Mathematische Zeitschrift manuscript No. (will be inserted by the editor)

Springer correspondence for disconnected groups

Gunter Malle^{1}, Karine Sorlin^{2}

1 Fachbereich Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40,

  D–34132 Kassel, Germany. e-mail: mallemathematik.uni-kassel.de

2 LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, F–80000 Amiens,

  France. e-mail: karine.sorlin@wanadoo.fr

Received: date / Revised version: date – \begin{matrix}c \\ ◯ \end{matrix} Springer-Verlag 2003

Abstract The Springer correspondence is a map from the set of unipotent conjugacy classes of a reductive algebraic group to the set of irreducible complex characters of the Weyl group. Here, we determine this map explicitly in the case of disconnected classical algebraic groups.

Mathematics Subject Classification (2000): 20G05, 20C33

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Disconnected groups and Springer correspondence . . . . . . . . . . . . 2 3 Reduction results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Some combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Restricted Springer correspondence for GL_{n}: 2 . . . . . . . . . . . . . . 19 6 Restricted Springer correspondence for GO_{2n} . . . . . . . . . . . . . . . 27

1 Introduction

We determine the Springer correspondence for disconnected reductive groups of classical types. The Springer correspondence was originally introduced by Tonny Springer as an injective map from the set of unipotent conjugacy classes of a connected reductive algebraic group G over an algebraic closed field to the set of irreducible characters of the Weyl group W (G) of G. This correspondence and its later generalization by Lusztig (see [10] for references) turned out to be very important for the computation of character values of reductive groups over finite fields via Shoji’s and Lusztig’s algorithm. In the connected case, the Springer correspondence is explicitly known in all cases.

The second author showed in [12] that the definition of the Springer correspondence can be extended to the case of disconnected groups G̃ obtained from a connected reductive algebraic group G by adjoining an automorphism σ of order a power of the characteristic. Here, the correspondence is an injection from the set of unipotent conjugacy classes in the coset Gσ to the irreducible characters of the fixed points W (G)^{σ} under σ of the Weyl group of G (see Section 2.2 for precise definitions).

It is not yet known whether the correspondence plays a role in the computation of character values of such disconnected groups, but experiments on small examples show that a straightforward generalization of Lusztig’s algorithm, with this new correspondence as input, yield the values of the Green functions computed by the first author [8] by different methods.

It turns out that two properties proved by the second author in [12], which we recall in Theorems 1 and 2, together with three reduction results, Theorems 3, 4 and 5, suffice to determine the Springer correspondence for disconnected groups of classical type. For the exceptional types, the Green functions were already obtained by the first author in [9].

Our proof makes essential use of results of Spaltenstein [13] on unipotent classes in disconnected groups, in particular the description of induction of classes (which we are able to make more explicit in Theorems 6 and 8) and a formula for the dimension of the variety of Borel subgroups containing a given unipotent element (see Propositions 4(b) and 5(b)).

2 Disconnected groups and Springer correspondence

2.1 Disconnected groups

In this article we will be interested in disconnected reductive groups of the form a semidirect product G.⟨σ⟩, where G is a connected reductive group and σ an automorphism of G of finite order.

Let us recall a few results about the automorphisms of connected reductive groups. Let G be a connected reductive group defined over an algebraically closed field K of characteristic p. An automorphism σ of G is called quasi-semisimple if there exists a σ-stable pair (T, B) where T is a maximal torus included in a Borel subgroup B of G. According to [3, Thm. 1.8], when σ is quasi-semisimple, the group G^{σ} of fixed points of G under σ is a reductive group, and there is a bijection between σ-stable maximal tori T of G contained in a σ-stable Borel subgroup of G and maximal tori of G^{σ} defined by T ↦ T ∩ (G^{σ})^{0}. Moreover, also according to [3, Thm. 1.33], when σ is quasi-semisimple unipotent, then G^{σ} is connected and the following equivalent properties hold true (σ is then said to be quasi-central):

(i) dim(G^{σ}) ≥ dim(G^{σ′}) for any quasi-semisimple 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 B of G and Borel subgroups of G^{σ} defined by B ↦ B ∩ (G^{σ})^{0}.

(iii) The Weyl group W (G^{σ}) of G^{σ} is isomorphic to the group W (G)^{σ} of fixed points under σ of the Weyl group of G.

Till the end of this section, we consider a disconnected reductive group Ĝ = G.⟨σ⟩, where G is a connected reductive group and σ is a unipotent quasi-semisimple automorphism of G. We say that an element of Ĝ is quasisemisimple if it induces a quasi-semisimple automorphism of G.

Regular quasi-semisimple elements of Gσ play an important rôle in the Springer correspondence for Gσ. An element x of Ĝ is said to be regular if it is regular in the connected component Gx of Ĝ which it belongs to, that is to say, if the dimension of its centralizer C_{G}(x) in G is minimal in Gx. By [12, Thm. 2.2] regular quasi-semisimple elements are open dense in Gσ.

2.2 Springer correspondence

In this section, we recall the construction of the Springer correspondence for Gσ.

Let us consider the following variety:

˜ (Gσ) := {(gσ, B) ∈ Gσ × B^{G}|B ∈ B_{gσ}^{G}},

where B^{G} is the variety of Borel subgroups of G and B_{gσ}^{G} the variety of Borel subgroups of G which are stable under conjugation by gσ. Let π : ˜ (Gσ) → Gσ be the projection on the first factor. Let (Gσ)_{reg} be the set of regular quasi-semisimple elements of Gσ and π_{0} : π^{−1}((Gσ)_{reg}) → (Gσ)_{reg} the restriction of π to π^{−1}((Gσ)_{reg}).

The group W (G)^{σ} of σ-stable elements of the Weyl group W (G) of G acts on π^{−1}((Gσ)_{reg}) from the right and π_{0} : π^{−1}((Gσ)_{reg}) → (Gσ)_{reg} is W (G)^{σ}-equivariant with respect to the trivial action of W (G)^{σ} on (Gσ)_{reg}. It is a Galois covering with group W (G)^{σ}. Thus W (G)^{σ} acts on L := (π_{0})_{∗}\overline{ℚ}_{l}, the direct image of the constant sheaf \overline{ℚ}_{l} on π^{−1}((Gσ)_{reg}) under the map π_{0}, where l denotes a prime different from p.

Let us now consider the intersection cohomology complex IC(Gσ, L) on \overline{(Gσ)}_{reg} = Gσ associated with L. As IC is a faithful functor from the category of local systems on (Gσ)_{reg} to the category M(Gσ) of perverse sheaves on Gσ, the action of W (G)^{σ} on L induces an action of W (G)^{σ} on IC(Gσ, L)[dim(G)].

Finally, IC(Gσ, L)[dim(G)] is isomorphic to π_{∗}\overline{ℚ}_{l}[dim(G)] in M(Gσ), where π_{∗}\overline{ℚ}_{l} is the direct image of the constant sheaf \overline{ℚ}_{l} on ˜ (Gσ) under the map π. Thus, W (G)^{σ} acts on π_{∗}\overline{ℚ}_{l}.

Let us now consider the restriction S of π_{∗}\overline{ℚ}_{l} to the variety U(Gσ) of unipotent elements of Gσ. Then, S[dim(U(Gσ))] is a semisimple perverse sheaf on U(Gσ) and by [12, Thm. 6.2] it decomposes as

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

where C runs over the unipotent conjugacy classes of Gσ, E runs over the G-equivariant irreducible local systems on C, 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). Moreover, the action of W (G)^{σ} on S induces an action of W (G)^{σ} on all V_{C,E}. By [12, Thm. 6.3(ii)] all non-zero W (G)^{σ}-modules V_{C,E} are irreducible, and the map (C, E) ↦ V_{C,E} induces a bijection from {(C, E)|V_{C,E} ≠ 0} to Irr(W (G)^{σ}) which is the Springer correspondence for Gσ.

For any unipotent class C in Gσ, the \overline{ℚ}_{l}-vector space V_{C,\overline{ℚ}_{l}} is different from zero [12, Cor. 6.5]. The Springer correspondence thus induces an injection from the set of unipotent conjugacy classes of Gσ to the set of irreducible representations of W^{σ} defined by C ↦ V_{C,\overline{ℚ}_{l}}, which we call the restricted Springer correspondence. It is this map which we shall determine in this paper for groups of classical type.

2.3 Induction of unipotent classes

Let L be a σ-stable Levi subgroup of a σ-stable parabolic subgroup P of G. Let C be a unipotent L-conjugacy class of Lσ. Then, there exists a unique G-conjugacy class C̃ in the coset Gσ such that C̃ ∩ CU_{P} is dense in CU_{P}, where U_{P} is the unipotent radical of P. The class C̃ is then called the unipotent class obtained by inducing the class C from Lσ to Gσ. In this situation we write C̃ = Ind_{L}^{G}(C).

Proposition 1. Let C be a unipotent L-conjugacy class of Lσ and let C̃ = Ind_{L}^{G}(C) be the unipotent G-conjugacy class of Gσ obtained by inducing the class C from Lσ to Gσ. Let u be in C and let ũ be in C̃. Then

dim(B_{u}^{L}) = dim(BG ; ũ).

Proof. This is Proposition II.3.2 of [13]. ⊓ ⊔

If C is the quasi-semisimple unipotent class of Lσ, then C̃ is called the Richardson class associated to P and contained in Gσ. We have the following transitivity:

Proposition 2. Let L be a σ-stable Levi subgroup of a σ-stable parabolic subgroup P of G and let L′ be a σ-stable Levi subgroup of a σ-stable parabolic subgroup P′ of L. Then for every unipotent L′-conjugacy class C′ of L′σ we have

Ind_{L}^{G} (Ind_{L′}^{L}(C′)) = Ind_{L′}^{G}(C′).

