{σ} are then locally trivial fibration over O_{w} with fiber isomorphic to B ∩ ^{w}B.

End of the proof.

Corollary 3.2.

(i) For all i > 0, we have

dim{g.σ ∈ G^{1}|dim(B_{g.σ}^{G}) ≥ i/2} < dim(G) − i.

(ii) For all i ≥ 0, we have

dim{g.σ ∈ U(G^{1})|dim(B_{g.σ}^{G}) ≥ i/2} ≤ dim(G) − i.

Proof: Let us prove (i). We suppose there exists i > 0 such that (i) is false. We consider V = {g.σ ∈ G^{1}|dim(B_{g.σ}^{G}) ≥ i/2}. We have dim(V) ≥ dim(G) − i and V is a closed subvariety of G^{1} which is stable by conjugation under G. We consider the variety Z of theorem 3.1. Let p_{1} : Z → G^{1} be the first projection. We have dim(p_{1}^{−1}(V)) ≥ dim(G) − i + 2 × i/2 = dim(G). But, according to 3.1, dim(Z) = dim(G), so p_{1}^{−1}(V) is a subvariety of Z of the same dimension as Z. Hence, p_{1}^{−1}(V) contains a irreducible component of Z, that is to say, there is w ∈ W^{σ} such that \overline{Z_{w}} ⊂ p_{1}^{−1}(V), where

Z_{w} = {(g.σ, xB, xwB) ∈ G^{1} × (G/B) × (G/wB)|g.σ ∈ NG̃(^{x}B) ∩ NG̃(^{xw}B)}.

Let t.σ be a regular element of T.σ, then (t.σ, B, wB) ∈ Z_{w} because T.σ ⊂ NG̃(B) ∩ NG̃(^{w}B). So, we have (t.σ, B, wB) ∈ p_{1}^{−1}(V), which means that t.σ ∈ V. But, as t.σ is regular then dim(B_{t.σ}^{G}) = 0 (proposition 2.1) which contradicts the definition of V.

The proof of the second assertion is similar. We suppose that there exists i ≥ 0 such that (ii) is false. We consider V′ = {g.σ ∈ G^{1}|dim(B_{g.σ}^{G}) ≥ i/2}. and p′_{1} : Z′ → U(G^{1}) be the projection on the first factor. Then, we have dim(p′_{1}^{−1}(V′)) > dim(G) − dim(T^{σ}) = dim(Z′) which is obviously false.

End of the proof.

4. Springer representations

Let us consider the following variety:

G̃^{1} = {(g.σ, xB) ∈ G^{1} × (G/B)|x^{−1}(g.σ)x ∈ B},

and π : G̃^{1} → G^{1} the projection on the first factor. In this section, we construct a representation of W^{σ} on the direct image π_{∗}\overline{ℚ}_{l} of the constant sheaf \overline{ℚ}_{l} on G̃^{1} (proposition 4.5), which we call the Springer representation of W^{σ} from G^{1}. Moreover, by proposition 4.7, it induces an action of W^{σ} on any cohomology group H^{i}(B_{g.σ}^{G}, \overline{ℚ}_{l}).

Proposition 4.1.

(i) G̃^{1} is a smooth variety.

(ii) π is a proper morphism.

(iii) π^{−1}(U(G^{1})) is a smooth variety and dim(π^{−1}(U(G^{1}))) = dim(G) − rk(G^{σ}).

Proof:

(i) The projection p_{2} : G̃^{1} → G/B is a locally trivial fibration with fiber isomorphic to B. Hence, G̃^{1} is a smooth variety.

(ii) G̃^{1} is a closed subset of G^{1} × (G/B). Hence, G̃^{1} → G^{1} × (G/B) is a proper morphism. Moreover, the projection on the first factor G^{1} ×(G/B) → G^{1} is proper (it is a universally closed morphism because G/B is complete and it is obviously separate). Hence, G̃^{1} → G^{1} is proper.

(iii) π^{−1}(U(G^{1})) is a locally trivial fibration over G/B with fiber isomorphic to L_{σ}(T)U via the second projection, whence the result.

End of the proof.

We denote by G_{reg}^{1} the set of regular quasi-semi-simple elements of G^{1}. Let us consider the following variety:

G̃_{reg}^{1} = {(g.σ, xT) ∈ G^{1} × (G/T)|x^{−1}(g.σ)x ∈ T_{reg}^{1}}.

Proposition 4.2. The map ϕ : G̃_{reg}^{1} → π^{−1}(G_{reg}^{1}) defined by (g.σ, xT) ↦ (g.σ, xB) is an isomorphism of varieties.

Proof: Let us first show that ϕ is injective. Let (g.σ, xT) and (g.σ, x′T) be in G̃_{reg}^{1} such that xB = x′B. Let us show that xT = x′T. The regular quasi-semi-simple element g.σ is in both quasi-tori NG̃(^{x}T,^{x}B) and NG̃(^{x′}T,^{x′}B). So, NG̃(^{x}T,^{x}B) = NG̃(^{x′}T,^{x′}B) (proposition 2.4), which means x^{−1}x′ ∈ N_{G}(T). But, we also have xB = x′B, so x^{−1}x′ ∈ T, which prove the injectivity of ϕ.

Let us now show that ϕ is surjective. Let (g.σ, xB) be in π^{−1}(G_{reg}^{1}), let us show that there exists z ∈ xB such that (g.σ, zT) ∈ G̃_{reg}^{1}. By hypothesis, g.σ is a quasi-semi-simple element of NG̃(^{x}B) which is regular in G^{1}. By proposition 2.4, there are t ∈ T^{σ} regular in G^{σ} and y ∈ G such that ^{y}(g.σ) = t.σ. Thus, t ∈ T^{σ} is regular in G^{σ} and ^{y^{−1}x}B ∈ B_{t.σ}^{G}. So, there exists w ∈ W^{σ} such that ^{y^{−1}x}B = ^{w}B (proposition 2.4, (iii)). Let us set z = yw̃ where w̃ is a representative of w in N_{G}(T)^{σ}. Then z^{−1}g.σz = w^{−1}y^{−1}g.σyw ∈ T.σ, hence, (g.σ, zT) ∈ G̃_{reg}^{1}.

Let ψ be the inverse map of ϕ. Let us show that ψ is a morphism of varieties. Let (g.σ, xB) be in G̃_{reg}^{1} and let (g.σ, zT) be the image of (g.σ, xB) by ψ. Then, ψ((^{x^{−1}}(g.σ), B)) = (^{x^{−1}}(g.σ), x^{−1}zT), that is to say that ^{x^{−1}z}T is the unique torus of B which is normalised by x^{−1}(g.σ). Thus, the problem is to prove that the map f, which associates to each element of B.σ ∩ G_{reg}^{1} the unique torus of B it normalises, is a morphism of varieties.

Let b.σ be a regular quasi-semi-simple element of B.σ. Then, there is a unique pair (u, t) ∈ T × U such that ut.σ = b.σ. Let us show that t.σ is also regular. Applying lemma 2.3, the map U → U defined by v ↦ v^{−1}^{b.σ}v is surjective. So, there is v ∈ U such that v^{−1}^{b.σ}v = u^{−1} and we have t.σ = u^{−1}b.σ = v^{−1}^{b.σ}vb.σ = v^{−1}b.σv. Hence, t.σ is regular. We then need the following lemma:

Lemma 4.3. Let t.σ a regular element of T.σ, then the map h_{t} : U → U defined by u ↦ u^{−1}^{t.σ}u is an isomorphism of varieties.

Proof of lemma 4.3: According to lemma 2.3, h_{t} is surjective. Moreover, applying proposition 1.2, we can write t = t_{1}t_{2}^{−1}σ(t_{2}) where t_{1} ∈ T^{σ} and t_{2} ∈ T. So, we have CG̃(t.σ) = t_{2}^{−1}CG̃(t_{1}.σ) = T^{σ}. Then, U^{t.σ} = 1, which shows that h_{t} is injective. As h_{t} is a bijective morphism U → U and U is an irreducible variety, we only have to check that h_{t} is separable and that U is normal. As U is smooth, U is normal. Let us show that h_{t} is separable. According to [9] (page 44), as h_{t} is a dominant morphism of irreducible varieties, it is sufficient to check that (dh_{t})_{0} : T (U)_{0} → T (U)_{0} is surjective. The tangent space T (U)_{0} is isomorphic to k^{|ϕ^{+}|}. Let u = ∏_{α∈Φ^{+}} x_{α}(k_{α}) be an element of U. For all α ∈ Φ^{+}, there is c_{α} ∈ K^{×} such that σ(x_{α}(k_{α})) = x_{σ(α)}(c_{α}k_{α}) for all k_{α} ∈ K. Then, h_{t}(u) = ∏_{α∈Φ^{+}} x_{α}(k_{α} − c_{σ^{−1}(α)}α(t)k_{σ^{−1}(α)})u_{∗}, where u_{∗} is a product of commutators. Hence, (dh_{t})_{0} is the linear map defined by (k_{α})_{α∈Φ^{+}} ↦ (k_{α} − c_{σ^{−1}(α)}α(t)k_{σ^{−1}(α)})_{α∈Φ^{+}}. Let us reorganize the coefficients k_{α} in σ-orbits. The matrix corresponding to h_{t} is then decomposed into blocks which look like:

A(α) = ( \begin{matrix}1 & & & ... & −c_{σ^{−1}(α)}α(t_{1}) \\ −c_{α}α(t_{1}) & 1 & & ... \\ & −c_{σ(α)}α(t_{1}) & 1 & ... \\ & & & ... \\ & & & −c_{σ^{l−2}(α)}α(t_{1}) & 1 \end{matrix} ) ,

where l is the cardinal number of the σ-orbit of α, and t = t_{1}L_{σ}(t_{2}), with t_{1} ∈ T^{σ} regular in G^{σ}. Then, det(A(α)) = 1 − c_{α}...c_{σ^{−1}(α)}α(t_{1})^{l}, and, as σ is quasi-central, we have c_{α}...c_{σ^{−1}(α)} = 1, according to [7] (theorem 1.15). However, t_{1} is a regular semi-simple element of G^{σ}, hence, α(t_{1}) ≠ 1 ([6] proposition 14.16). Moreover, t_{1} is semi-simple, σ is unipotent and l divides the order of σ. If r is the order of t_{1} then l and r are relatively prime, and according to Bezout, there are two integers a and b such that al + br = 1. Thus, we have α(t_{1})^{al} = α(t_{1})^{1−br} = α(t_{1}) ≠ 1, so, α(t_{1})^{l} ≠ 1. We have then proved that the determinant of A(α) is not equal to zero. Then, the jacobian at 0 of h_{t} is a product of matrices with non zero determinant, so that (dh_{t})_{0} is surjective.

End of the proof of lemma 4.3.

Let π_{reg} : π^{−1}(G_{reg}^{1}) → G_{reg}^{1} be the restriction of π to π^{−1}(G_{reg}^{1}). The group W^{σ} of σ-stable elements in the Weyl group W of G acts on G̃_{reg}^{1} on the right by w : (g.σ, xT) ↦ (g.σ, xwT) and then acts on π^{−1}(G_{reg}^{1}) via the isomorphism of proposition 4.2.

Proposition 4.4. π_{reg} : π^{−1}(G_{reg}^{1}) → G_{reg}^{1} is an unramified covering with Galois group W^{σ}.

Proof: The map π_{reg} : π^{−1}(G_{reg}^{1}) → G_{reg}^{1} is a finite morphism of varieties whose fibers have the same cardinality as W^{σ}. Indeed, if g.σ ∈ G_{reg}^{1}, then π^{−1}(g.σ) is conjugated to {(t_{0}.σ,^{w}B); w ∈ W^{σ}}, where t_{0} is a regular element of T^{σ}. Thus, according to [11] (proposition 2.9), π_{reg} : π^{−1}(G_{reg}^{1}) → G_{reg}^{1} is a flat morphism of varieties. As π_{reg} is also surjective, it is faithfully flat.

The group W^{σ} acts on π^{−1}(G_{reg}^{1}) via the isomorphism G̃_{reg}^{1} → π^{−1}(G_{reg}^{1}) of proposition 4.2 and the action of W^{σ} on G̃_{reg}^{1}. This is an action on the right over G_{reg}^{1} (indeed π_{reg} is W^{σ}- invariant with respect to the trivial action of W^{σ} on G_{reg}^{1}). Moreover, the map ϕ which sends (w, (g.σ, xT)) ∈ W^{σ} × G̃_{reg}^{1} to ((g.σ, xT), (g.σ, xwT)) is an isomorphism from W^{σ} × G̃_{reg}^{1} to G̃_{reg}^{1} ×_{G_{reg}^{1}} G̃_{reg}^{1}. Indeed, for all g.σ ∈ G_{reg}^{1}, there exists x ∈ G such that x^{−1}g.σx ∈ T_{reg}^{1} and {yT |y^{−1}g.σy ∈ T_{reg}^{1}} = {xwT |w ∈ W^{σ}}. Moreover, W^{σ} acts faithfully and transitively on this set.

End of the proof.

Let us denote by L = (π_{reg})_{∗}\overline{ℚ}_{l} the direct image of the constant sheaf \overline{ℚ}_{l} on π^{−1}(G_{reg}^{1}). We can deduce from the previous proposition that W^{σ} acts on the local system L = (π_{reg})_{∗}\overline{ℚ}_{l} on G_{reg}^{1}.

By theorem 2.2, the set G_{reg}^{1} of regular quasi-semi-simple elements is dense in G^{1}. Let us now consider the intersection cohomology complex IC(G^{1}, L) on \overline{G^{1}}_{reg} = G^{1} associated with L.

Definition 4.1. Let X be an algebraic variety over K. Let Y ⊂ X be a locally closed smooth irreducible subvariety of dimension d of X. We can associate to each local system L on Y an intersection cohomology complex IC(\overline{Y} , L)[d] and S = IC(\overline{Y} , L)[d] is then a perverse sheaf on Y which is characterized by the following properties:

(i) H^{i}S = 0 if i < −d,

(ii) H^{−d}S|_{Y} = L,

(iii) dim SuppH^{i}S < −i if i > −d,

(iv) dim SuppH^{i}DS < −i if i > −d.

As IC is a faithful functor from the category of local systems on G_{reg}^{1} to the category of perverse sheaves, the action of W^{σ} on L induces an action of W^{σ} on IC(G^{1}, L)[dim(G)]. Then, by the following proposition we get an action of W^{σ} on π_{∗}(\overline{ℚ}_{l}) which we call the Springer representation of W^{σ} from G^{1}.

Proposition 4.5. Let us set n = dim(G), then,

IC(G^{1}, L)[n] ≃ π_{∗}(\overline{ℚ}_{l})[n] in MG^{1},

where, π_{∗}(\overline{ℚ}_{l}) is the direct image of the constant sheaf \overline{ℚ}_{l} on G̃^{1} by π.

In order to prove the above proposition, we will need the following definition and the following proposition:

Definition 4.2. Let ϕ : Z → X be a proper morphism between two varieties of same dimension n. If X is smooth, ϕ is said semi-small, if for all i, we have

dim{a ∈ X|dim(ϕ^{−1}(a)) ≥ i/2} ≤ dim(X) − i,

The morphism ϕ is said small if the last inequality is strict.