Proof. This comes from [13, II.3.6]. ⊓ ⊔ 2.4 j-induction

We recall the Macdonald-Lusztig-Spaltenstein induction, also known as jinduction, in finite Coxeter groups. Let W be a finite Coxeter group and V a real vector space on which W acts as a reflection group. Then for every d ≥ 0 the space S^{d}(V) of homogeneous symmetric polynomials on V of degree d is naturally a W module. For an irreducible complex character χ of W let b(χ) denote the minimal d such that χ occurs in S^{d}(V). It is known that such a d always exists (see [5, 5.2]).

Let W′ be a reflection subgroup of W and ψ ∈ Irr(W′). Then the binvariant of any constituent χ of Ind_{W′}^{W}(ψ) satisfies b(χ) ⩾ b(ψ) [5, Lemma 5.2.5]. The Macdonald-Lusztig-Spaltenstein induction of ψ is defined as

j_{W′}^{W}(ψ) := ∑_{χ∈Irr(W):b(χ)=b(ψ)}⟨Ind_{W′}^{W}(ψ), χ⟩ χ.

That is, j_{W′}^{W}(ψ) consists of those constituents χ of Ind_{W′}^{W}(ψ) with b-invariant equal to b(ψ). Since any constituent of Ind_{W′}^{W}(ψ) has b-invariant at least b(ψ), we see that j-induction is transitive. It is known moreover that, in Coxeter groups of type B_{m}, the j-induction sends irreducibles to irreducibles (see Proposition 3(a)).

2.5 Some properties of the Springer correspondence

We recall here two properties of the Springer correspondence for Gσ which were proved by the second author [12]. For all unipotent element u ∈ Gσ, let d_{u} denote the dimension of B_{u}^{G}.

Theorem 1. Let C be the unipotent conjugacy class of u ∈ Gσ, let χ_{u} be the irreducible representation of W (G)^{σ} corresponding to the pair (C, \overline{ℚ}_{l}) by the Springer correspondence for Gσ. Then

d_{u} = b(χ_{u}).

Proof. This is Proposition 6.7(ii) in [12]. ⊓ ⊔

Theorem 2. Let L be a σ-stable Levi subgroup of a σ-stable parabolic subgroup P of G. Let C be the unipotent class of u ∈ Lσ and let ũ ∈ C̃ = Ind_{L}^{G}(C).

Let χ_{u} be the irreducible representation of W (L)^{σ} which corresponds to (C, \overline{ℚ}_{l}) by the Springer correspondence for Lσ.

Let χũ be the irreducible representation of W (G)^{σ} which corresponds to (C̃, \overline{ℚ}_{l}) by the Springer correspondence for Gσ. Then,

⟨Res_{W (L)^{σ}}^{W (G)^{σ}}(χũ), χ_{u}⟩ ≠ 0 .

Proof. This is Proposition 7.13(vi) in [12]. ⊓ ⊔

The preceding results can be conveniently reformulated in terms of jinduction.

Corollary 1. Let L be a σ-stable Levi subgroup of a σ-stable parabolic subgroup of G. Let C be the unipotent class of u ∈ Lσ and let ũ ∈ C̃ = Ind_{L}^{G}(C).

Let χ_{u}, χũ be the Springer correspondents of (C, \overline{ℚ}_{l}), (C̃, \overline{ℚ}_{l}) respectively. Assume that j_{W (L)^{σ}}^{W (G)^{σ}}(χ_{u}) is irreducible. Then

χũ = j_{W (L)^{σ}}^{W (G)^{σ}}(χ_{u}).

Proof. By Theorem 2, χũ is an irreducible component of Ind_{W (L)^{σ}}^{W (G)^{σ}}(χ_{u}). Moreover, as C̃ is the unipotent class of Gσ obtained by inducing C from Lσ to Gσ, we have dũ = d_{u} by Proposition 1. According to Theorem 1, we have d_{u} = b(χ_{u}) and dũ = b(χũ), so b(χ_{u}) = b(χũ). Thus indeed by definition, χũ occurs with non-zero multiplicity in j_{W (L)^{σ}}^{W (G)^{σ}}(χ_{u}). ⊓ ⊔

3 Reduction results

In this section we show that in order to determine the Springer correspondence for disconnected groups, it is sufficient to consider the case where the connected component is simple.

Theorem 3. Assume that G = G_{1} × G_{2} is a direct product and σ stabilizes both factors. Let CU(Gσ) be the set of unipotent classes of Gσ and for i = 1, 2, let CU(G_{i}σ) be the set of unipotent classes of G_{i}σ. Then there is a natural bijection between CU(G_{1}σ) × CU(G_{2}σ) and CU(Gσ). Moreover, the Springer correspondence commutes with this map.

Proof. The map CU(G_{1}σ) × CU(G_{2}σ) → CU(Gσ) defined by (C_{1}, C_{2}) ↦ C_{1}×C_{2} is clearly a bijection. Similarly, there is a bijection between B^{G_{1}}×B^{G_{2}} and B^{G} defined by (B_{1}, B_{2}) ↦ B_{1} × B_{2}.

The variety ˜ (Gσ), defined in 2.2, is the direct product ˜ (G_{1}σ) × ˜ (G_{2}σ), where, for i = 1, 2, ˜ (G_{i}σ) is defined by:

˜ (G_{i}σ) := {(gσ, B) ∈ G_{i}σ × B^{G_{i}}|B ∈ B_{gσ}^{G_{i}}},

where B^{G_{i}} is the variety of Borel subgroups of G_{i} and B_{gσ}^{G_{i}} the variety of Borel subgroups of G_{i} which are stable under conjugation by gσ and the projection π : ˜ (Gσ) → Gσ is the product morphism π_{1} × π_{2}. Then, π_{∗}\overline{ℚ}_{l} is the external product π_{1∗}\overline{ℚ}_{l} ⊠ π_{2∗}\overline{ℚ}_{l} of the direct images π_{i∗}\overline{ℚ}_{l} of the constant sheaves \overline{ℚ}_{l} on ˜ (G_{i}σ) under the maps π_{i} (for the external product commutes with the direct image functor [1, 4.2.7]) and the constant sheaf \overline{ℚ}_{l} on π^{−1}(Gσ) = π_{1}^{−1}(G_{1}σ) × π_{2}^{−1}(G_{2}σ) is the external product \overline{ℚ}_{l} ⊠ \overline{ℚ}_{l}.

As W (G) = W (G_{1}) × W (G_{2}) and as each W (G_{i}) is σ-stable, W (G)^{σ} = W (G_{1})^{σ}×W (G_{2})^{σ}. So, the Springer action of W (G)^{σ} on π_{∗}\overline{ℚ}_{l} is the product of the Springer action of W (G_{1})^{σ} on π_{1∗}\overline{ℚ}_{l} and of the Springer action of W (G_{2})^{σ} on π_{2∗}\overline{ℚ}_{l}.

The variety U(Gσ) is the direct product U(G_{1}σ)×U(G_{2}σ), where U(G_{i}σ) is the variety of the unipotent elements of G_{i}σ. Let S_{i} be the restriction of π_{i∗}\overline{ℚ}_{l} to U(G_{i}σ). Then, S_{i}[dim(U(G_{i}σ))] is a semisimple perverse sheaf on U(G_{i}σ) and it decomposes as

S_{i}[dim(U(G_{i}σ))] ≃ ⨁_{(C_{i},E_{i})} V_{C_{i},E_{i}} ⊗ IC(\overline{C_{i}}, E_{i})[dim(C_{i})]

where C_{i} runs over the unipotent conjugacy classes of G_{i}σ, E_{i} runs over the G_{i}-equivariant irreducible local systems on C_{i}, the \overline{ℚ}_{l}-vector spaces V_{C_{i},E_{i}} are irreducible representations of W (G_{i})^{σ} (when V_{C_{i},E_{i}} ≠ 0) and the map from {(C_{i}, E_{i})|V_{C_{i},E_{i}} ≠ 0} to Irr(W (G_{i})^{σ}) defined by (C_{i}, E_{i}) ↦ V_{C_{i},E_{i}} is the Springer correspondence for G_{i}σ.

But now denote by S the restriction of π_{∗}\overline{ℚ}_{l} to U(Gσ), like in 2.2. Then, S is the external product S_{1} ⊠ S_{2} (for the external product commutes with the reciprocal image functor, [1, 4.2.7]) and W (G)^{σ} acts on S via the actions of W (G_{1})^{σ} on S_{1} and the action of W (G_{2})^{σ} on S_{2}.

We set dim(U(Gσ)) = n and dim(U(G_{i}σ)) = n_{i} for i = 1, 2. We have S[n] = (S_{1} ⊠ S_{2})[n] in M(U(Gσ)). So we get two decompositions of S[n] as a semisimple perverse sheaf. One, given by 2.2, induces the Springer correspondence for Gσ:

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

where C runs over the unipotent conjugacy classes of Gσ, E runs over the G-equivariant irreducible local systems on C, and the other given by the decomposition of the perverse sheaves S_{i}[n_{i}]:

S[n] ≃ ⨁_{(C_{1},E_{1})} ⨁_{(C_{2},E_{2})} V_{C_{1},E_{1}}⊗V_{C_{2},E_{2}}⊗(IC(\overline{C_{1}}, E_{1})[dim(C_{1})]⊠IC(\overline{C_{2}}, E_{2})[dim(C_{2})]).

So the theorem will be proved if we prove the following lemma:

Lemma 1. Let C be a unipotent class of Gσ and let E be an irreducible Gequivariant local system on C, then, for i = 1, 2, there is a unipotent class C_{i} of G_{i}σ and an irreducible G_{i}-equivariant local system on C_{i} such that C = C_{1} × C_{2} and E = E_{1} ⊠ E_{2}. Moreover, we have

IC(\overline{C}, E)[dim(C)] = IC(\overline{C_{1}}, E_{1})[dim(C_{1})] ⊠ IC(\overline{C_{2}}, E_{2})[dim(C_{2})]

Proof. The first part of the lemma is clear, let us prove the last equality.