Proposition 4.6. Let ϕ : Z → X be a morphism between two varieties of same dimension n. We suppose that X and Z are smooth. Let us consider the constant sheaf \overline{ℚ}_{l} on Z, then, if ϕ is semi-small, ϕ_{∗}\overline{ℚ}_{l}[n] is a perverse sheaf on X, if ϕ is small, M = ϕ_{∗}\overline{ℚ}_{l}[n] verify the following properties:

(i) H^{i}M = 0 if i < −n,

(ii) dim SuppH^{i}M < −i if i > −n,

(iii) dim SuppH^{i}DM < −i if i > −n.

Proof: As the functor ϕ_{∗} is left exact and preserves injectives, the sheaves H^{i}(ϕ_{∗}(\overline{ℚ}_{l})) and R^{i}ϕ_{∗}(\overline{ℚ}_{l}) are isomorphic.

Let us first suppose that ϕ is small. Let us show that M verify (i). For all x ∈ X, the fiber R^{i}ϕ_{∗}(\overline{ℚ}_{l})_{x} at x of R^{i}ϕ_{∗}(\overline{ℚ}_{l}) is isomorphic to H^{i}(ϕ^{−1}(x), \overline{ℚ}_{l}), so we have R^{i}ϕ_{∗}(\overline{ℚ}_{l})_{x}[n] ≃ H^{i+n}(ϕ^{−1}(x), \overline{ℚ}_{l}), for all x ∈ X. Now, we have H^{i}(ϕ^{−1}(x), \overline{ℚ}_{l}) = 0 for all i < 0 and all x ∈ X. So, H^{i+n}(ϕ_{∗}(\overline{ℚ}_{l})_{x}) = 0 for all i < −n and for all x ∈ X, which leads to H^{i+n}(ϕ_{∗}(\overline{ℚ}_{l})) = 0 for all i < −n.

Let us now check (ii) and (iii). As Z is a smooth variety of dimension n, we have D(\overline{ℚ}_{l}[n]) = \overline{ℚ}_{l}[n], where D is the Verdier duality. Moreover, as ϕ is proper, D commute with ϕ_{∗} ([4] théoréme 10.11). Thus, we have D(M) = M. So, we just need to check (ii). Let us recall that R^{i}ϕ_{∗}(\overline{ℚ}_{l})_{x} ≃ H^{i}(ϕ^{−1}(x), \overline{ℚ}_{l}). If H^{i}(ϕ^{−1}(x), \overline{ℚ}_{l}) ≠ 0, then, i ≤ 2dim(ϕ^{−1}(x)). So, SuppH^{i}(ϕ_{∗}(\overline{ℚ}_{l})[n]) is a subset of {x ∈ X|i ≤ 2dim(ϕ^{−1}(x))}. Thus, (ii) and (iii) are direct consequences of the fact that ϕ is small.

The same proof shows that if ϕ is semi-small, then ϕ_{∗}\overline{ℚ}_{l}[n] is a perverse sheaf.

End of the proof.

Proof of proposition 4.5: Let us check that S′ = π_{∗}(\overline{ℚ}_{l})[n] satisfies properties (i) to (iv) which characterize S = IC(G^{1}, L)[dim(G)]. The map π : G̃^{1} → G^{1} is a proper morphism between two varieties of the same dimension and G̃^{1} is smooth. Moreover, for all x.σ ∈ G^{1}, we have π^{−1}(x.σ) ≃ B_{x.σ}^{G}, so, by corollary 3.2, π is a small morphism. Moreover, G̃^{1} is a smooth variety (proposition 4.1). The hypotheses of proposition 4.6 are satisfied, which proves (i), (iii) and (iv).

Let us show (ii). Let us consider the following cartesian diagram:

\begin{matrix}π^{−1}(G_{reg}^{1}) & ⟶^{π_{reg}} & G_{reg}^{1} \\ ↓i & & ↓i′ \\ G̃^{1} & ⟶^{π} & G^{1} \end{matrix} ,

where i and i′ are inclusions. According to the proper base change theorem applied to the proper morphism π : G̃^{1} → G^{1}, we have i′^{∗}R^{0}π_{∗}(\overline{ℚ}_{l}) ≃ π_{reg∗}(\overline{ℚ}_{l}). Moreover, H^{−n}K|_{G_{reg}^{1}} is isomorphic to i′^{∗}R^{0}π_{∗}(\overline{ℚ}_{l}) = π_{reg∗}(\overline{ℚ}_{l}) = L.

End of the proof of proposition 4.5.

Proposition 4.7. For all g.σ ∈ G^{1}, the group W^{σ} acts on H^{i}(B_{g.σ}^{G}, \overline{ℚ}_{l}).

Proof: According to the previous proposition, W^{σ} acts on π_{∗}(\overline{ℚ_{l}})[n] and then on any stalk H^{i}(π_{∗}(\overline{ℚ}_{l})[n])_{g.σ}, which is isomorphic to H^{i}(B_{g.σ}, \overline{ℚ}_{l}).

End of the proof.

5. The action of W^{σ} on H^{∗}(B_{σ}, \overline{ℚ}_{l})

From the previous section, we have an action of W^{σ} on H^{∗}(B_{σ}^{G}, \overline{ℚ}_{l}), where B_{σ}^{G} is the variety of σ-stable Borel subgroups of G. We will call this representation the Springer representation from G^{1}.

On another part, we can apply to G^{σ} the construction of the Springer representation for a connected reductive group described in [13] (section 4), we then have an action of the Weyl group W (G^{σ}) of G^{σ} on H^{∗}(B^{G^{σ}}, \overline{ℚ}_{l}), where B^{G^{σ}} is the variety of Borel subgroups of G^{σ}. We will call this representation the Springer representation from G^{σ}.

As σ is unipotent quasi-semi-simple, we have W (G^{σ}) ≃ W^{σ} and B_{σ}^{G} ≃ B^{G^{σ}}. Till the end of this article, we will identify the groups H^{∗}(B_{σ}^{G}, \overline{ℚ}_{l}) and H^{∗}(B^{G^{σ}}, \overline{ℚ}_{l}) and the Weyl groups W (G^{σ}) and W^{σ}. Hence, we get two Springer representations of W^{σ} on H^{∗}(B_{σ}^{G}, \overline{ℚ}_{l}), one from G^{1} and one from G^{σ}. We show in proposition 5.1 below that these representations coincide.

We set

G̃^{σ} = {(g, B′) ∈ G^{σ} × B^{σ}|g ∈ B′},

and we denote by π′ : G̃^{σ} → G^{σ} the projection on the first factor. Let us consider the set G_{reg}^{σ} of regular semi-simple elements of G^{σ}. Let G̃_{reg}^{σ}, π′_{0} and ĩ′ be defined by the following cartesian diagram:

\begin{matrix}G̃_{reg}^{σ} & ⟶^{ĩ′} & G̃^{σ} \\ ↓π′_{0} & & ↓π′ \\ G_{reg}^{σ} & ⟶^{i′} & (G^{σ})^{0} \end{matrix} ,

Let L′ be the local system (π′_{0})_{∗}\overline{ℚ}_{l} on G_{reg}^{σ}. As in the previous section, we have

IC(G^{σ}, L′)[n′] ≃ π′_{∗}(\overline{ℚ}_{l})[n′] in M(G^{σ}),

where n′ = dim(G^{σ}). This induces the Springer representation of W^{σ} from G^{σ} on H^{∗}(B_{σ}^{G}, \overline{ℚ}_{l}).

Proposition 5.1. The Springer representation of W^{σ} on H^{∗}(B_{σ}, \overline{ℚ}_{l}) from G^{1} and that from G^{σ} coincide.

Proof: Let us consider the following diagram:

\begin{matrix}G̃^{σ} & ⟶^{j̃} & G̃^{1} \\ ↓π′ & & ↓π \\ G^{σ} & ⟶^{j} & G^{1} \end{matrix} ,

where j is defined by g ↦ g.σ and j̃ is defined by (g, B̃) ↦ (g.σ, B) where B is the only σ-stable Borel subgroup of G such that B^{σ} = B̃. It is not a cartesian diagram but we have G̃^{σ} ≃ (G̃^{1})^{σ}. As j̃_{∗} and j̃^{∗} are adjoint, there exists an isomorphism α : Hom_{F(G̃^{σ})}(\overline{ℚ}_{l}, \overline{ℚ}_{l}) ≃ Hom_{F(G̃^{1})}(\overline{ℚ}_{l}, j̃_{∗}\overline{ℚ}_{l}) (where F(H) is the category of sheaves on H). Let us consider the adjunction morphism α(Id_{\overline{ℚ}_{l}}) : \overline{ℚ}_{l} → j̃_{∗}\overline{ℚ}_{l}. By functoriality, it induces a morphism of sheaves ϕ : π_{∗}\overline{ℚ}_{l} → π_{∗}j̃_{∗}\overline{ℚ}_{l}, and as π ∘ j̃ = j ∘ π′, ϕ is a morphism of sheaves π_{∗}\overline{ℚ}_{l} → j_{∗}π′_{∗}\overline{ℚ}_{l}.

Let us consider the following cartesian diagram:

\begin{matrix}G̃_{reg}^{1} & ⟶^{ĩ} & G̃^{1} \\ ↓π^{reg} & ↓π \\ G_{reg}^{1} & ⟶^{i} & G^{1} \end{matrix} ,

We still denote by L the local system π_{reg∗}\overline{ℚ}_{l}. According to proposition 4.5, π_{∗}\overline{ℚ}_{l}[n] is isomorphic to i_{!∗}L[n] in M(G^{1}). In the following lemma, we prove a similar result for j_{∗}π′_{∗}\overline{ℚ}_{l}[n].

Lemma 5.2. j_{∗}π′_{∗}\overline{ℚ}_{l}[n] ≃ i_{!∗}j_{reg∗}L′[n] in M(G^{1}), where L′ = π′_{0∗}\overline{ℚ}_{l}.

Proof of lemma 5.2: Let us consider the following diagram:

\begin{matrix}G_{reg}^{σ} & ⟶^{i′} & G^{σ} \\ ↓j_{reg} & & ↓j \\ G_{reg}^{1} & ⟶^{i} & G^{1} \end{matrix} .

The morphism j_{reg} is the open immersion defined by g ↦ g.σ, indeed, according to 2.4, g.σ ∈ (G^{σ})^{0} is quasi-semi-simple regular if and only if g is semi-simple regular in (G^{σ})^{0}. We need the following lemma:

Lemma 5.3. The functors T : D((G^{σ})^{0}) → D(G^{1}) and T_{reg} : D(G_{reg}^{σ}) → D(G_{reg}^{1}), defined by T (S) := j_{∗}S[n − n′] for all S ∈ D((G^{σ})^{0}) and T_{reg}(S) := j_{reg∗}S[n − n′] for all S ∈ D(G_{reg}^{σ}), are t-exact.

Proof of lemma 5.3: As j and j_{reg} are proper morphisms, j_{∗} and j_{reg∗} are exact functors and they commute with the Verdier duality. Thus, it is sufficient to check that for all perverse sheaf S on (G^{σ})^{0} (resp. G_{reg}^{σ}), we have dim(Supp(H^{i}(T (S)))) ≤ −i (resp. dim(Supp(H^{i}(T_{reg}(S)))) ≤ −i) for all i ≥ −n. We check below the case of the functor T, the case of T_{reg} is similar.

For all i ≥ −n,

dim(Supp(H^{i}(T (S)))) = dim(Supp(H^{i}(j_{∗}S[n − n′])))
= dim(Supp(H^{i+n−n′}(j_{∗}S)))
= dim(Supp(H^{i+n−n′}(S)))
≤ −i − (n − n′)

(because i + n − n′ ≥ −n′)

≤ −i

End of the proof of lemma 5.3.

Thus, by the previous lemma, j_{∗}π′_{∗}\overline{ℚ}_{l}[n] = T (π′_{∗}\overline{ℚ}_{l}[n′]) is a perverse sheaf on G^{1} and T induces an exact functor from M((G^{σ})^{0}) to M(G^{1}) denoted by ^{p}T and defined by ^{p}H^{0} ∘ T ∘ ϵ, where pH^{0} = τ_{≥0}τ_{≤0} and ϵ is the inclusion of the category of perverse sheaves into the derived category ([1] proposition 1.3.17).

j_{∗}π′_{∗}\overline{ℚ}_{l}[n] = ^{p}T (i′_{!∗}L′[n′])
= ^{p}T (Im(^{p}i′_{!}L′[n′] → ^{p}i′_{∗}L′[n′]))
= Im(^{p}T^{p}i′_{!}L′[n′] → ^{p}T^{p}i′_{∗}L′[n′])

Thus, in order to prove lemma 5.2, it is sufficient to show that ^{p}T^{p}i′_{!}L′[n′] = ^{p}i_{!}(j_{reg∗}L′[n]) and that ^{p}T^{p}i′_{∗}L′[n′] = ^{p}i_{∗}(j_{reg∗}L′[n]).

Let us show the first equality, the proof of the second one is similar.

pT^{p}i′_{!}L′[n′] = ^{p}T ∘ ^{p}i′_{!}(L′[n′])
= ^{p}(T ∘ i′_{!})(L′[n′])

([1] proposition 1.3.17)

= ^{p}(i_{!} ∘ T_{reg})(L′[n′])

(because j ∘ i′ = i ∘ j_{reg})

= ^{p}i_{!}^{p}T_{reg}L′[n′]

([1] proposition 1.3.17)

= ^{p}i_{!}(j_{reg∗}L′[n])

End of the proof of lemma 5.2.

So, the morphism ϕ[n] : π_{∗}\overline{ℚ}_{l}[n] → π_{∗}j̃_{∗}\overline{ℚ}_{l}[n] is a morphism between two extensions of perverse sheaves on G_{reg}^{1}. But the functor i_{!∗} induces an equivalence of categories from M(G_{reg}^{1}) to {i_{!∗}A′; A′ ∈ M(G_{reg}^{1})} whose reciprocal functor is i^{∗} ([1] proposition 2.1.9). Thus, i_{!∗} is a fully faithful functor (i_{!∗} : Hom(A′, B′) → Hom(i_{!∗}A′, i_{!∗}B′) is an isomorphism with reciprocal morphism i^{∗}) and in order to show that ϕ[n] is W^{σ}-equivariant, it is sufficient to show that this is also true for its restriction to G_{reg}^{1}. Thus, we have to show that i^{∗}π_{∗}α(Id_{\overline{ℚ}_{l}}) : i^{∗}π_{∗}\overline{ℚ}_{l} → i^{∗}π_{∗}j̃_{∗}\overline{ℚ}_{l} is W^{σ}-equivariant, that is to say, by the proper base change theorem applied to proper morphism π, that the morphism π_{reg∗}ĩ^{∗}α(Id_{\overline{ℚ}_{l}}) : π_{reg∗}\overline{ℚ}_{l} → π_{reg∗}ĩ^{∗}j̃_{∗}\overline{ℚ}_{l} is W^{σ}-equivariant. Nevertheless, π_{reg} is a galoisian covering with Galois group W^{σ} and the actions of W^{σ} on π_{reg∗}\overline{ℚ}_{l} and on π_{reg∗}ĩ^{∗}j̃_{∗}\overline{ℚ}_{l} come from those on \overline{ℚ}_{l} and on ĩ^{∗}j̃_{∗}\overline{ℚ}_{l}. So we just have to check that ĩ^{∗}α(Id_{\overline{ℚ}_{l}}) is W^{σ}-equivariant. Let us consider the following cartesian diagram:

G̃_{reg}^{σ} ⟶^{ĩ′} G̃^{σ}
↓j̃_{reg}  ↓j̃
G̃_{reg}^{1} ⟶^{ĩ} G̃^{1}

The functors j̃_{reg∗} and j̃_{reg}^{∗} are adjoint; let α_{reg} be the isomorphism Hom_{F(G̃_{reg}^{σ})}(\overline{ℚ}_{l}, \overline{ℚ}_{l}) ≃

Hom_{F(G̃_{reg}^{1})}(\overline{ℚ}_{l}, j̃_{reg∗}\overline{ℚ}_{l}). Let us show that ĩ^{∗}α(Id_{\overline{ℚ}_{l}}) : \overline{ℚ}_{l} → ĩ^{∗}j̃_{∗}\overline{ℚ}_{l} is the adjunction morphism

α_{reg}(Id_{\overline{ℚ}_{l}}) : \overline{ℚ}_{l} → j̃_{reg∗}\overline{ℚ}_{l}. We will then be able to conclude that ĩ^{∗}α(Id_{\overline{ℚ}_{l}}) is W^{σ}-equivariant: j̃_{reg} is an equivariant morphism from G̃_{reg}^{σ} to G̃_{reg}^{1} with respect to the action of W^{σ} on G̃_{reg}^{σ} (defined by w : (g, xT) ↦ (g, xwT)) and on G̃_{reg}^{1} (defined by w : (g.σ, xT) ↦ (g.σ, xwT)).

According to the proper base change theorem applied to the proper morphism j̃, the sheaves ĩ^{∗}j̃_{∗}\overline{ℚ}_{l} and j̃_{reg∗}\overline{ℚ}_{l} are isomorphic. For any etale morphism U → G̃_{reg}^{1}, we have α_{reg}(Id_{\overline{ℚ}_{l}})(U) : \overline{ℚ}_{l}(U) → \overline{ℚ}_{l}(U ×G̃_{reg}^{1} G̃_{reg}^{σ}). Moreover, as ĩ : G̃_{reg}^{1} → G̃^{1} is an open immersion, it is an etale

morphism and ĩ^{∗} is the restriction functor, so for any etale morphism U → G̃_{reg}^{1}, ĩ^{∗}α(Id_{\overline{ℚ}_{l}})(U) = α(Id_{\overline{ℚ}_{l}})(U) : \overline{ℚ}_{l}(U) → j̃_{∗}\overline{ℚ}_{l}(U), where j̃_{∗}\overline{ℚ}_{l}(U) = \overline{ℚ}_{l}(U ×G̃^{1} G̃^{σ}) and U ×G̃^{1} G̃^{σ} = U ×G̃_{reg}^{1} G̃_{reg}^{σ}.

Hence, we have ĩ^{∗}α(Id_{\overline{ℚ}_{l}}) = α_{reg}(Id_{\overline{ℚ}_{l}}).

The proposition is then proved if we prove that the stalk at σ of the W^{σ}-equivariant morphism ϕ is an isomorphism ϕ_{σ} : (π_{∗}\overline{ℚ}_{l})_{σ} → (π_{∗}j̃_{∗}\overline{ℚ}_{l})_{σ}.

Let V be the open variety G̃^{1} − G̃^{σ} and ĩ_{V} the open immersion V → G̃^{1}. Let us recall that j̃ is the closed immersion G̃^{σ} → G̃^{1}. We have the following exact sequence:

0 → ĩ_{V!}ĩ_{V}^{∗} \overline{ℚ}_{l} → \overline{ℚ}_{l} → j̃_{∗}j̃^{∗}\overline{ℚ}_{l} → 0

The functor π_{∗} is left-exact. Let us consider the right-derived functors R^{k}π_{∗} of π_{∗}, we then obtain the following long exact sequence:

... → R^{k}π_{∗}ĩ_{V!}ĩ_{V}^{∗} \overline{ℚ}_{l} → R^{k}π_{∗}\overline{ℚ}_{l} → R^{k}π_{∗}j̃_{∗}j̃^{∗}\overline{ℚ}_{l} → R^{k+1}π_{∗}ĩ_{V!}ĩ_{V}^{∗} \overline{ℚ}_{l} → ...

Let us apply the exact functor F ↦ F_{σ}, we then have the following long exact sequence:

... → (R^{k}π_{∗}ĩ_{V!}ĩ_{V}^{∗} \overline{ℚ}_{l})_{σ} → (R^{k}π_{∗}\overline{ℚ}_{l})_{σ} → (R^{k}π_{∗}j̃_{∗}j̃^{∗}\overline{ℚ}_{l})_{σ} → (R^{k+1}π_{∗}ĩ_{V!}ĩ_{V}^{∗} \overline{ℚ}_{l})_{σ} → ...

So, the fact that (R^{0}π_{∗}\overline{ℚ}_{l})_{σ} = (π_{∗}\overline{ℚ}_{l})_{σ} is isomorphic to (R^{0}π_{∗}j̃_{∗}j̃^{∗}\overline{ℚ}_{l})_{σ} = (π_{∗}j̃_{∗}\overline{ℚ}_{l})_{σ} will be proved if we prove that (R^{0}π_{∗}ĩ_{V!}ĩ_{V}^{∗} \overline{ℚ}_{l})_{σ} = 0 and (R^{1}π_{∗}ĩ_{V!}ĩ_{V}^{∗} \overline{ℚ}_{l})_{σ} = 0, that is to say that H_{c}^{0}(V ∩ π^{−1}(σ), \overline{ℚ}_{l}) = 0 and H_{c}^{1}(V ∩ π^{−1}(σ), \overline{ℚ}_{l}) = 0. But this is clear as π^{−1}(σ) ⊂ G̃^{σ} and V = G̃^{1} − G̃^{σ}.

End of the proof of proposition 5.1.

Corollary 5.4. The Springer representation of W^{σ} on H^{∗}(B_{σ}^{G}, \overline{ℚ}_{l}) from G^{1} is the regular representation of W^{σ}.

Proof: According to [13] (corollary 5.5), the Springer representation of W^{σ} on H^{∗}(B^{G^{σ}}, \overline{ℚ}_{l}) from G^{σ} is the regular representation of W^{σ}.

End of the proof.

6. Springer correspondence theorem

In this section, we define a Springer correspondence for G^{1} between the irreducible representations of W^{σ} and some pairs (C, E) where C is a unipotent G-conjugacy class of G^{1} and E a G-equivariant irreducible local system on C. Let us denote by N the set of all such pairs (C, E).

Proposition 6.1. ([14] proposition I.1.6)

The variety U(G^{1}) of unipotent elements of G^{1} is an irreducible closed subvariety of G^{1} with codimension equal to the rank of G^{σ}.

We will need the following theorem in order to prove the main result of this section which is theorem 6.3:

Theorem 6.2. Let ϕ : Z → X be a semi-small morphism between two smooth varieties which have the same dimension equal to n. We suppose there is an action of a group H on Z and on X and that ϕ is H-equivariant. Then,

(i) the perverse sheaf ϕ_{∗}\overline{ℚ}_{l}[dim(X)] is semi-simple in M(X),

(ii) ϕ_{∗}\overline{ℚ}_{l}[dim(X)] is decomposed as:

K ≃ ⨁_{(C,E)} IC(\overline{C}, E)[dim(C)] ⊗ V_{(C,E)},

where C runs over the set of H-stable smooth irreducible subvarieties of X and E the set of H-equivariant local systems over C and where V_{(C,E)} is a \overline{ℚ}_{l}-vector space whose dimension is equal to the multiplicity of the simple object corresponding to (C, E).

Proof of theorem 6.2:

(i) As Z is smooth, \overline{ℚ}_{l}[n] is a simple perverse sheaf with finite monodromy. Moreover ϕ is semi-small, so, according to proposition 4.6, ϕ_{∗}\overline{ℚ}_{l}[n] is a perverse sheaf. We can then apply the decomposition theorem ([1], theorem 6.2.5). The result is that ϕ_{∗}\overline{ℚ}_{l}[n] is a semi-simple object of M(X).

(ii) By (i), ϕ_{∗}\overline{ℚ}_{l}[n] is a direct sum of simple objects of M(X). But, according to [1] (4.3.1), a simple object of M(X) is of the form IC(Y, E)[dim(Y)] where Y is a locally closed smooth irreducible subvariety of X and E an irreducible local system on Y.

Moreover, as \overline{ℚ}_{l}[n] is a H-equivariant perverse sheaf and as ϕ is a H-equivariant morphism, ϕ_{∗}\overline{ℚ}_{l}[n] is also H-equivariant ([12] 1.4.1). Hence, if IC(Ȳ , E)[dim(Y)] is a summand of ϕ_{∗}\overline{ℚ}_{l}[n], then IC(Ȳ , E)[dim(Y)] is G-equivariant, which means that Y is a H-stable subvariety and E a H-equivariant local system.

Thus, ϕ_{∗}\overline{ℚ}_{l}[n] is a direct sum of perverse sheaves of the form IC(Ȳ , E)[dim(Y)], where Y is a H-stable irreducible smooth subvariety of X and E is an irreducible H-equivariant local system on Y.

End of the proof of theorem 6.2.

Theorem 6.3. Let n be the dimension of G and r′ the rank of G^{σ}.

(i) π_{∗}\overline{ℚ}_{l}[n − r′]|_{U(G^{1})} is a semi-simple object of M(U(G^{1})) and decomposes as:

π_{∗}\overline{ℚ}_{l}[n − r′]|_{U(G^{1})} ≃ ⨁_{(C,E)} V_{C,E} ⊗ IC(C̄, E)[dim(C)]

where (C, E) runs over the set N and V_{C,E} is a \overline{ℚ}_{l}-vector space whose dimension is equal to the multiplicity of the simple object corresponding to (C, E).

(ii) The action of W^{σ} on π_{∗}\overline{ℚ}_{l}|_{U(G^{1})} induces an action of W^{σ} on all V_{C,E}, moreover, any W^{σ}-module V_{C,E} is irreducible if V_{C,E} ≠ 0 and the map (C, E) ↦ V_{C,E} induces a bijection from {(C, E)|V_{C,E} ≠ 0} to Irr(W^{σ}).

We call Springer correspondence for G^{1} the bijection defined by (ii) in theorem 6.3.

Proof of theorem 6.3: The following proof is inspired by that for the connected case in [13] (proof of theorem 6.2).

(i) Let π_{u} be the restriction of π : G̃^{1} → G^{1} to π^{−1}(U(G^{1})). Let us consider the following diagram:

π^{−1}(U(G^{1})) ⟶^{i} G̃^{1}
↓π_{u}  ↓π
U(G^{1})  ⟶^{i′} G^{1}

Let us apply the proper base change theorem to π. We get π_{∗}\overline{ℚ}_{l}|_{U(G^{1})} ≃ (π_{u})_{∗}\overline{ℚ}_{l}. We then apply theorem 6.2 to the morphism π_{u}. Let us check that the hypotheses of this theorem are satisfied in the situation of theorem 6.3.

First, π_{u} : π^{−1}(U(G^{1})) → U(G^{1}) is a proper morphism (as proper morphisms are stable under base extension) between two varieties with the same dimension equal to n − r′ and U(G^{1}) is smooth. Moreover,

dim{g.σ ∈ U(G^{1})|2dim(B_{g.σ}^{G}) ≥ i} ≤ dim(U(G^{1})) − i

(corollary 3.2) and π_{u}^{−1}(g.σ) ≃ B_{g.σ}^{G}, so π_{u} is semi-small. Then, according to proposition 4.1, π^{−1}(U(G^{1})) is a smooth variety. Finally, π_{u} is a G-equivariant morphism.

Thus, A = (π_{u})_{∗}\overline{ℚ}_{l}[n − r′] is a G-equivariant semi-simple object of M(U(G^{1})) and A is a direct sum of IC(\overline{Y} , E)[dim(Y)], where Y is a G-stable smooth irreducible subvariety of U(G^{1}) and E is a G-equivariant irreducible local system on Y. The closure \overline{Y} of Y is then an irreducible closed subvariety of U(G^{1}).

According to [14] (théorème I.4.1), there is a finite number of unipotent conjugacy classes in G^{1}. Any unipotent conjugacy class C of G^{1} is a smooth subvariety of U(G^{1}), its closure \overline{C} is an irreducible closed subvariety of U(G^{1}) and a union of unipotent conjugacy classes. Moreover, the G-stable irreducible closed subvarieties of U(G^{1}) are the closures of the unipotent conjugacy classes.

Thus, the G-equivariant simple objects of M(U(G^{1})) are of the form IC(\overline{C}, E)[dim(C)], where C is a unipotent conjugacy class of G^{1} and E is a G-equivariant irreducible local system on C.

Thus, A is decomposed as:

A ≃ ⨁_{(C,E)} V_{C,E} ⊗ IC(C̄, E)[dim(C)],

where, (C, E) runs over all the pairs in N and we get 6.3 (i).

(ii) The action of W^{σ} on π_{∗}\overline{ℚ}_{l} induces an action of W^{σ} on A and the action of W^{σ} on A induces a map α : \overline{ℚ}_{l}[W^{σ}] → End(A). Let us prove that α is an isomorphism.

Let us first prove that α is injective. We consider the stalk H_{σ}^{i}(A) of A at σ ∈ U(G^{1}), We have H_{σ}^{i}(A) ≃ H^{i+γ}(B_{σ}, \overline{ℚ}_{l}), where γ = n − r′. So, we have a natural morphism β : End(A) → End(H^{∗}(B_{σ}, \overline{ℚ}_{l})), and β ∘ α is the action of W^{σ} on H^{∗}(B_{σ}, \overline{ℚ}_{l}) that we have studied in the previous section. Thus, by corollary 5.4, β ∘ α is injective which leads to the injectivity of α.

Let us now prove that α is surjective. As α is injective, we just have to check that dim(\overline{ℚ}_{l}[W^{σ}]) ≥ dim(End(A)), which means, by (i), that

(1)  ∑_{(C,E)} dim(V_{C,E})^{2} ≤ |W^{σ}|

Let x be in U(G^{1}) and let C_{x} be the conjugacy class of x under G. We set d_{x} = dim(B_{x}). As H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}) ≃ H_{x}^{2d_{x}}(A[−γ]), we have:

H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}) ≃ ⨁_{(C,E)} V_{C,E} ⊗ H_{x}^{2d_{x}−γ+dimC}IC(C̄, E)

As A is G-equivariant, C_{G}(x) acts on the stalk H_{x}^{2d_{x}}(A[−γ]). This induces an action of A_{G}(x) which coincides with the action of A_{G}(x) on H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}) induced by the action of C_{G}(x) on B_{x}.

As G is a connected group which acts transitively on the variety C_{x}, there is a bijection between G-equivariant irreducible local systems on C_{x} and irreducible representations of A_{G}(x) over \overline{ℚ}_{l} ([13] 3.5). Let ρ be an irreducible representation of A_{G}(x) and let E an irreducible Gequivariant local system on C_{x} such that ρ is the image of E by the above bijection. We consider the direct summand in (6.1) corresponding to (C_{x}, E). As, we have 2d_{x} − γ + dim(C_{x}) = 2dim(B_{x}^{G}) + dim(C_{x}) − (dim(G) − rk(G^{σ})) = 0 (theorem 3.1, (i)), this summand is equal to V_{C_{x},E} ⊗ H_{x}^{0}(IC(C̄_{x}, E_{ρ})), that is to say, to V_{C_{x},E_{ρ}} ⊗ (E)_{x} and (E)_{x} is an irreducible A_{G}(x)-module isomorphic to ρ^{γ} ([13] 3.5).

Let us now consider H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}) as a representation of A_{G}(x). Let m_{x,ρ} be the multiplicity of the irreducible representation ρ^{γ} in H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}). Then, we have dim(V_{C_{x},E}) ≤ m_{x,ρ}.