Let us set d = dim(C) and d_{i} = dim(C_{i}) for i = 1, 2. We denote by j_{i} the inclusion C_{i} ↪ \overline{C_{i}} for i = 1, 2 and we set j = j_{1} × j_{2}. Then, IC(\overline{C_{i}}, E_{i})[d_{i}] is the perverse extension j_{i!∗}E_{i}[d_{i}] of E_{i}[d_{i}] and IC(\overline{C}, E)[d] = j_{!∗}E[d].

According to [1, 4.2.8], ⊠ is a t-exact bifunctor. Then, it induces an exact bifunctor of perverse sheaves and

j_{1!∗}E_{1}[d_{1}]⊠j_{2!∗}E_{2}[d_{2}] = im(^{p}j_{1!}E_{1}[d_{1}]⊠^{p}j_{2!}E_{2}[d_{2}] → ^{p}j_{1∗}E_{1}[d_{1}]⊠^{p}j_{2∗}E_{2}[d_{2}]).

But the bifunctor ⊠ commutes with the usual operations by [1, 4.2.7]. Hence,

pj_{1!}E_{1}[d_{1}] ⊠ ^{p}j_{2!}E_{2}[d_{2}] = ^{p}j_{!}E[d] and ^{p}j_{1∗}E_{1}[d_{1}] ⊠ ^{p}j_{2∗}E_{2}[d_{2}] = ^{p}j_{∗}E[d].

So, we get j_{1!∗}E_{1}[d_{1}] ⊠ j_{2!∗}E_{2}[d_{2}] = j_{!∗}E[d]. ⊓ ⊔

This completes the proof of Theorem 3. ⊓ ⊔

Theorem 4. Let G = G_{1} × ⋯ × G_{r} be a direct product of r isomorphic copies of a connected reductive group over an algebraically closed field of characteristic p, where r is a power of p. Let σ be an automorphism of G_{1} such that σ^{r} is quasi-central unipotent. We also denote by σ the automorphism of G defined by (g_{1}, g_{2}, …, g_{r}) ↦ (σ(g_{2}), …, σ(g_{r}), σ(g_{1})). Then, the map

ϕ : Gσ → G_{1}σ^{r}, (u_{1}, …, u_{r})σ ↦ u_{1} ⋯ u_{r}σ^{r} ,

induces a natural bijection between the unipotent classes of Gσ and the unipotent classes of G_{1}σ^{r}. Moreover, the Springer correspondence commutes with this bijection.

Proof. Let us first show that ϕ induces a bijection between the conjugacy classes of Gσ and those of G_{1}σ^{r}. Let g = (g_{1}, …, g_{r}) and x = (x_{1}, …, x_{r}) be in G, then

xgσx^{−1} = (x_{1}g_{1}σ(x_{2})^{−1}, …, x_{r−1}g_{r−1}σ(x_{r})^{−1}, x_{r}g_{r}σ(x_{1})^{−1}),

and

ϕ(xgσx^{−1}) = x_{1}g_{1}σ(g_{2}) … σ^{r−1}(g_{r})σ^{r}(x_{1})^{−1}σ^{r} = x_{1}ϕ(gσ)x_{1}^{−1}.

So, ϕ induces a map Φ : C ↦ ϕ(C) from the conjugacy classes of Gσ to those of G_{1}σ^{r}. Let us prove that this is a bijection. Let g = (g_{1}, …, g_{r}) be in G. If we set x = (1, (g_{2} … σ^{r−2}(g_{r}))^{−1}, (g_{3} … σ^{r−3}(g_{r}))^{−1}, …, g_{r}^{−1}) then we have xgσx^{−1} = (g_{1}σ(g_{2}) … σ^{r−1}(g_{r}), 1, …, 1)σ. So, Φ is a bijection whose inverse associates to the conjugacy class of gσ^{r} in G_{1}σ^{r} the conjugacy class of (g, 1, …, 1)σ in Gσ.

The morphism ϕ may also be defined by gσ ↦ (gσ)^{r}. Let g be in G. As r is a power of the characteristic p of K, ϕ(gσ) is unipotent if and only if gσ is unipotent.

Thus, Φ : C ↦ ϕ(C) induces a bijection between unipotent conjugacy classes in Gσ and those in G_{1}σ^{r}.

Lemma 2. Let C be a unipotent conjugacy class of Gσ. Then we have:

dim U(Gσ) − dim(C) = dim U(G_{1}σ^{r}) − dim(ϕ(C)).

Proof. We have

dim U(Gσ) = dim(Gσ) − rank(G^{σ}) = dim(Gσ) − rank((G_{1})^{σ^{r}})

and dim U(G_{1}σ^{r}) = dim(G_{1}) − rank((G_{1})^{σ^{r}}). Thus, the equality in the lemma is equivalent to dim(C_{G}(gσ)) = dim(C_{G_{1}}(ϕ(gσ))), where gσ ∈ C. But, for gσ = (g_{1}, …, g_{r})σ ∈ C, the centralizer C_{G}(gσ) =

{(x, g_{2} … σ^{r−2}(g_{r})σ^{r−1}(x)(g_{2} … σ^{r−2}(g_{r}))^{−1}, …, g_{r}σ(x)g_{r}^{−1})|x ∈ C_{G_{1}}(ϕ(gσ))}

is clearly isomorphic to C_{G_{1}}(ϕ(gσ)), whence the result. ⊓ ⊔

We will use the same notations as in 2.2 for Gσ. As G^{σ} is isomorphic to (G_{1})^{σ^{r}}, the Weyl group W (G^{σ}) of G^{σ} is isomorphic to the Weyl group W ((G_{1})^{σ^{r}}) of (G_{1})^{σ^{r}}.

Let us introduce the following notations for G_{1}:

G̃_{1} = {(gσ^{r}, B) ∈ G_{1}σ^{r} × B^{G_{1}}|B ∈ B_{gσ^{r}}^{G^{1}}},

where B^{G_{1}} is the variety of Borel subgroups of G_{1}. Let π′ : G̃_{1} → G_{1}σ^{r} be the projection on the first factor. Let (G_{1}σ^{r})_{reg} be the set of regular quasisemisimple elements of G_{1}σ^{r} and π′_{0} : π^{′−1}((G_{1}σ^{r})_{reg}) → (G_{1}σ^{r})_{reg} the restriction of π′ to π^{′−1}((G_{1}σ^{r})_{reg}). Let us consider the following cartesian diagram:

π^{′−1}((G_{1}σ^{r})_{reg}) ⟶^{ĩ′} G̃_{1}
↓π′_{0}  ↓π′
(G_{1}σ^{r})_{reg}  ⟶^{i′} G_{1}σ^{r}

Let L′ be the local system (π′_{0})_{∗}\overline{ℚ}_{l} on (G_{1}σ^{r})_{reg}. According to 2.2, we have an action of the Weyl group W ((G_{1})^{σ^{r}}) on L′, and the fact that IC(G_{1}σ^{r}, L′)[n′] is isomorphic to π′_{∗}\overline{ℚ}_{l}[n′] in M(G_{1}σ^{r}) (with n′ = dim(G_{1})) induces a Springer action of W ((G_{1})^{σ^{r}}) on π′_{∗}\overline{ℚ}_{l}.

Let us consider the following commutative diagram

˜ (Gσ) ⟶^{ϕ̃} ˜ (G_{1})
↓π  ↓π′
Gσ ⟶^{ϕ} G_{1}σ^{r}

where ϕ̃ is defined by (gσ, B) ↦ (ϕ(gσ), p_{1}(B)) where p_{1} : G = G_{1} × … × G_{r} → G_{1} is the projection on the first factor.

Lemma 3. The map ϕ̃ is well-defined and the diagram above is cartesian.

Proof. Let gσ = (g_{1}, …, g_{r})σ be in Gσ and let B = B_{1} × … × B_{r} be a Borel subgroup of G which is stable under conjugation by gσ. Then, the Borel subgroup B_{1} of G_{1} is stable under conjugation by ϕ(gσ) = g_{1}σ(g_{2}) … σ^{r−1}(g_{r}). Hence, ϕ̃ is well-defined. Moreover, the map p_{1} induces a morphism of varieties f from B_{gσ}^{G} to B_{ϕ(gσ)}^{G_{1}} defined by B ↦ p_{1}(B). Let C be the conjugacy class of gσ. By Lemma 2, we have dim U(Gσ) − dim(C) = dim U(G_{1}σ^{r}) − dim(ϕ(C)). But, by [13, II.10.15], we have 2 dim B_{gσ}^{G} = dim(G) − rank(G^{σ}) − dim(C) = dim U(Gσ) − dim(C) and 2 dim B_{ϕ(gσ)}^{G_{1}} = dim U(G_{1}σ^{r}) − dim(ϕ(C)). Then, we have dim B_{gσ}^{G} = dim B_{ϕ(gσ)}^{G_{1}} and f is

an isomorphism. Hence, ˜ (Gσ) is isomorphic to (Gσ) ×_{(G_{1}σ^{r})} ˜ (G_{1}) and the diagram above is cartesian. ⊓ ⊔

By the proper base change theorem, we have ϕ^{∗}π′_{∗}\overline{ℚ}_{l} ≃ π_{∗}ϕ̃^{∗}\overline{ℚ}_{l} = π_{∗}\overline{ℚ}_{l}. Let us show that this isomorphism is W ((G_{1})^{σ^{r}})-equivariant for the Springer actions on π′_{∗}\overline{ℚ}_{l} and on π_{∗}\overline{ℚ}_{l}.

Let us consider the following commutative diagram:

(Gσ)_{reg} ⟶^{ϕ_{reg}} (G_{1}σ^{r})_{reg} ↓j_{reg}  ↓j′_{reg} Gσ ⟶^{ϕ} G_{1}σ^{r} ,

where ϕ_{reg} is the restriction of ϕ to (Gσ)_{reg}.

Lemma 4. The image of (Gσ)_{reg} by the morphism ϕ is (G_{1}σ^{r})_{reg}.

Proof. Let T := T_{1} × … × T_{r} be a σ-stable maximal torus of G included in a σ-stable Borel subgroup B = B_{1} × … × B_{r} of G. Then, T_{1} is a σ^{r}-stable maximal torus of G_{1} included in the σ^{r}-stable Borel subgroup B_{1} of G_{1} and we have T^{σ} = {(t_{1}, σ^{−1}(t_{1}), …, σ^{r−1}(t_{1}))|t_{1} ∈ (T_{1})^{σ^{r}}} ≃ (T_{1})^{σ^{r}}.

Let gσ be a quasi-semisimple regular element of Gσ. Let us prove that ϕ(gσ) is regular semisimple in G_{1}σ^{r}. By [12, 2.4] there exists an element t = (t_{1}, σ^{−1}(t_{1}), …, σ^{r−1}(t_{1})) ∈ T^{σ} which is regular in G^{σ}, such that gσ is conjugate to tσ under G. Moreover, ϕ(gσ) and ϕ(tσ) are conjugate under G_{1}. So, it is sufficient to prove that ϕ(tσ) = (t_{1})^{r}σ^{r} is quasi-semisimple and regular in G_{1}σ^{r}. It is quasi-semisimple as it stabilizes the pair (T_{1}, B_{1}). Let us prove it is regular in G_{1}σ^{r}.

As t ∈ T^{σ} is semisimple regular in G^{σ}, we have α(t_{1}) ≠ 1 for all roots α of (G_{1})^{σ^{r}} relative to (T_{1})^{σ^{r}} by [2, 14.4]. But, r is a power of the characteristic p of K. So we have α(t_{1}^{r}) = α(t_{1})^{r} ≠ 1, for all roots α of (G_{1})^{σ^{r}} relative to (T_{1})^{σ^{r}} and (t_{1})^{r} is regular in (G_{1})^{σ^{r}} by [2, 14.4]. Hence, by [12, 2.4], ϕ(tσ) = (t_{1})^{r}σ^{r} is regular in G_{1}σ^{r}. ⊓ ⊔

As π′_{∗}\overline{ℚ}_{l}[n′] ≃ IC(G_{1}σ^{r}, L′)[n′] (where n′ = dim(G_{1})) and π_{∗}\overline{ℚ}_{l}[n] ≃ IC(Gσ, L)[n] (where n = dim(Gσ)), and as IC is a fully faithful functor from the category of local systems on (G_{1}σ^{r})_{reg} (resp. on (Gσ)_{reg}) to the category of perverse sheaves on G_{1}σ^{r} (resp. on Gσ), it is sufficient to check that the isomorphism ϕ_{reg}^{∗}π′_{0∗}\overline{ℚ}_{l} ≃ π_{0∗}\overline{ℚ}_{l} (obtained by applying the functor j_{reg}^{∗} to the previous isomorphism) is W ((G_{1})^{σ^{r}})-equivariant.

Let us consider the following cartesian diagram:

π^{−1}((Gσ)_{reg}) ⟶^{ϕ̃_{reg}} π^{′−1}((G_{1}σ^{r})_{reg}) ↓π_{0}  ↓π′_{0} (Gσ)_{reg}  ⟶^{ϕ_{reg}} (G_{1}σ^{r})_{reg} ,

where ϕ̃_{reg} is the restriction of ϕ̃ to π^{−1}((Gσ)_{reg}). By the proper base change theorem, we have ϕ_{reg}^{∗}π′_{0∗}\overline{ℚ}_{l} ≃ π_{0∗}ϕ̃_{reg}^{∗}\overline{ℚ}_{l}. Now recall that the action of W ((G_{1})^{σ^{r}}) on L comes from the action of W ((G_{1})^{σ^{r}}) on π^{−1}((Gσ)_{reg}) and the fact that π_{0} is a Galois fibration with Galois group W ((G_{1})^{σ^{r}}). So the W ((G_{1})^{σ^{r}})-equivariance of the isomorphism ϕ^{∗}π′_{∗}\overline{ℚ}_{l} ≃ π_{∗}\overline{ℚ}_{l} is proved with the following lemma.

Lemma 5. The map ϕ̃_{reg} : π^{−1}((Gσ)_{reg}) → π^{′−1}((G_{1}σ^{r})_{reg}) is W ((G_{1})^{σ^{r}})- equivariant.

Proof. Let T = T_{1} × … × T_{r} be a σ-stable maximal torus of G and let B = B_{1} × … × B_{r} be a σ-stable Borel subgroup of G such that T ⊂ B.

Let us consider the following varieties:

˜ (Gσ)_{reg} = {(gσ, xT) ∈ Gσ × G/T |x^{−1}gσx ∈ T σ ∩ (Gσ)_{reg}}.

and

˜ (G_{1})_{reg} = {(g′σ^{r}, x_{1}T_{1}) ∈ G_{1}σ^{r} × G_{1}/T_{1}|x_{1}^{−1}g′σ^{r}x_{1} ∈ T_{1}σ^{r} ∩ (G_{1}σ^{r})_{reg}}.

According to [12, Prop. 4.2], π^{−1}((Gσ)_{reg}) (resp. π^{′−1}((G_{1}σ^{r})_{reg})) is isomorphic to ˜ (Gσ)_{reg} (resp. ˜ (G_{1})_{reg}) via the map (gσ, xT) ↦ (gσ, xB) (resp. (g′σ^{r}, x_{1}T_{1}) ↦ (g′σ^{r}, x_{1}B_{1})) and W ((G_{1})^{σ^{r}}) acts on π^{−1}((Gσ)_{reg}) (resp. on π^{′−1}((G_{1}σ^{r})_{reg})) via this isomorphism and the action of W ((G_{1})^{σ^{r}}) on ˜ (Gσ)_{reg} (resp. on ˜ (G_{1})_{reg}) defined by

w : (gσ, (x_{1}T_{1}, …, x_{r}T_{r})) ↦ (gσ, (x_{1}wT_{1}, …, x_{r}wT_{r}))

(resp. by w : (g′σ^{r}, x_{1}T_{1}) ↦ (g′σ^{r}, x_{1}wT_{1})).

But the map ˜ (Gσ)_{reg} → ˜ (G_{1})_{reg} defined by

((g_{1}, …, g_{r})σ, (x_{1}T_{1}, …, x_{r}T_{r})) ↦ (g_{1}σ(g_{2}) ⋯ σ^{r−1}(g_{r})σ^{r}, x_{1}T_{1})

is W ((G_{1})^{σ^{r}})-equivariant for the actions described above. This proves the lemma. ⊓ ⊔

The image under ϕ of the variety U(Gσ) of unipotent elements of Gσ is the variety U(G_{1}σ^{r}) of unipotent elements of G_{1}σ^{r}. Let us consider the restriction of ϕ^{∗}π′_{∗}\overline{ℚ}_{l} to U(Gσ). By commutativity, we have ϕ^{∗}π′_{∗}\overline{ℚ}_{l}|_{U(Gσ)} ≃ ϕ_{2}^{∗}(π′_{∗}\overline{ℚ}_{l}|_{U(G_{1}σ^{r})}), where ϕ_{2} is the restriction of ϕ to U(Gσ). Hence, we have ϕ_{2}^{∗}(π′_{∗}\overline{ℚ}_{l}|_{U(G_{1}σ^{r})}) ≃ π_{∗}\overline{ℚ}_{l}|_{U(Gσ)} and this isomorphism is W ((G_{1})^{σ^{r}})- equivariant for the Springer actions.

But, according to 2.2, we have the following decomposition formulas:

π′_{∗}\overline{ℚ}_{l}|_{U(G_{1}σ^{r})}[dim(U(G_{1}σ^{r}))] ≃ ⨁_{(C′,E′)} V′_{C′,E′} ⊗ IC(\overline{C}′, E′)[dim(C′)]

where C′ runs over the unipotent conjugacy classes of G_{1}σ^{r} and E′ over the G_{1}-equivariant irreducible local systems on C′ 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′), and

π_{∗}\overline{ℚ}_{l}|_{U(Gσ)}[dim(U(Gσ))] ≃ ⨁_{(C,E)} V_{C,E} ⊗ IC(\overline{C}, E)[dim(C)]

where C runs over the unipotent conjugacy classes of Gσ and E over the G-equivariant irreducible local systems on C.

Moreover, the action of W ((G_{1})^{σ^{r}}) on π_{∗}\overline{ℚ}_{l}|_{U(Gσ)} (resp. π′_{∗}\overline{ℚ}_{l}|_{U(G_{1}σ^{r})}) induces an action of W ((G_{1})^{σ^{r}}) on all V_{C,E} (resp. V′_{C′,E′}) and the map (C, E) ↦ V_{C,E} (resp. (C′, E′) ↦ V′_{C′,E′}) induces a bijection from

{(C, E)|V_{C,E} ≠ 0} (resp. {(C′, E′)|V′_{C′,E′} ≠ 0})

to Irr(W ((G_{1})^{σ^{r}})) which is the Springer correspondence for Gσ (resp. for G_{1}σ^{r}).

Thus, according to Lemma 2, ϕ_{2}^{∗}(π′_{∗}\overline{ℚ}_{l}|_{U(G_{1}σ^{r})})[dim(U(Gσ))] is isomorphic to

⨁_{(C′,E′)} V′_{C′,E′} ⊗ ϕ_{2}^{∗}(IC(\overline{C}′, E′))[dim(Φ^{−1}(C′))].

So, the theorem will be proved with the following lemma:

Lemma 6. For every unipotent class C′ of G_{1}σ^{r} and every irreducible G_{1}- equivariant local system E′ on C′, (ϕ_{2}|_{C})^{∗}E′ is an irreducible G-equivariant local system on C = Φ^{−1}(C′) and all irreducible G-equivariant local systems on C can be obtained in that way. Moreover,

ϕ_{2}^{∗}(IC(\overline{C′}, E′))[dim(C)] ≃ IC(\overline{C}, (ϕ_{2}|_{C})^{∗}E′)[dim(C)]

in M(U(Gσ)).