We have then proved that, for any unipotent conjugacy class C_{x} of G^{1}, and for any irreducible representation ρ of A_{G}(x), we have dim(V_{C_{x},E_{ρ}}) ≤ m_{x,ρ}.

However, we have

dim(End_{A_{G}(x)}(H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}))) = ∑_{ρ∈A_{G}(x)^{∧}} m_{x,ρ}^{2}

And

dim(End_{A_{G}(x)}(H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}))) = dim(End_{\overline{ℚ}_{l}}(H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}))^{A_{G}(x)})

The space H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}) has a basis parametrised by the set I(B_{x}) of irreducible components of B_{x}, and the action of A_{G}(x) on H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}) is induced by its action on I(B_{x}). So, the basis of End_{\overline{ℚ}_{l}}(H^{2d_{x}}(B_{x}, \overline{ℚ}_{l})) is indexed by I(B_{x}) × I(B_{x}) and the action of A_{G}(x) on End_{\overline{ℚ}_{l}}(H^{2d_{x}}(B_{x}, \overline{ℚ}_{l})) is induced by its diagonal action on I(B_{x}) × I(B_{x}). Thus, the dimension of End_{\overline{ℚ}_{l}}(H^{2d_{x}}(B_{x}, \overline{ℚ}_{l}))^{A_{G}(x)} is equal to the number of A_{G}(x)-orbits of I(B_{x}) × I(B_{x}).

We then have the following equality, if we associate, to each pair (C, E) ∈ N a pair (x, ρ) with x ∈ C and ρ ∈ A_{G}(x)^{∧} is the image of C by the above bijection:

∑_{(x,ρ)} dim(V_{C_{x},E_{ρ}})^{2} ≤ ∑_{(x,ρ)} m_{(x,ρ)}^{2}

= ∑_{x∈U(G^{1})/∼} |I(B_{x}) × I(B_{x})/A_{G}(x)|
= |W^{σ}|

The last equality is a result from [14] (II.2.11), hence we have equality in (1) and α is surjective. End of the proof of theorem 6.3.

Corollary 6.4. For any unipotent conjugacy class C_{u.σ} in G^{1}, we have

H^{2d_{u.σ}}(B_{u.σ}, \overline{ℚ}_{l}) ≃ ⊕_{ρ∈A_{G}(u.σ)^{∧}}V_{(C_{u.σ},E_{ρ})} ⊗ ρ^{∨}

as W^{σ} × A_{G}(u.σ)-modules.

Corollary 6.5. For any unipotent conjugacy class C_{u.σ} in G^{1}, the \overline{ℚ}_{l}-vector space V_{(C_{u.σ},\overline{ℚ}_{l})} is different from zero. The Springer correspondence induces then an injection from the set of unipotent conjugacy classes of G^{1} to the set of irreducible representations of W^{σ} defined by C_{u.σ} ↦ V_{(C_{u.σ},\overline{ℚ}_{l})}.

Proof: The constant sheaf \overline{ℚ}_{l} on the unipotent conjugacy class C_{u.σ} corresponds to the trivial representation of A_{G}(u.σ). Thus, we have the following isomorphism of W^{σ}-modules: V_{(C_{u.σ},\overline{ℚ}_{l})} ≃ H^{2d_{u.σ}}(B_{u.σ}, \overline{ℚ}_{l})^{A_{G}(u.σ)}, whence the corollary as we have H^{2d_{u.σ}}(B_{u.σ}, \overline{ℚ}_{l})^{A_{G}(u.σ)} ≠ 0.

End of the proof.

According to corollary 5.4, the Springer action of W^{σ} from G^{1} on H^{∗}(B^{G^{σ}}, \overline{ℚ}_{l}) is the regular representation of W^{σ}. Thus, for all i odd, we have H^{i}(B^{G^{σ}}, \overline{ℚ}_{l}) = 0, and for any i = 2k, H^{i}(B^{G^{σ}}, \overline{ℚ}_{l}) is W^{σ}-isomorphic to a space of homogeneous polynomial functions of degree k. Let n_{k}(ρ) be the multiplicity of an irreducible representation ρ in this space. Then, the polynomial function defined by P_{ρ}(t) = ∑_{k} n_{k}(ρ)t^{k} is the fake degree of ρ ([5] section 11.1). The following corollary can be deduced from theorem 6.3.

Corollary 6.6.

n_{k}(ρ_{(C_{u.σ},E)}) = dim(H_{σ}^{2k−2d_{u.σ}}(IC(\overline{C_{u.σ}}, E))).

Let us denote by a_{ρ} the valuation of P_{ρ} for any irreducible representation ρ of W^{σ}.

Proposition 6.7.

(i) If k < d_{u.σ}, then n_{k}(ρ_{(C_{u.σ},E)}) = 0.

(ii) If k = d_{u.σ} and if E = \overline{ℚ}_{l}, then n_{k}(ρ_{(C_{u.σ},E)}) ≠ 0, which means that d_{u.σ} = a_{ρ}, where ρ = ρ_{(C_{u.σ},\overline{ℚ}_{l})}.

Proof: (i) is an immediate consequence of the previous corollary, for if j < 0, we have H_{σ}^{2j−2d_{u.σ}}(IC(\overline{C_{u.σ}}, E)) = 0.

In order to prove (ii), we have to check that for all unipotent conjugacy class, we have H_{σ}^{0}(IC(\overline{C}, \overline{ℚ}_{l})) ≠ 0.

Let C be a unipotent conjugacy class. Then, its closure \overline{C} is stratified by the unipotent conjugacy classes which are included in \overline{C}. Moreover, Cl_{G}(σ) is its smallest stratum and is closed. We set V = \overline{C} − Cl_{G}(σ) and we denote by j the immersion V ↪ \overline{C}. Then, dim(H^{0}(j_{!}j^{∗}(IC(\overline{C}, \overline{ℚ}_{l})))) is the number of connected components of C whose closure in V is closed in \overline{C} and dim(H^{0}(IC(\overline{C}, \overline{ℚ}_{l}))) is the number of connected components of C and is then equal to 1, as C is connected. As C is dense in \overline{C} and as Cl_{G}(σ) is not empty, we have dim(H^{0}(j_{!}j^{∗}(IC(\overline{C}, \overline{ℚ}_{l})))) = 0. But there is an injection from the cokernel of the injective morphism

0 → H^{0}(j_{!}j^{∗}(IC(\overline{C}, \overline{ℚ}_{l}))) → H^{0}(IC(\overline{C}, \overline{ℚ}_{l}))

to H^{0}(IC(\overline{C}, \overline{ℚ}_{l})|_{Cl_{G}(σ)}), which is then different from zero. Moreover, IC(\overline{C}, \overline{ℚ}_{l}) is G-equivariant, so we have H_{σ}^{0}(IC(\overline{C}, \overline{ℚ}_{l})) ≠ 0.

End of the proof.

7. Restriction of the Springer correspondence to a σ-stable Levi subgroup included in a σ-stable parabolic subgroup

Let L be a σ-stable Levi subgroup included in a σ-stable parabolic subgroup P. We denote by P the variety of parabolic subgroups which are conjugate to P. Let W (L) be the Weyl group of L. Let C^{L} be a unipotent class of L^{1} and C the unipotent class of G^{1} obtained by inducing C^{L} from L^{1} to G^{1} (definition 7.1). The purpose of this section is to prove that the irreducible representation ρ_{(C^{L},\overline{ℚ}_{l})} of W (L)^{σ}, image of (C^{L}, \overline{ℚ}_{l}) by the Springer correspondence for L^{1} is an irreducible component of the restriction to W (L)^{σ} of the irreducible representation ρ_{(C,\overline{ℚ}_{l})} of W^{σ}, image of (C, \overline{ℚ}_{l}) by the Springer correspondence from G^{1} (proposition 7.13).

7.1. Notations. The variety G̃^{1} is defined, like in section 4, by:

G̃^{1} = {(g.σ, B) ∈ G^{1} × B^{G}|B ∈ B_{g.σ}^{G}},

where, B_{g.σ}^{G} is the variety of Borel subgroups of G which are stable under conjugation by g.σ. We still denote by π the projection G̃^{1} → G^{1} on the first factor, Ũ(G^{1}) = π^{−1}(U(G^{1})) and π_{u} the restriction of π to Ũ(G^{1}).

Notation 7.1. We put A = π_{u∗}\overline{ℚ}_{l}.

We denote by L̃^{1}, B_{l.σ}^{L}, π(L), Ũ(L^{1}), π(L)_{u}, the analogous objects relative to L.

Notation 7.2. We put A(L) = π(L)_{u∗}\overline{ℚ}_{l}.

According to theorem 6.3, we have in M(U(G^{1})):

(2)  A[dim(U(G^{1}))] ≃ ⨁_{(C,E)} IC(\overline{C}, E)[dim(C)] ⊗ V_{(C,E)}

where C runs over the set of unipotent classes of G^{1} and E the set of irreducible G-equivariant local systems on C.

And by 6.3 in L, we have in M(U(L^{1})):

(3)  A(L)[dim(U(L^{1}))] ≃ ⨁_{(C^{L},F)} IC(\overline{C^{L}}, F)[dim(C^{L})] ⊗ V_{(C^{L},F)}

where C^{L} runs over the set of unipotent classes of L^{1} and F the set of irreducible L-equivariant local systems on C^{L}.

The variety G̃^{1P} is defined by

G̃^{1P} = {(g.σ, P′) ∈ G^{1} × P|P′ ∈ P_{g.σ}},

where P_{g.σ} is the variety of parabolic subgroups conjugate to P and stable under conjugation by g.σ. The variety G̃^{1P} can be identified with G ×^{P} P^{1} by the map (g.σ,^{x}P) ↦ (x,^{x^{−1}}(g.σ)). Let ξ : G̃^{1P} → G^{1} be the projection on the first factor and η : G̃^{1} → G̃^{1P} the morphism defined by (g.σ, B′) ↦ (g.σ, P′), where P′ is the only parabolic subgroup in P such that B′ ⊂ P′. Then, we have π = ξ ∘ η.

Let Ũ(G^{1})^{P} be the inverse image of U(G^{1}) under ξ. Then, the variety Ũ(G^{1})^{P} can be identified with G ×^{P} U(P^{1}) by the map (g.σ,^{x}P) ↦ (x,^{x^{−1}}(g.σ)). We denote by ξ_{u} the restriction of ξ to Ũ(G^{1})^{P} and by η_{u} the restriction of η to Ũ(G^{1}).

Notation 7.3. We denote by A^{P} the complex of sheaves η_{u∗}\overline{ℚ}_{l} on Ũ(G^{1})^{P}.

Finally, let V be the variety G ×^{P} L^{1}, where the action of P on the left on L^{1} is defined by p : l.σ ↦ π_{P} (p(l.σ)p^{−1}), where π_{P} is the projection NG̃(P) → NG̃(L, P) whose kernel is U_{P}. Let us identify a point of V with its image under the map (x, l.σ) ↦ (^{x}(l.σU_{P}),^{x}P). Then, V is the set of the pairs (g.σ, P′) with P′ ∈ P such that g.σ ∈ NG̃(P′)/U_{P′}, where U_{P′} is the unipotent radical of P′. Let r : V → P be the projection on the second factor defined by (g.σ, P′) ↦ P′. We denote by V_{u} the subvariety G ×^{P} U(L^{1}) of V and by r_{u} the restriction of r to V_{u}.

Let Ṽ be the variety G ×^{B} (L ∩ B)^{1} where the action of B on (L ∩ B)^{1} on the left is defined by l : b.σ ↦ π_{P} (l(b.σ)l^{−1}) (the image of B^{1} by π_{P} is (L ∩ B)^{1}, where we have put (L ∩ B)^{1} = (L ∩ B).σ).

We identify Ṽ with the set of (g.σ, B′, P′) where P′ ∈ P, B′ is a Borel subgroup of P′ and g.σ ∈ NG̃(B′)/U_{P′}. Then, we can define a morphism ζ : Ṽ → V by (y, B′, P′) ↦ (y, P′). We set Ṽ_{u} = ζ^{−1}(V_{u}).

Notation 7.4. We denote by K the direct image of the constant sheaf \overline{ℚ}_{l} on Ṽ under ζ_{u}.

Let us consider the following diagram:

Diagram 7.1.

\xymatrix{L̃^{1} = L ×^{L∩B} (L ∩ B)^{1} \ar[r]^{ĩ} \ar[d]^{π(L)} & Ṽ = G ×^{B} (L ∩ B)^{1} \ar[d]^{ζ} & G̃^{1} = G ×^{B} B^{1} \ar[l]_{q̃} \ar[d]^{η} \\ L^{1} \ar[r]_{i} & V = G ×^{P} L^{1} & G̃^{1P} = G ×^{P} P^{1} \ar[l]^{q} \ar[d]^{ξ} \\  &  & G^{1}}

The morphism i is defined by l.σ ↦ (1, l.σ) (this is the inclusion of the fiber r^{−1}(P)); the morphism q is defined by (x, p.σ) ↦ (x, π_{P} (p.σ)).

We denote by ĩ the morphism defined by (l.σ, B̃′) ↦ (l.σU_{P} , B′, P) where B′ is the Borel subgroup included in P such that B′ ∩ L = B̃′. The morphism q̃ is defined by (g.σ, B′) ↦ (\overline{g.σ}, B′, P′), where P′ is the only parabolic subgroup which is conjugate to P and such that B′ ⊂ P′ and where we denote by \overline{g.σ} the image of g.σ in NG̃(P′)/U_{P′}.

In the following diagram, the morphisms indexed by u are the restrictions to the unipotent varieties of the corresponding morphisms of diagram 7.1:

Diagram 7.2.

\xymatrix{Ũ(L^{1}) \ar[r]^{ĩ_{u}} \ar[d]^{π(L)_{u}} & Ṽ_{u} \ar[d]^{ζ_{u}} & Ũ(G^{1}) \ar[l]_{q̃_{u}} \ar[d]^{η_{u}} \\ U(L^{1}) \ar[r]_{i_{u}} & V_{u} & Ũ(G^{1})^{P} \ar[l]^{q_{u}} \ar[d]^{ξ_{u}} \\  &  & U(G^{1})}

Using diagram 7.2 we will compare A^{P} = η_{u∗}\overline{ℚ}_{l} and A(L) = π(L)_{u∗}\overline{ℚ}_{l} with the help of K = ζ_{u∗}\overline{ℚ}_{l}.

7.2. Application of the decomposition theorem to A^{P} and to K. The following proposition shows that the hypotheses of theorem 6.2 are verified for the morphisms η_{u} : Ũ(G^{1}) → Ũ(G^{1})^{P} and ζ_{u} : Ṽ_{u} → V_{u}.

Proposition 7.1. (i) The variety Ũ(G^{1})^{P} is smooth and

dim(Ũ(G^{1})^{P}) = dim(U(G^{1})) = dim(G) − rk(G^{σ}).

The varieties V_{u} and Ṽ_{u} are smooth and

dim(V_{u}) = dim(Ṽ_{u}) = dim(P) + dim(U(L^{1})).

(ii) η and ζ are proper.

(ii’) η_{u} and ζ_{u} are proper.

(iii) The morphisms η_{u} and ζ_{u} are semi-small.

Proof:

(i) Let us prove that the projection p_{2} : Ũ(G^{1})^{P} → P on the second factor is a locally trivial fibration whose fiber is isomorphic to U(P^{1}) = U(L^{1})U_{P}. We will be able to conclude that Ũ(G^{1})^{P} is a smooth variety and that its dimension is equal to dim(G/P) + dim(L) − rk(L^{σ}) + dim(U_{P}) = n − r′, where n = dim(G) and r′ = rk(G^{σ}) as rk(L^{σ}) = rk(G^{σ}).

Let P′ be in P, then there is x in G such that P′ = ^{xw_{0}}P, where w_{0} is the longest element of W and V (P′) = {^{xuw_{0}}P; u ∈ U} is an open neighbourhood of P′ because Uw_{0}B a dense open set of G and Uw_{0}B/P is a dense open set of G/P. Moreover, V (P′) ≃ U_{P} as we have w_{0}(u) ∈ U^{−} for all u ∈ U.

Let us prove that the morphism ϕ : U_{P} × U(L^{1})U_{P} → p_{2}^{−1}(V (P′)) defined by (u, p.σ) ↦ (^{xuw_{0}}(p.σ),^{xuw_{0}}P) is an isomorphism. Let (g.σ, P_{1}) be in p_{2}^{−1}(V (P′)), then there is only one u ∈ U_{P} such that P_{1} = ^{xuw_{0}}P and if we set ψ((g.σ, P_{1})) = (u,^{(xuw_{0})^{−1}}(g.σ)), then ψ and ϕ are inverse morphisms of varieties. So, p_{2} : Ũ(G^{1})^{P} → P is a locally trivial fibration whose fiber is isomorphic to U(P^{1}).

Similarly, we have V_{u} = G ×^{P} U(L^{1}) and Ṽ_{u} = G ×^{B} U((L ∩ B)^{1}). So, r_{u} : V_{u} → P is a locally trivial fibration whose fiber is isomorphic to U(L^{1}) and r̃_{u} : Ṽ_{u} → B is a locally trivial fibration whose fiber is isomorphic to U((L ∩ B)^{1}).

Then, these varieties are smooth. Moreover, they have the same dimension, as dim(V_{u}) = dim(P) + dim(U(L^{1})) and,

dim(Ṽ_{u}) = dim(B) + dim(U((L ∩ B)^{1}))
= dim(G) − dim(B) + dim(L_{σ}(T)) + dim(U) − dim(U_{P})
= dim(G) − dim(B) + dim(T) − dim(T^{σ}) + dim(U) − dim(P) + dim(L)
= dim(G) − rk(L^{σ}) − dim(P) + dim(L)
= dim(P) + dim(U(L^{1}))

(ii) G̃^{1} is a closed subset of G^{1} × B and η = (Id × f) ∘ i, where i : G̃^{1} ↪ G^{1} × G/B and where f associates to any B′ ∈ B the only parabolic subgroup P′ ∈ P such that B′ ⊂ P′. The morphism i is proper as it is a closed immersion Moreover, the morphism f : B → P is proper as it is separated, of finite type and universally closed as B^{G} is complete. Thus, η = (Id × f) ∘ i is proper.

Let us now prove that ζ is proper. We consider the following cartesian diagram:

\begin{matrix}V ×_{P} B & → & B \\ ↓f′ & & ↓f \\ V & →^{r} & P \end{matrix},

The variety V ×_{P} B is the set {(g.σ, P′, B′)|g.σ ∈ NG̃(P′)/U_{P′}|f(B′) = P′}. Recall that f is a proper, then, by base extension, f′ is proper. Moreover, Ṽ = {(g.σ, B′, P′)|g.σ ∈ NG̃(B′)/U_{P′}|f(B′) = P′} is a closed subset of V ×_{P} B and the closed immersion i : Ṽ ↪ V ×_{P} B is proper. So, ζ = f′ ∘ i is proper.

(ii’) is a consequence of (ii) as Ṽ_{u} is a closed smooth subvariety of Ṽ and Ũ(G^{1}) is a closed smooth subvariety of G̃^{1}.

In order to prove (iii), we need the following lemma:

Lemma 7.2.

a) The morphism Ũ(G^{1})^{P} →^{q_{u}} V_{u} is a fibration whose fiber at v ∈ V_{u} is isomorphic to U_{P}.

b) For any y ∈ G×^{P} (C^{L}U_{P}) (resp. y ∈ G×^{P} C^{L}), we have η_{u}^{−1}(y) ≃ B_{t}^{L} (resp. ζ_{u}^{−1}(y) ≃ B_{t}^{L}), where t is in C^{L}.

c) For any unipotent L-conjugacy class C^{L} of L^{1}, the variety G ×^{P} (C^{L}U_{P}) is a double fibration G ×^{P} (C^{L}U_{P}) →^{q} G ×^{P} C^{L} →^{r} P, where the fiber of q|_{G×^{P} (C^{L}U_{P})} is U_{P} and the fiber of r|_{G×^{P} C^{L}} is C^{L}.

d) the codimension of G ×^{P} (C^{L}U_{P}) in Ũ(G^{1})^{P} is equal to that of C^{L} in U(L^{1}) and to that of G ×^{P} C^{L} in V_{u}.

Proof of lemma 7.2:

a) is clear using the identification Ũ(G^{1})^{P} = G ×^{P} U(P^{1}), as we have U(P^{1}) ≃ U(L^{1})U_{P} and V_{u} = G ×^{P} U(L^{1}).

b) Let C^{L} be a unipotent L-class in L^{1}. Then, if l_{1}.σ and l_{2}.σ are in C^{L}, the varieties B_{l_{1}.σ}^{L} are B_{l_{2}.σ}^{L} are isomorphic.

Let (y, P′) be in G ×^{P} C^{L}. Then, there is x ∈ G and l.σ ∈ C^{L} such that P′ = ^{x}P and x^{−1}yx = l.σU_{P} and the fiber of ζ_{u} at (y, P′) is isomorphic to the variety B_{l.σ}^{P} of Borel subgroups of G which are contained in P and stable under conjugation by l.σ. Whence the result for ζ_{u} as B′ ↦ B′ ∩ L is an isomorphism from B_{l.σ}^{P} onto B_{l.σ}^{L}.

Let (g.σ, P′) be in G ×^{P} (C^{L}U_{P}), then there is x ∈ G and p.σ ∈ C^{L}U_{P} such that P′ = ^{x}P and x^{−1}g.σx = p.σ and the fiber of η_{u} at (g.σ, P′) is isomorphic to the variety B_{p.σ}^{P}. Moreover, if we set l.σ = π_{P} (p.σ), then, B_{p.σ}^{P} ≃ B_{l.σ}^{L}. Whence the result for η_{u}.

c) is clear.

d) is a consequence of a) and c). Indeed, by a), we have

dim(Ũ(G^{1})^{P}) = dim(U(L^{1})) + dim(U_{P}) + dim(P)

and

dim(V_{u}) = dim(Ũ(G^{1})^{P}) − dim(U_{P})

and according to c), we have

dim(G ×^{P} (C^{L}U_{P})) = dim(U_{P}) + dim(C^{L}) + dim(P)

and

dim(G ×^{P} C^{L}) = dim(G ×^{P} (C^{L}U_{P})) − dim(U_{P}).

So,

dim(Ũ(G^{1})^{P}) − dim(G ×^{P} (C^{L}U_{P})) = dim(U(L^{1})) − dim(C^{L}) = dim(V_{u}) − dim(G ×^{P} C^{L}).

End of the proof of lemma 7.2

Let us prove that η_{u} is semi-small. It is a proper morphism between two equidimensional varieties. Let us set

f(i) = dim{y ∈ Ũ(G^{1})^{P} |dim(η_{u}^{−1}(y)) ≥ i/2}.

By b) in lemma 7.2, the dimension of the fiber η_{u}^{−1}(y) is constant for y ∈ G ×^{P} (C^{L}U_{P}). Let us set d_{C^{L}} = dim(η_{u}^{−1}(y)), where y is in G ×^{P} (C^{L}U_{P}). Let C_{i}^{L} be the element whose dimension is maximal in the set of unipotent L-conjugacy classes

C^{L} of L^{1} such that d_{C^{L}} ≥ i/2; then f(i) is equal to the dimension of G ×^{P} (C_{i}^{L}U_{P}).

Let us now put

g(i) = dim{t ∈ U(L^{1})|dim(B_{t}^{L}) ≥ i/2}.

The dimension of B_{t}^{L} is constant for t ∈ C^{L} where C^{L} is a unipotent L-conjugacy class in L^{1}. So, g(i) is equal to the dimension of the unipotent L-conjugacy classes of L^{1} whose dimension is maximal in the set of classes C^{L} such that dimB_{t}^{L} ≥ i/2 for t ∈ C^{L}. Moreover, by b) in lemma 7.2, we have d_{C^{L}} = dimB_{t}^{L}, for t ∈ C^{L} and by d) in lemma 7.2, we have codim_{Ũ(G^{1})^{P}} (G ×^{P} (C^{L}U_{P})) = codim_{U(L^{1})}(C^{L}). So f(i) = g(i) + dim(Ũ(G^{1})^{P}) − dim(U(L^{1})). But, applying corollary 3.2, we have g(i) ≤ dim(U(L^{1})) − i, hence, f(i) ≤ dim(Ũ(G^{1})^{P}) − i and η_{u} is semismall.

The proof for ζ_{u} is similar but f(i) has to be replaced by:

h(i) = dim{y ∈ V_{u}|dim(ζ_{u}^{−1}(y)) ≥ i/2}.

End of the proof of proposition 7.1.

According to proposition 7.1, the hypotheses of theorem 6.2 are verified for η_{u} and ζ_{u}. Hence, we have the following decomposition formula in M(U(G^{1})):

(4)  A^{P} [dim(U(G^{1}))] = η_{u∗}\overline{ℚ}_{l}[dim(U(G^{1}))] ≃ ⨁_{(C′,E′)} IC(\overline{C′}, E′)[dim(C′)] ⊗ V′_{(C′,E′)}

where C′ runs over the set of G-stable irreducible smooth varieties of Ũ(G^{1})^{P} and E′ over the set of G-equivariant irreducible local systems on C′.

Moreover, K[dim(V_{u})] = ζ_{u∗}\overline{ℚ}_{l}[dim(V_{u})] is a G-equivariant semi-simple object of M(V_{u}) and if IC(\overline{Y} , F′)[dim(Y)] is a simple object whose multiplicity in K[dim(V_{u})] is non zero, then, Y is a G-stable irreducible smooth subvariety of V_{u} and F′ is a G-equivariant irreducible local system on Y. Then, \overline{Y} is a G-stable irreducible closed subset of V_{u} = G ×^{P} U(L^{1}). So, we have \overline{Y} = G×^{P} X, where X is a L-stable irreducible closed subset of U(L^{1}), that is to say, the closure of a unipotent L-conjugacy class inL^{1}. We then get the following decomposition formula for K[dim(V_{u})] = ζ_{u∗}\overline{ℚ}_{l}[dim(V_{u})] in M(V_{u}):

(5)  K[dim(V_{u})] ≃ ⨁_{(C^{L},F′)} IC(G ×^{P} \overline{C^{L}}, F′)[dim(G ×^{P} C^{L})] ⊗ V′′_{(C^{L},F′)}

where C^{L} runs over the set of unipotent L-conjugacy classes in L^{1}, and F′ over the set of G-equivariant irreducible local systems on G ×^{P} C^{L}.

7.3. Identification of the decompositions of A^{P} and of A(L). The purpose of this subsection is to prove proposition 7.4 which shows how we can identify the decompositions (3) of A(L) and (4) of A^{P} via the functors q_{u}^{∗} et i_{u}^{∗}. We need the following lemma:

Lemma 7.3.

a) For any unipotent L-conjugacy class C^{L} of L^{1}, the map F ↦ (i|_{C^{L}})^{∗}F is a bijection from the set of G-equivariant irreducible local systems on G ×^{P} C^{L} to the set of L-equivariant irreducible local systems on C^{L}.

b) Let C^{L} be a unipotent L-conjugacy class in L^{1}. The image of G ×^{P} (C^{L}U_{P}) under q is G×^{P} C^{L} and if F is a G-equivariant irreducible local system on G×^{P} C^{L}, then, (q|_{G×^{P} (C^{L}U_{P})})^{∗}F is a G-equivariant irreducible local system on G ×^{P} (C^{L}U_{P}), where q is defined in diagram 7.1.

Proof:

a) Let C^{L} = Cl_{L}(x.σ) be a unipotent L-conjugacy class in L^{1}. Then, y = (1, x.σ) is in G ×^{P} C^{L}.

According to [13] (3.5), as L is a connected group which acts transitively on C^{L}, there is a bijection f_{1} between L-equivariant irreducible local systems on C^{L} and irreducible finite dimensional \overline{ℚ}_{l}-representations of A_{L}(x.σ) = C_{L}(x.σ)/C_{L}(x.σ)^{0}. If L is a L-equivariant irreducible local system on C^{L}, then, A_{L}(x.σ) acts on L_{x.σ}, this is an irreducible representation ρ of A_{L}(x.σ) and the bijection f_{1} is defined by L ↦ ρ.

Similarly, the action of G on G ×^{P} C^{L}, defined by g : (h, l.σ) ↦ (gh, l.σ) for all g ∈ G, is transitive. Indeed, let (h, l.σ) be G ×^{P} C^{L}, then, there is l′ ∈ L such that l.σ = ^{l′}(x.σ) and (h, l.σ) = (hl′, x.σ) = (hl′).(1, x.σ). Hence, there is a bijection f_{2} between G-equivariant irreducible local systems on G ×^{P} C^{L} and irreducible finite dimensional \overline{ℚ}_{l}-representations of A_{G}(y) = Stab_{G}(y)/Stab_{G}(y)^{0}. If L′ is a G-equivariant irreducible local system on G ×^{P} C^{L}, then, A_{G}(y) acts on L′_{y}, this is an irreducible representation ρ′ of A_{G}(y) and the bijection f_{2} is defined by L′ ↦ ρ′.

But, we have Stab_{G}(y) = C_{L}(x.σ)U_{P}, whence A_{G}(y) = A_{L}(x.σ), for U_{P} is connected, and there is a bijection (f_{1}^{−1} ∘ f_{2}) between G-equivariant irreducible local systems on G ×^{P} C^{L} and L-equivariant irreducible local systems on C^{L}.

Let us prove that this bijection is induced by the functor (i|_{C^{L}})^{∗}. Let L′ be a G-equivariant irreducible local system on G ×^{P} C^{L}, it is sent by the bijection f_{2} to the representation ρ of A_{G}(y) on L′_{y} and (i|_{C^{L}})^{∗}L′ is a L-equivariant irreducible local system on C^{L} which is sent by the bijection f_{1} to the same representation ρ of A_{G}(y) (= A_{L}(x.σ)) on L′_{y} (= (i|_{C^{L}})^{∗}(L′)_{x.σ} for i(x.σ) = y). Indeed, as i|_{C^{L}} is L-equivariant, it is compatible with the actions of A_{L}(x.σ).

Whence the result.

b) Let us consider the variety G ×^{P} (C^{L}U_{P}), then q|_{G×^{P} (C^{L}U_{P})} : G ×^{P} C^{L}U_{P} → G ×^{P} C^{L} is a fibration whose fiber is isomorphic to U_{P}. Let us set d = dim(U_{P}). The morphism q|_{G×^{P} (C^{L}U_{P})} is smooth with connected fibers whose dimension is equal to d, so the image of an irreducible local system by the functor (q|_{G×^{P} (C^{L}U_{P})})^{∗} is still an irreducible local system. Moreover, q|_{G×^{P} (C^{L}U_{P})} is G-equivariant. So, for all G-equivariant irreducible local system L on G×^{P} C^{L}, the local system (q|_{G×^{P} (C^{L}U_{P})})^{∗}(L) is still irreducible and G-equivariant.