Proof. As ϕ_{2}|_{C} is a smooth morphism with connected fibers, (ϕ_{2}|_{C})^{∗}E′ is an irreducible local system on C for all irreducible local systems on C′. Let us consider the following diagram,

\xymatrix{G × C \ar[r]^{p_{1}×ϕ_{2}|_{C}} \ar[d]_{p} \ar[d]^{m} & G_{1} × C′ \ar[d]_{m′} \ar[d]^{p′} \\ C \ar[r]_{ϕ_{2}|_{C}} & C′}

where p and p′ are the projections on the second factor, m (resp. m′) the action of G on C (resp. of G_{1} on C′) defined by

(g_{1}, g_{2}, ..., g_{r}) : (l_{1}, l_{2}, ..., l_{r})σ ↦ (g_{1}l_{1}σ(g_{2})^{−1}, g_{2}l_{2}σ(g_{3})^{−1}, ..., g_{r}l_{r}σ(g_{1})^{−1})σ

(resp. by g_{1} : lσ^{r} ↦ g_{1}lσ^{r}g_{1}^{−1}) and where p_{1} : G → G_{1} is defined by (g_{1}, …, g_{r}) ↦ g_{1}. As this diagram is commutative, (ϕ_{2}|_{C})^{∗}E′ is a G-equi-variant local system for all G_{1}-equivariant local systems E′ on C′.

Moreover, as G (resp. G_{1}) acts transitively on C (resp. on C′), there is a 1 − 1 correspondence between G-equivariant (resp. G_{1}-equivariant) irreducible local systems on C (resp. on C′) and irreducible representations of C_{G}(gσ)/C_{G}(gσ)^{0}, where gσ ∈ C (resp. of C_{G_{1}}(g′σ^{r})/C_{G_{1}}(g′σ^{r})^{0}, where g′σ^{r} ∈ C′).

But, according to the proof of Lemma 2, C_{G}(gσ) ≃ C_{G_{1}}(g′σ^{r}). Thus, the functor (ϕ_{2}|_{C})^{∗} induces a bijection between G_{1}-equivariant irreducible local systems on C′ and G-equivariant irreducible local systems on C.

The second part of the lemma comes from [1, 4.2.5]. Indeed, as ϕ_{2}|_{C} is a smooth morphism with connected fibers, the image of IC(\overline{C′}, E′)[dim(C)] under (ϕ_{2}|_{C})^{∗} is IC(\overline{C}, (ϕ_{2}|_{C})^{∗}E′)[dim(C)]. ⊓ ⊔

This completes the proof of Theorem 4. ⊓ ⊔

Finally, we deal with groups with non-trivial center. The following result is due to the referee; it replaces a weaker statement in a previous version:

Theorem 5. Let S be a σ-stable subgroup of the center of G. We set \overline{G} = G/S and let π : G → \overline{G} be the natural projection. Then, σ induces an automorphism of \overline{G} that we still denote by σ, and the map

ϕ : Gσ → \overline{G}σ, gσ ↦ (gS)σ,

induces a natural bijection between the unipotent classes of Gσ and the unipotent classes of \overline{G}σ. Moreover, the Springer correspondence for Gσ is completely described by the Springer correspondence for \overline{G}σ.

Proof. First, it is clear that ϕ induces a surjection Φ between the unipotent conjugacy classes of Gσ and those of \overline{G}σ defined by C ↦ ϕ(C). Moreover, by the following lemma, Φ is a bijection.

Lemma 7. Let hσ be a unipotent element of Gσ and let s be in S such that hsσ is unipotent. Then hσ and hsσ are G-conjugate.

Proof. We can suppose that h lies in a σ-stable Borel subgroup B. Then hs is also contained in B. Let T be a σ-stable torus of B and let U be the unipotent radical of B. Then, as hσ and hsσ are unipotent, there are t, t′ ∈ T and v, v′ ∈ U such that h = t^{−1}σ(t)v and hs = t^{′−1}σ(t′)v′. This implies that s = (t^{−1}t′)^{−1}σ(t^{−1}t′) and that hσ and hsσ are conjugate. ⊓ ⊔

By [12, Cor. 6.4], the Springer correspondence for Gσ is given via the decomposition

H^{2d_{uσ}}(B_{uσ}^{G}, \overline{ℚ}_{l}) = ⨁_{ρ∈Irr(A_{G}(uσ))} V_{C_{uσ},E_{ρ}} ⊗ ρˆ ,  (1)

where C_{uσ} ∋ uσ is a unipotent conjugacy class in Gσ,

A_{G}(uσ) := C_{G}(uσ)/C_{G}(uσ)^{∘}

and ρˆ is the dual representation of ρ.

Let us consider a unipotent element uσ ∈ Gσ. We set \overline{u}σ = ϕ(u)σ. Via the map B ↦ π(B), B_{uσ}^{G} can be naturally identified to B\overline{G} ; \overline{u}σ. We then have d_{uσ} = d_{\overline{u}σ}.

Lemma 8. The map π induces a surjection C_{G}(uσ) → C\overline{G}(\overline{u}σ).

Proof. Let us consider the subset S′ of S defined by

S′ = {s ∈ S|sσ(s) … σ^{r−1}(s) = 1}

where r is the order of σ. Then the map

ψ : S′ → S′, s ↦ sσ(s)^{−1}

is bijective. Indeed, if ψ(s) = 1, then s is a σ-stable element of S′ which means that s^{r} = 1. But s is semisimple and r is the order of σ which is unipotent. Hence we get s = 1.

Let us now consider the set E of elements x ∈ G such that xuσx^{−1} ∈ uσS. Clearly, we have π(E) = C\overline{G}(\overline{u}σ). We will prove the lemma if we prove that π(E) = π(C_{G}(uσ)).

Let x ∈ E and s ∈ S be such that xuσx^{−1} = uσs. Then xuσx^{−1} is unipotent and r is a power of the characteristic, so

(uσs)^{r} = (uσ(u) … σ^{r−1}(u))(sσ(s) … σ^{r−1}(s))

is unipotent. As this is the Jordan decomposition of the unipotent element (uσs)^{r}, the semisimple part sσ(s) … σ^{r−1}(s) is trivial. Hence, s is in S′.

By the first part there is s′ ∈ S′ such that s = s′σ(s′)^{−1}, so x′ = xσ(s′)^{−1} ∈ C_{G}(uσ). Hence, for any x ∈ E, there is x′ ∈ C_{G}(uσ) such that π(x) = π(x′). This proves the lemma. ⊓ ⊔ Now the surjection C_{G}(uσ) → C\overline{G}(\overline{u}σ) induces a surjection

π : A_{G}(uσ) → A\overline{G}(\overline{u}σ).

But the action of A_{G}(uσ) on H^{2d_{uσ}}(B_{uσ}^{G}, \overline{ℚ}_{l}) factors through the action of A\overline{G}(\overline{u}σ) on H^{2d_{uσ}}(B\overline{G} ; \overline{u}σ, \overline{ℚ}_{l}) via π. This implies that V_{C_{uσ},E_{ρ}} is different from zero only when V_{C_{\overline{u}σ},E_{ρ′}} is different from zero and ρ = ρ′ ∘ π. Since the Springer correspondence for Gσ is given via the decomposition formula (1), where uσ runs over a set of representatives of unipotent classes of Gσ, the Springer correspondence for Gσ is completely described by the Springer correspondence for \overline{G}σ.

This completes the proof of Theorem 5. ⊓ ⊔

With Theorems 3, 4 and 5 we have reduced the determination of the Springer correspondence for disconnected groups of the form G.⟨σ⟩ to the case where the connected component G is simply connected almost simple and σ induces a graph automorphism on the Dynkin diagram of G.

Indeed, let G be connected reductive. Then G = D(G).Z(G)^{∘} with D(G) ∩ Z(G)^{∘} finite, where D(G) is the derived group of G and Z(G)^{∘} is the identity component of the center of G (see for example [2, 0.40]). Then there exists an isogeny D(G) × Z(G)^{∘} → G and for G′ = D(G) again there is an isogeny G̃′ → G′ where G̃′ is simply connected semisimple. All these maps can be chosen to be σ-equivariant for an automorphism σ of G (see Steinberg [14, 9.16]), hence by Theorem 5 the determination of the Springer correspondence is reduced to the case of simply connected groups. In turn, these are direct products of almost simple simply connected groups.

If G is a direct product of simply connected almost simple groups, permuted by σ, Theorem 3 gives a reduction to the case where σ acts transitively on the factors. Finally, Theorem 4 leads to the case where G is simply connected almost simple.

There are two families of such groups: G of type A_{n−1} (n ≥ 3) and G of type D_{n} (n ≥ 4), both in characteristic 2, extended by the non-trivial graph automorphism of order 2; and two exceptional cases: G of type E_{6} in characteristic 2 with σ the non-trivial graph automorphism of order 2, and G of type D_{4} in characteristic 3 with the non-trivial graph automorphism of order 3. We consider the classical types in Sections 5–6. Note that by Theorem 5 we may assume that G = GL_{n} for type A_{n−1}, respectively G = SO_{2n} for type D_{n}.

4 Some combinatorics

The unipotent classes and the Springer correspondence are both best described by symbols, combinatorial objects introduced by Lusztig.

4.1 Symbols

The following definition of symbols is taken from Lusztig and Spaltenstein [7, 1.1]. For non-negative integers r, s and for d ∈ ℤ, we consider the following set X̃_{d}^{r,s} of symbols: The elements S ∈ X̃_{d}^{r,s} are pairs S = (S_{0}, S_{1}) of finite sequences S_{i} = (λ_{i,j} | 1 ≤ j ≤ t_{i}) of non-negative integers, subject to the conditions

λ_{i,j} ≥ λ_{i,j−1} + r + s for i ∈ {0, 1}, 2 ≤ j ≤ t_{i},
λ_{1,1} ≥ s,
t_{0} − t_{1} = d.

Note that for s = 0 the second condition is empty. We will also write |S_{i}| = t_{i} for the number of entries of S_{i}. We call d = d(S) = |S_{0}| − |S_{1}| the defect of S.

For any symbol S = (S_{0}, S_{1}) define the contents by

|S| := |S_{0}| + |S_{1}|

and the rank by

rank(S) := ∑_{λ∈S} λ − (r + s) ⌊\frac{(|S| − 1)^{2}}{4} ⌋ − s ⌊\frac{|S|}{2}⌋ ,

where ⌊x⌋ := max{n ∈ ℤ | n ≤ x} denotes the Gauß brackets. Clearly, rank(S) ∈ ℕ_{0} for any S ∈ X̃_{d}^{r,s}.

On X̃_{d}^{r,s} we have the shift operation which to S = (S_{0}, S_{1}) associates

S′ = ({0} ∪ (S_{0} + r + s), {s} ∪ (S_{1} + r + s)) ∈ X̃_{d}^{r,s}.

We denote by X_{d}^{r,s} the classes in X̃_{d}^{r,s} under the equivalence relation defined by the symmetric and transitive hull of the shift operation.

It is clear that the rank is invariant under shift and also the contents is invariant modulo 2 under shift. We can thus write X_{d,n}^{r,s} for the set of classes of symbols in X_{d}^{r,s} of rank n.

An element S ∈ X_{d,n}^{r,s} with d ∈ {0, 1} is called distinguished if λ_{0,j} ≤ λ_{1,j} and λ_{1,j} ≤ λ_{0,j+1} for all relevant j. Note that this definition is independent of the chosen representative in its class. The subset of X_{d,n}^{r,s} of distinguished symbols is denoted by D_{d,n}^{r,s}.

4.2 The genus

Let the genus of a symbol S be defined by

g(S) := ∑_{\{λ,λ′\}} min(λ, λ′) − \{ \begin{matrix}(r\binom{t}{2} \frac{4t+1}{3} + s\binom{t+1}{2} \frac{4t−1}{3}) & if |S| = 2t + 1, \\ (r\binom{t}{2} \frac{4t−5}{3} + s\binom{t}{2} \frac{4t+1}{3}) & if |S| = 2t \end{matrix}

(see [4, (2.22)]). Here the sum extends over all unordered pairs λ = λ_{i,j}, λ′ = λ_{k,l} of entries of S with (i, j) ≠ (k, l). Thus we have g(S) ∈ ℕ_{0} for any S ∈ X̃_{d}^{r,s}. Furthermore, the following is easily checked:

Lemma 9. The genus is invariant under shift, hence well-defined on X_{d}^{r,s}.

4.3 Cuspidal symbols, addition

The kernel of a symbol S = (S_{0}, S_{1}) is the symbol ker(S) = (S′_{0}, S′_{1}) with

S′_{0} = (0, r + s, …, (t_{0} − 1)(r + s)), S′_{1} = (s, r + 2s, …, (t_{1} − 1)r + t_{1}s).

Thus the defect d(ker(S)) = d(S) remains unchanged while rank(ker(S)) ≤ rank(S). If equality holds, then ker(S) = S. Such symbols S will be called cuspidal.

For any given defect there exists an addition

+ : X_{d,n}^{r,s} × X_{d,n′}^{r′,s′} ⟶ X_{d,n+n′}^{r+r′,s+s′}

defined by component-wise addition of representatives of the same length. For fixed r, s, d, n let X(S′) be the subset of (classes of) symbols in X_{d,n}^{r,s}, with kernel S′ ∈ X_{d,k}^{r,s} of rank rank(S′) = k. Then addition of S′ defines a natural bijection

X_{d,n−k}^{0,0} ⟶^{∼} X(S′) ⊆ X_{d,n}^{r,s}, S ↦ S + S′.

On the other hand, the set X_{d,n−k}^{0,0} can be naturally identified with the set of pairs of partitions of n − k, so that we have a natural parametrization of X(S′) by pairs of partitions of n − k.

If s = 0 then the cyclic group of order 2 acts on X_{d}^{r,0}∪X_{−d}^{r,0} by interchanging the S_{i}, that is, (S_{0}, S_{1}) ↦ (S_{1}, S_{0}). We then denote by S^{tr} the symbol (S_{1}, S_{0}) obtained by interchanging the two rows of S = (S_{0}, S_{1}) ∈ X_{d}^{r,0}. Let Y_{d}^{r} denote the set of equivalence classes of X_{±d}^{r,0} modulo this action. Again, the rank, the genus and the contents are well defined on elements of Y_{d}^{r}, and the kernels of equivalent symbols are equivalent. The defect is well-defined up to sign. As before, we denote by Y_{d,n}^{r} the subsets of symbol-classes of rank n. We say that S ∈ Y_{d,n}^{r} is distinguished if one of S, S^{tr} is distinguished, and write D_{d,n}^{r} for the set of distinguished elements in Y_{d,n}^{r}.

4.4 j-induction in W (B_{m})

We need an explicit description of j-induction in Weyl groups of type B_{m}. The irreducible characters of W (B_{m}) are naturally indexed by pairs of partitions (μ_{1}, μ_{2}) of m (see [5, Thm. 5.5.6], for example). The b-invariant of the irreducible character χ ∈ Irr(W (B_{m})) indexed by (μ_{1}, μ_{2}) is then given by

b(χ) = 2 ∑_{{μ_{1i},μ_{1j}}} min(μ_{1i}, μ_{1j}) + 2 ∑_{{μ_{2i},μ_{2j}}} min(μ_{2i}, μ_{2j}) + |μ_{2}| ,

where the sums run over unordered pairs of entries of μ_{1}, μ_{2} respectively (see [5, 5.5.3]). Let (μ_{1}, μ_{2}) ⊢ m be a pair of partitions of m. By adding parts 0 to μ_{1} or μ_{2} we may assume that μ_{1} and μ_{2} have the same number of parts. Then (μ_{1}, μ_{2}) is an element of X̃_{0,m}^{0,0}, and defines a unique element of X_{0,m}^{0,0}. In particular, this defines a natural bijection

Irr(W (B_{m})) ⟶ X_{0,m}^{0,0}.

Similarly, by adding parts of length 0 we may assume that μ_{1} contains precisely one more part than μ_{2}, and this defines a further natural bijection

Irr(W (B_{m})) ⟶ X_{1,m}^{0,0}.

We denote the sign-representation of the symmetric group 𝔖_{k} by sgn. The following is well-known:

Proposition 3. Let χ ∈ Irr(W (B_{k})) with k ≤ m. Then we have:

(a) The j-induction of χ ⊗ sgn from W (B_{k}) × 𝔖_{m−k} to W (B_{m}) is irreducible.

(b) If χ is parametrized by S = (S_{0}, S_{1}) ∈ X_{d,k}^{0,0}, d ∈ {0, 1}, then the symbol of

χ̃ := j_{W (B_{k})×𝔖_{m−k}}^{W (B_{m})}(χ ⊗ sgn)

is obtained from S by increasing each of the ⌊(m − k + 1)/2⌋ largest entries of S_{0} and each of the ⌊(m − k)/2⌋ largest entries of S_{1} by 1 (that is, by increasing each of the rightmost m − k entries of S by 1).

Proof. The irreducibility is proved in [5, Thm. 5.2.6 and Rem. 5.6.7], for example. For the second statement note that the symbol described in the assertion is contained in the induction

Ind_{W (B_{k})×𝔖_{m−k}}^{W (B_{m})}(χ ⊗ sgn)

of χ ⊗ sgn by [5, Prop. 6.4.7]. Moreover, our above formula shows that its b-invariant equals the one of χ ⊗ sgn. Thus we are done by part (a). ⊓ ⊔

We can formally define a j-restriction from W (B_{m}) to W (B_{k}) × 𝔖_{m−k} by

j-Res_{W (B_{k})×𝔖_{m−k}}^{W (B_{m})}(χ) := ∑_{ψ}⟨χ, j_{W (B_{k})×𝔖_{m−k}}^{W (B_{m})}(ψ)⟩ ψ,

where the sum runs over all irreducible characters ψ of W (B_{k}) × 𝔖_{m−k}. By the previous proposition, all coefficients in the sum are 0 or 1. But in general, the j-restriction is not irreducible if it is non-zero. The smallest example is given by the case m = 3, k = 1, χ parametrized by (2, 1). Then χ is j-induced from both (−, 1) ⊗ 2 and (1, −) ⊗ 1^{2} in Irr(W (B_{1}) × 𝔖_{2}). Nevertheless, Proposition 3 shows the following:

Corollary 2. The j-restriction j-Res_{W (B_{k})×𝔖_{m−k}}^{W (B_{m})}(χ) of an irreducible character χ ∈ Irr(W (B_{m})) contains at most one constituent of the form χ′ ⊗sgn with χ′ ∈ Irr(W (B_{k})).

5 Restricted Springer correspondence for GL_{n}: 2

Let GL_{n}, n ≥ 3, be the general linear group over an algebraically closed field of characteristic 2, G(n) the extension of GL_{n} by the non-trivial graph automorphism σ of order 2. In this section we determine the restricted Springer correspondence for the unipotent conjugacy classes in the coset G(n) \ GL_{n}.

Both the classes and the correspondence are best described in terms of symbols, which we introduce first.

5.1 Unipotent classes

We start by recalling the parametrization of the unipotent conjugacy classes in G(n)\GL_{n}. The classes are parametrized by pairs (λ, ϵ) where λ = (λ_{1} ≤ λ_{2} ≤ …) ⊢ n is a partition of n into non-zero parts such that any even part occurs an even number of times, and ϵ : {λ_{i}} → {0, 1, ω} is a map from the set of distinct parts of λ to {0, 1, ω} which is equal to ω on parts of even length, different from ω on parts of odd length, and equal to 1 on parts of odd length which appear an odd number of times (see [13, I.2.7]).