End of the proof

Let C^{L} be a unipotent L-conjugacy class in L^{1}. We denote by Φ(C^{L}) the bijection obtained using lemma 7.3 a) and b) between L-equivariant irreducible local systems on C^{L} and G-equivariant irreducible local systems on G ×^{P} C^{L}U_{P}.

Proposition 7.4.

(i) We have the following isomorphisms in the derived category.

A^{P} ≃ q_{u}^{∗}K,
A(L) ≃ i_{u}^{∗}K.

(ii) EndK ≃ EndA^{P} via the functor q_{u}^{∗} and EndK ≃ EndA(L) via the functor i_{u}^{∗}.

(iii) The perverse sheaf A^{P} [dim(U(G^{1}))] is decomposed in M(Ũ(G^{1})^{P}) as:

(6) A^{P} [dim(U(G^{1}))] ≃ ⨁_{(C^{L},E′)} IC(G ×^{P} (\overline{C^{L}}U_{P}), E′)[dim(G ×^{P} (\overline{C^{L}}U_{P}))] ⊗ V′_{(C^{L},E′)}

where C^{L} runs over the set of unipotent L-conjugacy classes of L^{1} and E′ is the image under Φ(C^{L}) of an L-equivariant local system E on C^{L}.

Moreover, if V_{(C^{L},E)} is the \overline{ℚ}_{l}-vector space corresponding to the simple object IC(\overline{C^{L}}, E)[dim(C^{L})] in the decomposition (3) of A(L)[dim(U(L^{1}))], then V_{(C^{L},E)} is isomorphic to V′_{(C^{L},E′)} as \overline{ℚ}_{l}-vector spaces.

Proof:

Let us consider the constant sheaf \overline{ℚ}_{l} on Ṽ_{u}. Then, we have the following isomorphisms in the derived category: A^{P} ≃ η_{u∗}q̃_{u}^{∗}\overline{ℚ}_{l} and A(L) ≃ π(L)_{u∗}ĩ_{u}^{∗}\overline{ℚ}_{l}. Moreover, we have K = ζ_{u∗}\overline{ℚ}_{l}. As π(L)_{u} and η_{u} are proper, (i) is a consequence of the proper base change theorem.

Let us prove the other assertions. We have the following decomposition formulas (cf. 7.1, formula (3) and 7.2, formulas (5) and (4)).

A(L) ≃ ⨁_{(C^{L},F)} IC(\overline{C^{L}}, F)[2d_{C^{L}}] ⊗ V_{(C^{L},F)}

where C^{L} runs over the set of unipotent L-conjugacy classes of L^{1} and F runs over the set of L-equivariant irreducible local systems on G ×^{P} C^{L} and where we set d_{C^{L}} = 1/2(dim(C^{L}) − dim(U(L^{1}))).

K ≃ ⨁_{(C^{L},F)} IC(G ×^{P} \overline{C^{L}}, F)[d_{C^{L}}] ⊗ V′′_{(C^{L},F)}

where C^{L} runs over the set of unipotent L-conjugacy classes of L^{1}, and F the set of G-equivariant irreducible local systems on G ×^{P} C^{L} and where d_{C^{L}} = 1/2(dim(G ×^{P} C^{L}) − dim(V_{u})) (lemma 7.2, d)).

And,

A^{P} ≃ ⨁_{(C′,E′)} IC(\overline{C′}, E′)[2d_{C′}] ⊗ V′_{(C′,E′)}

where C′ runs over the set of G-stable irreducible smooth subvarieties of Ũ(G^{1})^{P} and E′ the set of G-equivariant irreducible local systems on C′ and where we set d_{C′} = 1/2(dim(C′) − dim(Ũ(G^{1})^{P})).

But, as A(L) is isomorphic to i_{u}^{∗}C, we get another decomposition of A(L):

A(L) ≃ ⨁_{(C^{L},F)} i_{u}^{∗}IC(G ×^{P} \overline{C^{L}}, F)[2d_{C^{L}}] ⊗ V′′_{(C^{L},F)}

And as A^{P} is isomorphic to q_{u}^{∗}C, we get another decomposition of A^{P}:

A^{P} ≃ ⨁_{(C^{L},F)} q_{u}^{∗}IC(G ×^{P} \overline{C^{L}}, F)[2d_{C^{L}}] ⊗ V′′_{(C^{L},F)}

where we have d_{G×^{P} (C^{L}U_{P})} = d_{C^{L}} (lemma 7.2, d)).

Then, applying lemma 7.5, for any unipotent class C^{L} of L^{1} and for any G-equivariant irreducible local system F′ on G ×^{P} C^{L}, we can identify the \overline{ℚ}_{l}-vector spaces V_{(C^{L},(i_{C^{L}})^{∗}F)} and V′′_{(C^{L},F)}. Moreover, for all G-stable irreducible smooth variety C′ of G̃^{1P} and for all G-equivariant irreducible local system E′, we have V′_{(C′,E′)} = 0 except if there is a unipotent L-conjugacy class C^{L} of L^{1} such that C′ = G×^{P} (C^{L}U_{P}) and a G-equivariant irreducible local system F on G×^{P} C^{L} such that E′ = (q_{G×^{P} C^{L}})^{∗}F in which case we have V′_{(C′,E′)} = V′′_{(C^{L},F)}. Thus, proposition 7.4 will be proved if we prove lemma 7.5.

Lemma 7.5.

(i) For any unipotent L-conjugacy class C^{L} of L^{1} and for any local system L on G ×^{P} C^{L}, the functor q^{∗} sends IC(G ×^{P} \overline{C^{L}}, L) over IC(G ×^{P} (\overline{C^{L}}U_{P}), (q|_{G×^{P} (C^{L}U_{P})})^{∗}(L)).

(ii) For any unipotent L-conjugacy class C^{L} of L^{1} and for all G-equivariant irreducible local systems L on G ×^{P} C^{L}, the functor i^{∗} maps IC(G ×^{P} \overline{C^{L}}, L) to IC(\overline{C^{L}}, (i|_{C^{L}})^{∗}L).

Proof of lemma 7.5:

Set d = dim(G×^{P} (C^{L}U_{P}))−dim(G×^{P} C^{L}). As q|_{G×^{P} (\overline{C^{L}}U_{P})} is a smooth morphism of relative

dimension d with connected fibers, the image of the simple object IC(G×^{P} \overline{C^{L}}, L)[dim(G×^{P} C^{L})] of M(G ×^{P} \overline{C^{L}}) by q^{∗}[d] is the simple object IC(G ×^{P} (\overline{C^{L}}U_{P}), (q|_{G×^{P} (C^{L}U_{P})})^{∗}L)[dim(G ×^{P} (C^{L}U_{P}))] of M(G ×^{P} \overline{C^{L}}U_{P}) ([1], 4.2.6), whence the result.

Let us now prove (ii).

We denote by i′ the closed immersion from \overline{C^{L}} to G × \overline{C^{L}} and by p_{2} the projection on the second factor from G × \overline{C^{L}} to \overline{C^{L}}. Let pr be the quotient morphism from G × \overline{C^{L}} to G ×^{P} \overline{C^{L}}. We have p_{2} ∘ i′ = Id\overline{C^{L}} and i|\overline{C^{L}} = pr ∘ i′. We get the following diagram:

\xymatrix{\overline{C^{L}} \ar[dr]_{i\overline{C^{L}}} \ar[r]^{i′} & G × \overline{C^{L}} \ar[d]^{pr} \ar[l]^{p_{2}} \\  & G ×^{P} \overline{C^{L}}}

Let L′ be a local system on C^{L}. As p_{2} is a smooth morphism of relative dimension dim(G) with connected fibers, we have p_{2}^{∗}IC(\overline{C^{L}}, L′) = IC(G × \overline{C^{L}}, (p_{2}|_{G×C^{L}})^{∗}L′). So we get

i′^{∗}IC(G × \overline{C^{L}}, (p_{2}|_{G×C^{L}})^{∗}L′) = IC(\overline{C^{L}}, L′).

Moreover, let L be a G-equivariant irreducible local system on G×^{P} C^{L} then, as pr is a smooth morphism of relative dimension dim(P) with connected fibers, we have pr^{∗}IC(G ×^{P} \overline{C^{L}}, L) = IC(G × \overline{C^{L}}, (pr|_{C^{L}})^{∗}L). Then we get

(i|\overline{C^{L}})^{∗}IC(G ×^{P} \overline{C^{L}}, L) = i′^{∗}IC(G × \overline{C^{L}}, (pr|_{G×C^{L}})^{∗}L).

So (ii) will be proved if we get

(pr|_{G×C^{L}})^{∗}L = (p_{2}|_{G×C^{L}})^{∗}(i|_{C^{L}})^{∗}(L).

Let us consider the following commutative diagram:

\xymatrix{G × L × G × C^{L} \ar[r]^{p̃r} \ar[d]_{p} \ar[d]^{m} & G × G ×^{P} C^{L} \ar[d]_{m′} \ar[d]^{p′} \\ G × C^{L} \ar[r]_{pr|_{G×C^{L}}} & G ×^{P} C^{L}}

where p and p′ are the projections on the second factor, m (resp. m′) the action of G × L on G × C^{L} (resp. of G on G ×^{P} C^{L}) defined by (g, l) : (x, h.σ) ↦ (gx,^{l}(h.σ)) (resp. by g : (x, h.σ) ↦ (gx, h.σ)) and where p̃r : G × L × G × C^{L} → G × G ×^{P} C^{L} is defined by (g, l, x, h.σ) ↦ (l^{−1}g, x, h.σ).

As L is a G-equivariant local system on G ×^{P} C^{L}, we have m′^{∗}L = p′^{∗}L, and as the above diagram is commutative, we get m^{∗}(pr|_{G×C^{L}})^{∗}L = p^{∗}(pr|_{G×C^{L}})^{∗}L. So, (pr|_{G×C^{L}})^{∗}L is a G × Lequivariant local system on G × C^{L} and is of the form L_{1} ⊠ L_{2} where L_{1} is a G-equivariant local system on G and L_{2} is a L-equivariant local system on C^{L}. But any G-equivariant local system on G is isomorphic to \overline{ℚ}_{l}. Hence, we have

(pr|_{G×C^{L}})^{∗}L = \overline{ℚ}_{l} ⊠ L_{2} = (p_{2}|_{G×C^{L}})^{∗}L_{2}.

We also have

(i|_{C^{L}})^{∗}(L) = (i′|_{C^{L}})^{∗}(pr|_{G×C^{L}})^{∗}L = (i′|_{C^{L}})^{∗}(p_{2}|_{G×C^{L}})^{∗}L_{2} = L_{2}.

Whence, the result.

End of the proof of lemma 7.5. End of the proof of proposition 7.4.

7.4. Action of W (L)^{σ} on A^{P}. In this subsection, we first construct an action of W (L)^{σ} on the direct image η_{∗}\overline{ℚ}_{l} of the constant sheaf \overline{ℚ}_{l} on G̃^{1} and an action of W (L)^{σ} on the direct image ζ_{∗}\overline{ℚ}_{l} of the constant sheaf \overline{ℚ}_{l} on Ṽ (propositions 7.6 and 7.7). The action of W (L)^{σ} on η_{∗}\overline{ℚ}_{l} induces an action on A^{P} = η_{u∗}\overline{ℚ}_{l}. In proposition 7.9, we compare the Springer action of W (L)^{σ} from L^{1} on A(L) = π(L)_{u∗}\overline{ℚ}_{l} and the above action of W (L)^{σ} on A^{P}.

For any σ-stable subgroup H of G, we denote by H_{reg}^{1} the set of quasi-semi-simple elements of H^{1} which are regular in G^{1}. As L^{σ} and G^{σ} have same rank, x ∈ L^{1} is quasi-semi-simple regular in G^{1} if and only if it is quasi-simple regular in L^{1}. So, if H is a σ-stable subgroup of L, then, H_{reg}^{1} is the set of quasi-semi-simple elements of H^{1} which are regular in L^{1}.

Let us consider the following diagram:

Diagram 7.3.

\xymatrix{π(L)^{−1}(L_{reg}^{1}) \ar[r]^{ĩ_{reg}} \ar[d]^{π(L)_{reg}} & ζ^{−1}(V_{reg}) \ar[d]^{ζ_{reg}} & η^{−1}(G̃_{reg}^{1P}) \ar[l]_{q̃_{reg}} \ar[d]^{η_{reg}} \\ L_{reg}^{1} \ar[r]_{i_{reg}} & V_{reg} = G ×^{P} L_{reg}^{1} & G̃_{reg}^{1P} = G ×^{P} P_{reg}^{1} \ar[l]^{q_{reg}} \ar[d]^{ξ_{reg}} \\  &  & G_{reg}^{1}}

where we have denoted by the index reg the restrictions of the morphisms of diagram 7.1 to the regular elements.

Proposition 7.6.

(i) The restriction η_{reg} of η : G̃^{1} → G̃^{1P} to η^{−1}(G̃_{reg}^{1P}) = π^{−1}(G_{reg}^{1}) is a galoisian covering with Galois group W (L)^{σ}.

(ii) The restriction ζ_{reg} of ζ : Ṽ → V to ζ^{−1}(V_{reg}) is a galoisian covering with Galois group W (L)^{σ}.

Proof:

(i) Let us consider the following cartesian diagram:

η^{−1}(G_{reg}^{1P}) ⟶ G̃^{1}
↓η^{reg}  ↓η
G̃_{reg}^{1P}  ⟶ G̃^{1P}

The morphism η_{reg} is defined by (g.σ, B′) ↦ (g.σ, f(B′)), where f(B′) is the only parabolic subgroup P′ of P such that B′ ⊂ P′.

Let (g.σ, P′) be in G̃_{reg}^{1P} and let us consider the fiber at (g.σ, P′). Quasi-semi-simple regular elements of G^{1} can be obtained under conjugation by G from elements of T_{reg}^{σ}.σ, where T_{reg}^{σ} is the set of elements of T^{σ} which are regular in G^{σ}. So, we can suppose that g.σ ∈ T^{σ}.σ and that P′ = P.

Then, we have B_{g.σ}^{G} = {^{w}B; w ∈ W^{σ}}. But, if B^{P} is the set of Borel subgroups of G which are included in P, we have B^{P} = {^{w}B; w ∈ W (L)}. Hence, η_{reg}^{−1}((g.σ, P′)) ≃ B_{g.σ}^{P} = {^{w}B; w ∈ W (L)^{σ}}.

The morphism η_{reg} : G̃_{reg}^{1} → G̃_{reg}^{1P} is a finite morphism of varieties and the cardinality of its fibers is equal to |W (L)^{σ}|. It is then a flat morphism and as it is surjective, it is faithfully flat.