This combinatorial parametrization seems not very well adapted to the description of induction of classes or to the Springer correspondence. Consider the following operation: Let (λ, ϵ) be the parameter of a unipotent class C in G(n)\GL_{n}. We subdivide the sequence λ_{1} ≤ λ_{2} ≤ … into blocks of length 1 or 2, such that λ_{i} lies in a block of length 1 if and only if λ_{i} is odd and ϵ(λ_{i}) ≠ 0. Then define

c_{i} := (λ_{i} − 1)/2 + 2(i − 1)  if {λ_{i}} is a block,
\begin{matrix}c_{i} & := & λ_{i}/2 + 2(i − 1), \\ c_{i+1} & := & (λ_{i} − 2)/2 + 2i = c_{i} + 1 \end{matrix} \} if \{λ_{i}, λ_{i+1}\} is a block, ϵ(λ_{i}) = ω,
\begin{matrix}c_{i} & := & (λ_{i} + 1)/2 + 2(i − 1), \\ c_{i+1} & := & (λ_{i} − 3)/2 + 2i = c_{i} \end{matrix} \}  if \{λ_{i}, λ_{i+1}\} is a block, ϵ(λ_{i}) = 0.

We write ρ(C) for the symbol ((c_{1}, c_{3}, …), (c_{2}, c_{4}, …)). We set m := ⌊\frac{n}{2}⌋ and define d ∈ {0, 1} by n ≡ d (mod 2).

Proposition 4. (a) The map C ↦ ρ(C) is a bijection from the set of unipotent classes in G(n) \ GL_{n} to D_{d,m}^{3,1}.

(b) For any unipotent class C in G(n) \ GL_{n} we have

d_{u} = g (ρ (C)) for u ∈ C.

Proof. It follows easily from the definitions that ρ(C) is a distinguished symbol in X_{d}^{3,1}. Since (on average over a block) the c_{i} are equal to (λ_{i} − 5)/2 + 2i, and by definition the number of parts of λ is congruent to n modulo 2, the rank of ρ(C) is equal to ⌊\frac{n}{2}⌋ by (4.1). Conversely, given a distinguished symbol S = ((c_{1}, c_{3}, …), (c_{2}, c_{4}, …)) ∈ D_{d,m}^{3,1}, the entries c_{1} ≤ c_{2} ≤ … may be uniquely subdivided into blocks of size 1 or 2 such that the difference between two blocks is at least 2, and inside a block at most 1. Inspired by the procedure in [7, 3.1] for connected groups we define

\begin{matrix}λ_{i} & := & 2c_{i} − 4i + 5, \\ ϵ(λ_{i}) & := & 1 \end{matrix} \} if \{c_{i}\} is a block,
\begin{matrix}λ_{i} = λ_{i+1} & := & 2c_{i} − 4i + 4, \\ ϵ(λ_{i}) & := & ω \end{matrix} \} if \{c_{i}, c_{i+1} = c_{i} + 1\} is a block,
\begin{matrix}λ_{i} = λ_{i+1} & := & 2c_{i} − 4i + 3, \\ ϵ(λ_{i}) & := & 0 \end{matrix} \} if \{c_{i}, c_{i+1} = c_{i}\} is a block.

It is clear that this map is the inverse of ρ, hence ρ is a bijection as claimed.

For part (b) we check that Spaltenstein’s formula for d_{u} [13, Prop. II.6.3] translates to our formula

g(S) := ∑_{λ,λ′} min(λ, λ′) − \{ \begin{matrix}t(8t^{2} − 3t − 2)/3 & if d = 1, \\ t(t − 1)(8t − 7)/3 & if d = 0, \end{matrix}

for the genus under the map ρ. Indeed, for the empty partition both formulas yield 0. Now assume that the assertion has been proved for all classes (λ, ϵ) with partitions λ with at most k parts. Let λ be a partition of n into k parts, where we first assume that both n and k = 2t + 1 are odd. We compare the behavior of the two formulas when parts are added to λ. Let λ̃ be the partition of n + 2l obtained from λ by adding two even parts λ_{k+1} = λ_{k+2} = l ≥ λ_{k}. Then our formula gives

g(ρ((λ̃, ϵ̃))) = g(ρ((λ, ϵ))) + n + \frac{λ_{k+1}}{2} ,

(where ϵ̃ is obtained from ϵ by adding ϵ̃(λ_{k+1}) = ω). On the other hand [13, Prop. II.6.3] shows that the value of d_{u} for the corresponding unipotent classes (λ, ϵ) and (λ̃, ϵ̃) increases by the same amount.

Exactly the same happens if λ̃ is the partition of n + 2l obtained from λ by adding two odd parts λ_{k+1} = λ_{k+2} = l ≥ λ_{k} with ϵ(λ_{k+1}) = 0. Finally, if λ̃ is the partition of n + l obtained from λ by adding an odd part λ_{k+1} = l ≥ λ_{k} with ϵ(λ_{k+1}) = 1, then both formulas increase by (n − 1)/2. This completes the induction step. A similar calculation works when n or k or both are even. ⊓ ⊔

In the connected case, the analogue of Proposition 4(a) was shown in [7, Lemma 2.2], and the analogue of part (b) in [4, Prop. 2.23].

5.2 Induced classes

Spaltenstein [13, II.7] gave an algorithm for the determination of the induction of unipotent classes. We rephrase his description in terms of the associated symbols and obtain a closed expression for such an induction.

We first recall the induction of unipotent classes in GL_{k}. The unipotent classes in GL_{k} are parametrized by partitions of k. By [13, II.5.14 and II.7.1] the class with label μ ⊢ k is induced from the trivial class of the Levi subgroup GL_{μ_{1}^{∗}} × … × GL_{μ_{s}^{∗}}, where μ^{∗} is the dual partition of μ. In particular, every unipotent class of GL_{k} is a Richardson class, that is, it is induced from the trivial class of a suitable Levi subgroup.

Now let L be a σ-stable Levi subgroup of GL_{n}. Then L has the shape

L = GL_{k} × (GL_{n_{1}} × … × GL_{n_{s}})^{2}, n = k + 2(n_{1} + … + n_{s}),

where σ acts as the graph automorphism on the first factor and interchanges the two factors GL_{n_{1}} × … × GL_{n_{s}}. Thus a unipotent class C in Lσ may be described by a tuple (C_{0}, C_{1}, …, C_{s}) of unipotent classes C_{0} of G(k) \ GL_{k}, respectively C_{i} of GL_{n_{i}}, 1 ≤ i ≤ s. Since induction of classes is transitive and all classes of GL_{n_{i}} are Richardson classes, it is sufficient to describe induction in the case that C_{i} = 1 for 1 ≤ i ≤ s. Moreover, again by transitivity and induction on s, we may assume that s = 1.

We need to reformulate a result of Spaltenstein in terms of symbols. Let (λ, ϵ) be the label of a unipotent class in G(n)\GL_{n}. Write c_{i} for the number of parts in λ equal to i. Consider the following two types of operations:

(a_{1}) if c_{i} ≥ 2, then decrease λ_{i}^{∗} by 2, (a_{2}) if i is odd, ϵ(i) = 1 and λ_{i−1}^{∗} = λ_{i}^{∗}, then decrease λ_{i}^{∗} and λ_{i−1}^{∗} by 1 each.

Denote a partition obtained by one of these two operations from λ by λ′. Moreover, define ϵ′ by ϵ′(j) = ϵ(j) if λ_{j}^{∗} = (λ′)_{j}^{∗}, and ϵ′(j) ≠ 1 if ϵ(j) = 0, ϵ′(j) ≠ 0 if ϵ(j) = 1.

Lemma 10. Assume that (λ′, ϵ′) is obtained from (λ, ϵ) by application of (a_{1}) or (a_{2}) above for some index i. Then the symbol ρ((λ′, ϵ′)) is obtained by decreasing exactly one of the entries of the symbol ρ((λ, ϵ)) by 1.

Proof. Subdivide the parts of λ into blocks as in (5.1). The operation (a_{2}) then corresponds to replacing a block {i} (with ϵ(i) = 1) by the block {i−2}, with ϵ(i − 2) = 1. On the associated symbol, this corresponds to decreasing the corresponding entry by 1. For the operation (a_{1}) we have to distinguish several cases. If i is odd, ϵ(i) = 1, then there exist at least two blocks {i}, and (a_{1}) replaces them by a block {i−1, i−1}, with ϵ(i−1) = ω. If i is odd, but ϵ(i) = 0, then a block {i, i} is replaced by a block {i − 1, i − 1} with ϵ(i − 1) = ω. Finally, if i is even, hence ϵ(i) = ω, then a block {i, i} is replaced by a block {i − 1, i − 1} with ϵ(i−1) = 0. In all three cases, this corresponds to decreasing one entry in the symbol by 1. ⊓ ⊔

The following is now a consequence of results of Spaltenstein:

Theorem 6. Let C be a unipotent class in G(k) \ GL_{k}. Then the symbol ρ(C̃) of the induced class

C̃ := Ind_{G(k)×GL_{(n−k)/2}≀2}^{G(n)}(C × {1})

is obtained from the symbol ρ(C) by increasing each of the largest (n − k)/2 entries by 1.

Proof. The situation of the theorem is considered in [13, 7.3]: Let C be parametrized by (λ, ϵ) and C̃ by (λ̃, ϵ̃). Then there exists a sequence

(λ, ϵ) = (λ̃^{k}, ϵ̃^{k}), (λ̃^{k+1}, ϵ̃^{k+1}), …, (λ̃^{n}, ϵ̃^{n}) = (λ̃, ϵ̃)

of unipotent classes, such that each class (λ̃^{k+i}, ϵ̃^{k+i}) is obtained by inducing C × {1} from G(k) × GL_{i} ≀ 2 to G(k + 2i). The values b_{i} := d_{u}(λ̃^{k+i}, ϵ̃^{k+i}) then satisfy b_{k+i} − b_{k+i−1} = i − 1, for 1 ≤ i ≤ n − k.