The action of W (L)^{σ} on π^{−1}(G_{reg}^{1}) is obtained through the isomorphism G̃_{reg}^{1} → π^{−1}(G_{reg}^{1}) of proposition 4.2 from the restriction to W (L)^{σ} of the action of W (G)^{σ} on G̃_{reg}^{1}. This is an action on the right on G̃_{reg}^{1P}. Indeed, η_{reg} is W (L)^{σ}-equivariant with respect to the action of W (L)^{σ} on G̃_{reg}^{1} ≃ π^{−1}(G_{reg}^{1}) defined by w : (g.σ, xT) ↦ (g.σ, xwT) and with respect to the trivial action of W (L)^{σ} on G̃_{reg}^{1P}, for we have f(^{w}B) = P for all w ∈ W (L)^{σ}. Moreover, if ((g.σ, xT), (g′.σ, x′T)) is in G̃_{reg}^{1} ×G̃_{reg}^{1P} G̃_{reg}^{1}, then, η_{reg}((g.σ, xT)) = η_{reg}((g′.σ, x′T)), which means that g′.σ = g.σ, that there is w ∈ W (L)^{σ} such that x′T = xwT and that ^{x}P = ^{x′}P. Hence,

G̃_{reg}^{1} ×G̃_{reg}^{1P} G̃_{reg}^{1} = {((g.σ, xT), (g.σ, xwT))|(g.σ, xT) ∈ G̃_{reg}^{1} and w ∈ W (L)^{σ}}

Thus, the map from W (L)^{σ} × G̃_{reg}^{1} to G̃_{reg}^{1} ×_{G_{reg}^{1P}} G̃_{reg}^{1} defined by (w, (g.σ, xT)) ↦ ((g.σ, xT), (g.σ, xwT)) is an isomorphism.

(ii) Let us now consider the following cartesian diagram:

ζ^{−1}(V_{reg}) = G ×^{B} (L ∩ B)_{reg}^{1} ⟵^{q̃_{reg}} G̃_{reg}^{1} = G ×^{B} B_{reg}^{1}
↓ζ_{reg}  ↓η^{reg}
V_{reg} = G ×^{P} L_{reg}^{1}  ⟵^{q_{reg}} G̃_{reg}^{1P} = G ×^{P} P_{reg}^{1}

We have B_{reg}^{1} = (L ∩ B)_{reg}^{1}U_{P} and P_{reg}^{1} = L_{reg}^{1}U_{P} for B_{reg}^{1} = T_{reg}^{1}U (theorem 2.2). Then, the morphism q_{reg} and q̃_{reg} are fibration whose fibers are isomorphic to U_{P}. Moreover, the action of W (L)^{σ} on the fibers is trivial. Hence, the action of W (L)^{σ} on G̃_{reg}^{1} induces an action of W (L)^{σ} on ζ^{−1}(V_{reg}) which is trivial and the action of W (L)^{σ} on G̃_{reg}^{1P} induces an action of W (L)^{σ} on G ×^{P} L_{reg}^{1} such that ζ_{reg} is a galoisian covering with Galois group W (L)^{σ}.

End of the proof.

We can deduce from proposition 7.6 that W (L)^{σ} acts on the local system η_{reg∗}\overline{ℚ}_{l} on G̃_{reg}^{1P} and that W (L)^{σ} acts on the local system ζ_{reg∗}\overline{ℚ}_{l} on V_{reg} = G ×^{P} L_{reg}^{1}.

As L_{reg}^{1} is a dense subset of L^{1}, the subvariety V_{reg} = G ×^{P} L_{reg}^{1} is dense in V = G ×^{P} L^{1}. Moreover, P_{reg}^{1} = L_{reg}^{1}U_{P} is dense in P^{1}, so the subvariety G̃_{reg}^{1P} = G ×^{P} P_{reg}^{1} is dense in G̃^{1P} = G ×^{P} P^{1}. Let us consider the intersection cohomology complexes IC(G̃^{1P} , η_{reg∗}\overline{ℚ}_{l}) and IC(V, ζ_{reg∗}\overline{ℚ}_{l}). As IC is a fully faithful functor and as W (L)^{σ} acts on η_{reg∗}\overline{ℚ}_{l} and on ζ_{reg∗}\overline{ℚ}_{l}), we get an action of W (L)^{σ} on these two intersection cohomology complexes.

Proposition 7.7. We have the following isomorphisms:

(i) η_{∗}\overline{ℚ}_{l}[n] ≃ IC(G̃^{1P} , η_{reg∗}\overline{ℚ}_{l})[n], in M(G̃^{1P}) where n = dim(G̃^{1P}).

(ii) ζ_{∗}\overline{ℚ}_{l}[n′] ≃ IC(V, ζ_{reg∗}\overline{ℚ}_{l})[n′], in M(V) where n′ = dim(V).

Proof: This is a consequence of the following result and of proposition 4.6:

Lemma 7.8. (i) The variety G̃^{1}^{P} is smooth and dim(G̃^{1}^{P}) = dim(G̃^{1}).

The varieties V_{u} and Ṽ_{u} are smooth and dim(V_{u}) = dim(Ṽ_{u}) = dim(P) + dim(L).

(ii) The morphisms η and ζ are small.

Proof of lemma 7.8: The proof of (i) is similar to that of proposition 7.1 (i).

Let us prove (ii): η and ζ are proper morphisms between equidimensional varieties.

28 Springer Correspondence in non connected reductive groups

Let us consider the following cartesian diagram:

G ×^{B} (L ∩ B)^{1} ⟵^{q̃} G̃^{1} = G ×^{B} B^{1}
↓ζ  ↓η
G ×^{P} L^{1}  ⟵^{q} G̃^{1P} = G ×^{P} P^{1}

The morphims q and q̃ are fibrations whose fibers are isomorphic to U_{P}.

For any integer i, we set

f(i) = dim{(x, l.σ) ∈ G ×^{P} L^{1}|dimζ^{−1}((x, l.σ)) ≥ i/2}

and

g(i) = dim{(x, p.σ) ∈ G̃^{1P} |dimη^{−1}((x, p.σ)) ≥ i/2}

For (x, p.σ) ∈ G̃^{1P}, we have η^{−1}((x, p.σ)) ≃ B_{p.σ}^{P}. Moreover, if π_{P} is the projection NG̃(P) → NG̃(L, B), we have B_{p.σ}^{P} ≃ B_{π_{P} (p.σ)}^{L}. Then, for (x, l.σ) ∈ G ×^{P} L^{1}, we have ζ^{−1}((x, l.σ)) ≃ B_{l.σ}^{P} ≃ B_{l.σ}^{L}. Hence,

g(i) = dim{(x, p.σ) ∈ G̃^{1P} |dimη^{−1}((x, p.σ)) ≥ i/2}
= dim{(x, p.σ) ∈ G̃^{1P} |dimB_{p.σ}^{P} ≥ i/2}
= dim{(x, p.σ) ∈ G̃^{1P} |dimB_{π_{P} (p.σ)}^{L} ≥ i/2}
= dim{(x, l.σ) ∈ G ×^{P} L^{1}|dimB_{l.σ}^{L} ≥ i/2} + dim(U_{P})
= f(i) + dim(U_{P})

Moreover,

f(i) = dim{(x, l.σ) ∈ G × L^{1}|dimB_{l.σ}^{L} ≥ i/2} − dim(P)
= dim{l.σ ∈ L^{1}|dimB_{l.σ}^{L} ≥ i/2} + dim(G) − dim(P)
≤ dim(L^{1}) − i + dimP
= dim(G ×^{P} L^{1}) − i

So, ζ is a small morphism and we have g(i) ≤ dim(G̃^{1P}) − i, whence η is also small.

End of the proof of lemma 7.8.

By lemma 7.8 and by proposition 4.6, η_{∗}\overline{ℚ}_{l}[n] (resp. ζ_{∗}\overline{ℚ}_{l}[n′]) verifies properties (i), (iii) and (iv) of definition 4.1 which characterize IC(G̃^{1P} , η_{reg∗}\overline{ℚ}_{l})[n] (resp. IC(G ×^{P} L^{1}, ζ_{reg∗}\overline{ℚ}_{l})[n′]).

Then, we just have to check that H^{0}η_{∗}\overline{ℚ}_{l}|G̃_{reg}^{1P} = η_{reg∗}\overline{ℚ}_{l} and that H^{0}ζ_{∗}\overline{ℚ}_{l}|_{G×^{P} L_{reg}^{1}} = ζ_{reg∗}\overline{ℚ}_{l}, which comes from the proper base change theorem.

End of the proof of proposition 7.7.

By proposition 7.7, we have an action of W (L)^{σ} on η_{∗}\overline{ℚ}_{l} and on ζ_{∗}\overline{ℚ}_{l}. Applying the proper base change theorem to η, we get η_{∗}\overline{ℚ}_{l}|_{Ũ(G^{1})^{P}} ≃ η_{u∗}\overline{ℚ}_{l}. The action of W (L)^{σ} on η_{∗}\overline{ℚ}_{l} induces then an action of W (L)^{σ} on A^{P} = η_{u∗}\overline{ℚ}_{l}. Proposition 7.9.

a) The action of W (L)^{σ} on A^{P} induces an action of W (L)^{σ} on the vector spaces V′_{(C^{L},E′)} of the decomposition (6) in proposition 7.4. We denote by ρ′_{(C^{L},E′)} the representation of W (L)^{σ} on V′_{(C^{L},E′)}.

b) The map (C^{L}, E′) ↦ ρ′_{(C^{L},E′)} can be identified with the Springer correspondence for L^{1}: if the local system E′ is the image under Φ of a L-equivariant irreducible local system F on C^{L} and if ρ_{(C^{L},F)} ∈ Irr(W (L)^{σ}) is the image of the pair (C^{L}, F) by the Springer correspondence for L^{1}, then we have ρ_{(C^{L},F)} = ρ′_{(C^{L},E′)}.

c) The action of W (L)^{σ} described in a) induces an isomorphism \overline{ℚ}_{l}[W (L)^{σ}] ≃ EndA^{P}.

Proof: a) The group W (L)^{σ} acts on A^{P} by automorphisms of A^{P}. So, it induces an action of W (L)^{σ} on the vector spaces V′_{(C^{L},E′)}.

Let us prove b). Applying theorem 6.3 to L^{1}, the Springer action of W (L)^{σ} from L^{1} on π(L)_{∗}\overline{ℚ}_{l}[dim(L)] induces an action on A(L) by restriction of π(L)_{∗}\overline{ℚ}_{l} to U(L^{1}). To compare the action of W (L)^{σ} on A^{P} and that on A(L) is equivalent to compare the action of W (L)^{σ} on η_{∗}\overline{ℚ}_{l} and that on π(L)_{∗}\overline{ℚ}_{l}.

Let us apply the proper base change theorem to the morphisms π(L) and η. We get the following isomorphims in the derived category: q^{∗}ζ_{∗}\overline{ℚ}_{l} ≃ η_{∗}\overline{ℚ}_{l} et i^{∗}ζ_{∗}\overline{ℚ}_{l} ≃ π(L)_{∗}\overline{ℚ}_{l}.

Let us check that these isomorphisms are compatible with the actions of W (L)^{σ}, the Springer action on π(L)_{∗}\overline{ℚ}_{l} and the actions on η_{∗}\overline{ℚ}_{l} and ζ_{∗}\overline{ℚ}_{l} described in propositions 7.6 and 7.7.

The functor IC is fully faithful and we have the following isomorphisms in the category of perverse sheaves, first η_{∗}\overline{ℚ}_{l}[dim(G^{1})] ≃ IC(G̃^{1P} , η_{reg∗}\overline{ℚ}_{l})[dim(G^{1})], then π(L)_{∗}\overline{ℚ}_{l}[dim(L^{1})] ≃ IC(L^{1}, π(L)_{reg∗}\overline{ℚ}_{l})[dim(L^{1})], finally ζ_{∗}\overline{ℚ}_{l}[dim(G ×^{P} L^{1})] ≃ IC(G ×^{P} L^{1}, η_{reg∗}\overline{ℚ}_{l})[dim(G ×^{P} L^{1})]. So, it is sufficient to check that q_{reg}^{∗}ζ_{reg∗}\overline{ℚ}_{l} ≃ η_{reg∗}\overline{ℚ}_{l} et i_{reg}^{∗}ζ_{reg∗}\overline{ℚ}_{l} ≃ π(L)_{reg∗}\overline{ℚ}_{l} are compatible with the actions of W (L)^{σ}.

But the actions of W (L)^{σ} on η_{reg∗}\overline{ℚ}_{l}, ζ_{reg∗}\overline{ℚ}_{l} and π(L)_{reg∗}\overline{ℚ}_{l} come from the fact that η_{reg∗}, ζ_{reg∗} and π(L)_{reg∗} are galoisian coverings with Galois group W (L)^{σ} and from the action of W (L)^{σ} on the varieties π^{−1}(G_{reg}^{1}), ζ^{−1}(V_{reg}) and π(L)^{−1}(L_{reg}^{1}). So, it is sufficient to prove that the morphisms ĩ_{reg} and q̃_{reg} are W (L)^{σ}-equivariant. The morphism q̃_{reg} is W (L)^{σ}-equivariant by definition of the action of W (L)^{σ} on G ×^{B} (L ∩ B)_{reg}^{1}. Moreover, L̃_{reg}^{1} = {(l.σ, xT) ∈ L^{1} × L/T |x^{−1}l.σx ∈ T_{reg}^{1}} and the action of W (L)^{σ} on L̃_{reg}^{1} is defined by w : (l.σ, xT) ↦ (l.σ, xwT) and the action of W (L)^{σ} on G ×^{B} (L ∩ B)_{reg}^{1} comes from that on G̃_{reg}^{1} = {(g.σ, xT) ∈ G^{1} × G/T |x^{−1}g.σx ∈ T_{reg}^{1}} where the action of W (L)^{σ} is defined by w : (g.σ, xT) ↦ (g.σ, xwT). So, ĩ_{reg} is also W (L)^{σ}-equivariant. Whence the result.

c) is a consequence of b) as, applying theorem 6.3 to L^{1}, we get \overline{ℚ}_{l}[W (L)^{σ}] ≃ EndA(L). End of the proof.

7.5. Action of W (L)^{σ} on A induced from that on A^{P}. The complexes A and A^{P} are related by the formula A = ξ_{u∗}A^{P} and the action of W (L)^{σ} on A^{P} described in 7.4 induces an action of W (L)^{σ} on A by functoriality. Let us compare this action with the action of W (L)^{σ} by restriction of the Springer action of W (G)^{σ} on A from G^{1} (proposition 7.11).

Proposition 7.10. For all pair (C^{L}, E′) such that V′_{(C^{L},E′)} ≠ 0, the direct image ξ_{u∗}IC(G ×^{P}

(\overline{C}^{L}U_{P}), E′)[dim(G ×^{P} (C^{L}U_{P}))] is a semi-simple object of M(U(G^{1})) and is decomposed as:

(7) ξ_{u∗}(IC(G ×^{P} \overline{C}^{L}U_{P} , E′)[dim(G ×^{P} (C^{L}U_{P}))]) ≃ ⨁_{(C,E)} IC(\overline{C}, E)[dim(C)] ⊗ V_{(C,E)}^{(C^{L},E′)}

where C runs over the set of unipotent G-conjugacy classes in G^{1}, E runs over the set of Gequivariant irreducible local systems on C and where V_{(C,E)}^{(C^{L},E′)} is a \overline{ℚ}_{l}-vector space whose dimension is equal to the multiplicity of the simple object corresponding to (C, E).

Proof: The perverse sheaf IC(G ×^{P} \overline{C}^{L}U_{P} , E′)[dim(G ×^{P} (C^{L}U_{P}))] is a direct summand of A^{P} [dim(U(G^{1}))]. So, ξ_{u∗}(IC(G ×^{P} \overline{C}^{L}U_{P} , E′)[dim(G ×^{P} (C^{L}U_{P}))]) is a direct summand of ξ_{u∗}A^{P} [dim(U(G^{1}))] = A[dim(U(G^{1}))]. But, A[dim(U(G^{1}))] is semi-simple in M(U(G^{1})), so all its direct summands are direct sums of simple objects of M(U(G^{1})) which are direct summands of A[dim(U(G^{1}))].

End of the proof.

If we put together the equality A = ξ_{u∗}A^{P}, the decomposition formula (6) of A^{P} and the decomposition formula (7) of the previous proposition, we get the following double decomposition of the perverse sheaf A[dim(U(G^{1})]:

(8)  A[dim(U(G^{1})] = ⨁_{(C,E)} ⨁_{(C^{L},E′)} IC(\overline{C}, E)[dim(C)] ⊗ V_{(C,E)}^{(C^{L},E′)} ⊗ V′_{(C^{L},E′)},

where C runs over the set of unipotent G-conjugacy classes of G^{1} and E runs over the set of Gequivariant irreducible local systems on C, where C^{L} runs over the set of unipotent L-conjugacy classes of L^{1} and E′ runs over the set of G-equivariant irreducible local systems on G×^{P} (C^{L}U_{P}) and where V_{(C,E)}^{(C^{L},E′)} is the \overline{ℚ}_{l}-vector space defined in the previous proposition.

Hence, the action of W (L)^{σ} on A induced by that on A^{P} is given by the action of W (L)^{σ} on the vector spaces V′_{(C^{L},E′)} which are W (L)^{σ}-modules, by proposition 7.9.

Proposition 7.11. The action of W (L)^{σ} on A = ξ_{u∗}A^{P} induced by that on A^{P} by functoriality of ξ_{u∗} is the restriction to W (L)^{σ} of the Springer action of W (G)^{σ} on A.

Proof: Let us consider the following diagram:

Ũ(G^{1}) ↪^{j̃_{u}}  G̃^{1}  ↩^{j̃_{reg}} G̃_{reg}^{1}
↓η_{u}  ↓η  ↓η^{reg}
Ũ(G^{1})^{P} ↪^{j_{u}^{P}} (G̃^{1})^{P} ↩^{j_{reg}^{P}} G̃_{reg}^{1P}
↓ξ_{u}  ↓ξ  ↓ξ_{reg}
U(G^{1}) ↪^{j_{u}} G^{1}  ↩^{j_{reg}} G_{reg}^{1}

Both actions of W (L)^{σ} on A come from actions on π_{∗}\overline{ℚ}_{l}. Let us compare the action of W (L)^{σ} on π_{∗}\overline{ℚ}_{l}[n] ≃ IC(G^{1}, π_{reg∗}\overline{ℚ}_{l})[n] which is the restriction of the Springer action of W (G)^{σ} from G^{1} (which is induced by functoriality of IC from the restriction to W (L)^{σ} of the action of W (G)^{σ} on π_{reg∗}\overline{ℚ}_{l}) and the action of W (L)^{σ} which comes from the isomorphism π_{∗}\overline{ℚ}_{l}[n] ≃ ξ_{∗}IC(G̃^{1P} , η_{reg∗}\overline{ℚ}_{l})[n].

As IC is a fully faithful functor, it is sufficient to compare the restriction to W (L)^{σ} of the action of W (G)^{σ} on π_{reg∗}\overline{ℚ}_{l} and the action of W (L)^{σ} induced by functoriality of ξ_{reg∗} from that on η_{reg∗}\overline{ℚ}_{l}.

But both these actions comes from the same action of W (L)^{σ} on G̃_{reg}^{1}, whence the result.

End of the proof.

Hence, the restriction to W (L)^{σ} of the Springer action of W (G)^{σ} from G^{1} on A is given by an action on the vector spaces V′_{(C^{L},E′)} of the double decomposition formula (7) of A. Moreover, if C is a unipotent G-conjugacy class of G^{1} and if E is a G-equivariant irreducible local system on C, such that (C, E) is associated by the Springer correspondence from G^{1} to an irreducible representation V_{(C,E)} of W (G)^{σ}, then, V_{(C,E)} is decomposed as a direct sum of W (L)^{σ}-modules as:

V_{(C,E)} = ⨁_{(C^{L},E′)} V_{(C,E)}^{(C^{L},E′)} ⊗ V′_{(C^{L},E′)},

where C^{L} runs over the set of unipotent L-conjugacy classes of L^{1} and E′ the set of G-equivariant irreducible local systems on G ×^{P} C^{L}U_{P} and where the vector spaces V_{(C,E)}^{(C^{L},E′)} are those of proposition 7.10.

7.6. Induced unipotent classes. We come to the purpose of section 7 which is assertion (vi) in proposition 7.13. Let us first recall the definition of an induced unipotent class:

Definition 7.1. ([14] page 60)

Let C^{L} be a unipotent L-conjugacy class in L.σ, then, the unipotent G-conjugacy class C is said to be obtained by induction of C^{L} from L.σ to G.σ if C ∩C^{L}U_{P} is dense in C^{L}U_{P}, where U_{P} is the unipotent radical of P. If t.σ is in C, we can choose x.σ ∈ t.σU_{P} such that C = Cl_{G}(x.σ).

Let us then recall a few properties of induced unipotent classes.

Proposition 7.12. ([14] page 60)

Let t.σ be a unipotent element of L^{1}, let x.σ be in t.σU_{P} such that C = Cl_{G}(x.σ) is obtained by inducing C^{L} = Cl_{L}(t.σ) from L.σ to G.σ. Then,

(i) dim(C_{G}(x.σ)) = dim(C_{L}(t.σ)) and dim(B_{t.σ}^{L}) = dim(B_{x.σ}^{G}).

(ii) C ∩ C^{L}U_{P} is a P -conjugacy class in NG̃(P).

(iii) C_{G}(x.σ)^{0} ⊂ P and, if g ∈ G is such that g^{−1}(x.σ)g ∈ C^{L}U_{P}, then, there is z in C_{G}(x.σ) such that ^{z}P = ^{g}P.

(iv) \overline{C} = ∪_{g∈G}^{g}(\overline{C}^{L}U_{P}).

Proposition 7.13. Let t.σ be a unipotent element of L^{1}, let x.σ be in t.σU_{P} such that C = Cl_{G}(x.σ) is obtained by inducing C^{L} = Cl_{L}(t.σ) from L.σ to G.σ. Then,

(i) ξ_{u} sends G ×^{P} (\overline{C}^{L}U_{P}) on \overline{C}.

(ii) dim(G ×^{P} (C^{L}U_{P})) = dim(C).

(iii) ξ_{u}^{−1}(C) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) is a G-orbit included and dense in G ×^{P} (C^{L}U_{P}).

(iv) ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) is finite and its cardinal is equal to [C_{G}(x.σ) : C_{P} (x.σ)].

(v) Let ρ_{(C^{L},\overline{ℚ}_{l})} denote the image of (C^{L}, \overline{ℚ}_{l}) by the Springer correspondence from L^{1}; then the multiplicity of ρ_{(C^{L},\overline{ℚ}_{l})} in H^{i}(B_{x.σ}, \overline{ℚ}_{l}) is equal to [C_{G}(x.σ) : C_{P} (x.σ)] if i = 2dim(B_{x.σ}) and is equal to zero otherwise.

(vi) If we denote by ρ_{(C,\overline{ℚ}_{l})} the image of (C, \overline{ℚ}_{l}) by the Springer correspondence from G^{1}, we have:

⟨ρ_{(C^{L},\overline{ℚ}_{l})}, ρ_{(C,\overline{ℚ}_{l})}|_{W (L)^{σ}}⟩_{W (L)^{σ}} = 1.

Proof:

(i) comes from the equality \overline{C} = ∪_{g∈G}^{g}(\overline{C}^{L}U_{P}) (proposition 7.12, (iv)).

(ii) is clear as G ×^{P} (C^{L}U_{P}) → G ×^{P} C^{L} → P is a double fibration with fibers isomorphic respectively to U_{P} and C^{L}, and as we have dim(C_{G}(x.σ)) = dim(C_{L}(t.σ)) (proposition 7.12, (i)).

(iii) As ξ_{u} is G-equivariant, any G-orbit is sent on a G-conjugacy class in U(G^{1}). The variety G×^{P} (\overline{C}^{L}U_{P}) is a union of G-orbits. Let O be a G-orbit in G×^{P} (\overline{C}^{L}U_{P}) which is sent on C, then, it is sent surjectively onto C and dim(O) ≥ dim(C). But, by (ii), dim(O) ≤ dim(G×^{P} (C^{L}U_{P})) = dim(C), so dim(O) = dim(G ×^{P} (C^{L}U_{P})). Then, as G ×^{P} (\overline{C}^{L}U_{P}) is irreducible (for \overline{C}^{L} is irreducible), we can deduce that \overline{O} = G ×^{P} (\overline{C}^{L}U_{P}) and that ξ_{u}^{−1}(C) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) = O. Hence, ξ_{u}^{−1}(C) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) is only one G-orbit.

Let us suppose that ξ_{u}^{−1}(C) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) is not included in G ×^{P} (C^{L}U_{P}). Then, as ξ_{u} is equivariant, ξ_{u}^{−1}(C) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) is included in (G ×^{P} (\overline{C}^{L}U_{P})) − (G ×^{P} (C^{L}U_{P})), which is impossible for dimension reasons.

(iv) ξ_{u}^{−1}(x.σ)∩(G×^{P} (\overline{C}^{L}U_{P})) is the closure of ξ_{u}^{−1}(x.σ)∩(G×^{P} (C^{L}U_{P})) in ξ_{u}^{−1}(x.σ) ≃ P_{x.σ}. But, we have:

ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (C^{L}U_{P})) = {(x.σ,^{g}P)|g^{−1}(x.σ)g ∈ C^{L}U_{P} }

So, we can deduce from the (iii) of proposition 7.12 that:

ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (C^{L}U_{P})) = {(x.σ,^{g}P)|g ∈ C_{G}(x.σ)}

Hence, ξ_{u}^{−1}(x.σ)∩(G×^{P} (C^{L}U_{P})) ≃ C_{G}(x.σ)/(C_{G}(x.σ)∩P) = C_{G}(x.σ)/C_{P} (x.σ), where C_{P} (x.σ) is a subgroup of C_{G}(x.σ) of finite index as we have C_{G}(x.σ)^{0} ⊂ C_{P} (x.σ).

Thus, ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (C^{L}U_{P})) is finite and its cardinal is equal to [C_{G}(x.σ) : C_{P} (x.σ)], so ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (C^{L}U_{P})) is equal to ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})).

(v) Applying the double decomposition formula (8) of A, we get:

H^{i}(B_{x.σ}, \overline{ℚ}_{l}) = ⨁_{(C,E)} ⨁_{(C^{L},E′)} H_{x.σ}^{i−2d_{C}}(IC(\overline{C}, E)) ⊗ V_{(C,E)}^{(C^{L},E′)} ⊗ V′_{(C^{L},E′)},

where E runs over the set of G-equivariant irreducible local systems on C and E′ the set of G-equivariant irreducible local systems on G×^{P} (C^{L}U_{P}), where V′_{(C^{L},E′)} is a W (L)^{σ}-module and where for all unipotent G-conjugacy class C, we set d_{C} = 1/2(dim(U(G^{1})) − dim(C)).

Whence,

H^{i}(B_{x.σ}, \overline{ℚ}_{l})^{ρ′_{(C^{L},\overline{Q}_{l})}} = ( ⨁_{(C,E)} H_{x.σ}^{i−2d_{C}}(IC(\overline{C}, E)) ⊗ V_{(C,E)}^{(C^{L},\overline{ℚ}_{l})}) ⊗ V′_{(C^{L},\overline{ℚ}_{l})}

Moreover, by proposition 7.10, we have:

ξ_{u∗}(IC(G ×^{P} (\overline{C}^{L}U_{P}), \overline{ℚ}_{l})[dim(G ×^{P} (C^{L}U_{P}))]) ≃ ⨁_{(C,E)} IC(C̄, E)[dim(C)] ⊗ V_{(C,E)}^{(C^{L},\overline{ℚ}_{l})}.

Hence,

H^{i}(B_{x.σ}, \overline{ℚ}_{l})^{ρ′_{(C^{L},\overline{Q}_{l})}} = H^{i−2d_{C^{L}}} (ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})), IC(G ×^{P} (\overline{C}^{L}U_{P}), \overline{ℚ}_{l})) ⊗ V′_{(C^{L},\overline{ℚ}_{l})}.

But, by (iv), ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) is a finite set of cardinality [C_{G}(x.σ) : C_{P} (x.σ)]. So, H^{i−2d_{C^{L}}} (ξ^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})), IC(G ×^{P} (\overline{C}^{L}U_{P}), \overline{ℚ}_{l})) is equal to zero except for i = 2d_{C^{L}}, in which case, it is a vector space of dimension [C_{G}(x.σ) : C_{P} (x.σ)].

Moreover, according to (ii),

d_{C^{L}} = 1/2(dim(U(G^{1})) − dim(G ×^{P} (C^{L}U_{P})))
= 1/2(dim(U(G^{1})) − dim(C))
= dim(B_{x.σ})

Consequently, the multiplicity of ρ′_{(C^{L},\overline{ℚ}_{l})} in H^{i}(B_{x.σ}, \overline{ℚ}_{l}) is equal to [C_{G}(x.σ) : C_{P} (x.σ)] if i = 2dim(B_{x.σ}) and is equal to zero otherwise. Hence, we have ρ′_{(C^{L},\overline{ℚ}_{l})} = ρ_{(C^{L},\overline{ℚ}_{l})} (proposition 7.9, b)).

Whence (v).

(vi) By the proof of (iv), we have ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})) = \{(x.σ,^{x}P)|g ∈ C_{G}(x.σ)\}. Moreover, we have C_{G}(x.σ)^{0} ⊂ C_{P} (x.σ) (proposition 7.12, (iii)). Hence, A_{G}(x.σ)

= C_{G}(x.σ)/C_{G}(x.σ)^{0} acts transitively on ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})). Thus, the dimension of H^{0}(ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})), \overline{ℚ}_{l})^{A_{G}(x.σ)} is equal to 1.

Moreover, according to the proof of (v),

(9)  H^{2d_{C^{L}}} (B_{x.σ})^{ρ_{(C^{L},\overline{Q}_{l})}} = H^{0}(ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})), \overline{ℚ}_{l}) ⊗ V_{(C^{L},\overline{ℚ}_{l})}

Then, by proposition 7.11, the action of W (L)^{σ} on H^{2d_{C^{L}}} (B_{x.σ}) induced from that on A^{P} by functoriality of ξ_{u∗} can be identified with the restriction to W (L)^{σ} of the Springer action of W^{σ} from G^{1}. Hence, according to corollary 6.4,

(10)  H^{2d_{C^{L}}} (B_{x.σ}, \overline{ℚ}_{l}) = ⨁_{ρ∈A_{G}(x.σ)^{∧}} V_{(C,E_{ρ})} ⊗ ρ^{∨}

So, we can deduce from equalities (9) and (10) that:

(V_{(C,\overline{ℚ}_{l})})^{ρ_{(C^{L},\overline{Q}_{l})}} = (H^{0}(ξ_{u}^{−1}(x.σ) ∩ (G ×^{P} (\overline{C}^{L}U_{P})), \overline{ℚ}_{l}) ⊗ V_{(C^{L},\overline{ℚ}_{l})})^{A_{G}(x.σ)},

and we get dim((V_{(C,\overline{ℚ}_{l})})^{ρ_{(C^{L},\overline{Q}_{l})}}) = 1.

End of the proof.

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K.S.: LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, F–80000 Amiens, France.

E-mail address: karine.sorlin@wanadoo.fr