By [13, 6.28] the parameter (λ̃^{k+i−1}, ϵ̃^{k+i−1}) can be obtained from the parameter (λ̃^{k+i}, ϵ̃^{k+i}) by one of the operations in [13, 6.7], which are nothing else but the operations (a_{1}), (a_{2}) introduced above. Hence, by Lemma 10, ρ((λ̃^{k+i−1}, ϵ̃^{k+i−1})) is obtained from ρ((λ̃^{k+i}, ϵ̃^{k+i})) by decreasing one of the entries by 1.

Furthermore, the definition of the genus of a symbol together with Proposition 4(b) yields that decreasing by 1 the (l − i + 1)th entry of a symbol with l entries decreases d_{u} exactly by i − 1. Thus the symbol for (λ̃^{k+i−1}, ϵ̃^{k+i−1}) is obtained from the symbol of (λ̃^{k+i}, ϵ̃^{k+i}) by decreasing the (l − i + 1)th entry by 1. In conclusion, ρ((λ̃^{n}, ϵ̃^{n})) is obtained from ρ((λ, ϵ)) by increasing each of the largest (n − k)/2 entries by 1. ⊓ ⊔

Note the formal similarity of this result with the description of j-induc-tion in Proposition 3. This will be the basis for the determination of the Springer correspondence in Theorem 7.

This allows to give an explicit description of rigid unipotent classes of G(n), that is, unipotent classes which are not induced from a unipotent class of a proper σ-stable Levi subgroup (compare with [13, II.7.9]):

Corollary 3. A unipotent class C in G(n) \ GL_{n} is rigid if and only if the associated symbol ρ(C) ∈ D_{d}^{3,1} has the following property:

– for all 1 ≤ j ≤ |ρ(C)|, either c_{j} = c_{j−1} or c_{j} −c_{j−2} ≤ 4 or c_{j+1} −c_{j−1} ≤ 4,

where we have set c_{−1} := −4, c_{0} := −3 (that is, reducing all entries c_{j}, c_{j+1}, … by 1 does not lead to a symbol in D_{d}^{3,1}).

Example 1. The symbols in D_{1,3}^{3,1} satisfying the above criterion are

\binom{0 4}{4} , \binom{0 4 8}{2 7} , \binom{1 5 9}{1 5} , \binom{0 4 8 12}{2 6 10} ,

so the rigid unipotent classes for G(7) are 3_{0}^{2}1, 2^{2}1^{3}, 31_{0}^{4}, 1^{7}. Here, following the convention of Spaltenstein, we denote the unipotent classes by the corresponding partitions λ, where a part λ_{i} with an index 0 indicates that ϵ(λ_{i}) = 0. The symbols in D_{0,4}^{3,1} satisfying the above criterion are

( \begin{matrix}0 & 4 \\ 3 & 7 \end{matrix} ) , ( \begin{matrix}0 & 4 & 9 \\ 2 & 6 & 10 \end{matrix} ) , ( \begin{matrix}0 & 4 & 8 & 12 \\ 2 & 6 & 10 & 14 \end{matrix} ) , ( \begin{matrix}1 & 5 & 9 & 13 \\ 1 & 5 & 9 & 13 \end{matrix} ) ,

so the rigid unipotent classes for G(8) are 32^{2}1, 2^{2}1^{4}, 1^{8}, 1_{0}^{8}.

Note that the induction of unipotent classes does not satisfy a Harish-Chandra theory with the rigid classes as cuspidal objects. Indeed, as we saw

before, the unipotent class C with symbol ( \begin{matrix}0 & 4 \\ 3 & 7 \end{matrix} ) is rigid in G(8). By Theo-

rem 6 the induction of C × \{1\} to G(12) is the class C̃ with symbol ( \begin{matrix}0 & 5 \\ 3 & 8 \end{matrix} ).

But that class is also induced from the unipotent class with symbol ( \begin{matrix}0 & 4 \\ 2 & 7 \end{matrix} )

in G(6), which in turn is induced from the rigid class with symbol ( \begin{matrix}0 & 4 \\ 2 & 6 \end{matrix} ) in

G(4). Thus C̃ is induced from two different rigid classes. This observation is the key to the explicit determination of the Springer correspondence in the next section.

5.3 The restricted Springer correspondence

The centralizer W^{σ} of the graph automorphism in the Weyl group W of GL_{n} is a Weyl group of type B_{m}, where m = ⌊n/2⌋, if n is odd, respectively of type C_{m} if n is even. Recall that the unipotent classes in G(n) \ GL_{n} are parametrized by D_{d,m}^{3,1}, where d ≡ n (mod 2), d ∈ {0, 1}, while the irreducible characters of W^{σ} of type B_{m} are indexed by X_{d,m}^{0,0}. We denote by

S_{d} := ( \begin{matrix}0 & 4 & 8 & … \\ 1 & 5 & 9 & … \end{matrix} )

the unique (cuspidal) symbol in X_{d,0}^{3,1} for d ∈ {0, 1}.

Theorem 7. (a) The map

D_{1,m}^{3,1} ⟶ X_{1,m}^{0,0}, S ↦ S − S_{1},

represents the restricted Springer correspondence for the outer unipotent classes of G(2m + 1) in characteristic 2.

(b) The map

D_{0,m}^{3,1} ⟶ X_{0,m}^{0,0}, S ↦ [S − S_{0}]^{tr} ,

represents the restricted Springer correspondence for the outer unipotent classes of G(2m) in characteristic 2.

Proof. The proof is by induction on n. For n = 2 there exist just two outer unipotent classes with symbols ((0), (2)), ((1), (1)), with d_{u} equal to 0, 1 respectively, and W (B_{1}) has two irreducible characters, (1, −), (−, 1) with b-values 0, 1, so by Theorem 1 the Springer correspondence is given by

( \begin{matrix}0 \\ 2 \end{matrix} ) ↦ (1, −), ( \begin{matrix}1 \\ 1 \end{matrix} ) ↦ (−, 1),

as claimed. Similarly for n = 3 we have two outer unipotent classes with symbols ((1), −), ((0, 4), (2)) and d_{u} = 0, 1, so again by Theorem 1 the Springer correspondence is given by

( 1 ) ↦ (1, −), \binom{0 4}{2} ↦ (−, 1).

Now assume that the assertion is proved for G(k) for all k < n. Let d ∈ {0, 1} be defined by n ≡ d (mod 2). Let C be an outer unipotent class of G(n), with symbol S = ρ(C) = ((c_{1}, c_{3}, …), (c_{2}, c_{4}, …)) with l entries. Define c_{−1} := −4, c_{0} := −3. Since S ∈ D_{d}^{3,1} we then have

c_{−1} ≤ c_{0} ≤ … ≤ c_{l}, and c_{i} − c_{i−2} ≥ 4 for all i ∈ {1, …, l}.

Let 1 ≤ j ≤ l be the largest index i such that c_{i} − c_{i−2} > 4 and c_{i} > c_{i−1}. (Thus j is maximal subject to the condition that decreasing c_{j} by 1 still gives a distinguished symbol in D_{d}^{3,1}.) If no such entry exists then S = S_{d} has rank 0, which is not the case. Denote by S̃ the symbol obtained from S by increasing the entries c_{j+1}, …, c_{l} by 1 each. By Theorem 6 the symbol S̃ parametrizes the induced class

C̃ = Ind_{G(n)×GL_{l−j}≀2}^{G(n+2(l−j))} (C × {1}) .

By the choice of j, decreasing the l − j + 1 largest entries of S̃ by 1 leads to a distinguished symbol S′. (In fact, S′ is obtained from S by decreasing the entry c_{j} by 1.) So again by Theorem 6 the class C̃ is also induced from the class C′ ×{1} of the Levi subgroup G(n−2)×GL_{l−j+1} ≀2 of G(n+2(l −j)), where C′ has symbol ρ(C′) = S′.

\begin{matrix}G(n + 2(l − j)) \\ ↗ & ↖ \\ G(n) × GL_{l−j} ≀ 2 & G(n − 2) × GL_{l−j+1} ≀ 2 \end{matrix}  C̃ ; ↗ ↖ ; C × \{1\} C′ × \{1\}

By induction, the Springer correspondent χ′ of C′ is already known to be indexed by S′ − S_{d} if d = 1, respectively its transpose if d = 0, so by Corollary 1 the Springer correspondent of C̃ is the j-induced of χ′. Since increasing the s rightmost entries of a symbol commutes with addition of S_{d}, Proposition 3 shows that the Springer correspondent χ̃ of C̃ has symbol S̃ − S_{d}, respectively its transpose.

Now the Springer correspondent of the unipotent class C × {1} of the Levi subgroup G(n) × GL_{l−j} ≀ 2 must be of the form χ ⊗ sgn for some χ ∈ Irr(W (B_{m})), m = ⌊n/2⌋, by the first reduction theorem 3. Proposition 3 and Corollary 2 now show that χ is uniquely determined, and its symbol is obtained from the symbol for χ̃ by decreasing the rightmost l − j entries of S̃ − S_{d} by 1. Again by the commutation of this operation with addition of S_{d}, we deduce that the Springer correspondent of C must have symbol S − S_{d} respectively its transpose, as claimed. ⊓ ⊔

Example 2. We give tables for the restricted Springer correspondence in G(n) \ GL_{n}, 3 ≤ n ≤ 6. Recall the convention for the notation for unipotent classes introduced in Example 1.

Table 1 Restricted Springer correspondence for G(3)

\begin{matrix}C & d_{u} & ρ(C) & χ_{u} \\ 3 & 0 & ^{( 1 )} & (1, −) \\ 1^{3} & 1 & \binom{0 4}{2} & (−, 1) \end{matrix}

Table 2 Restricted Springer correspondence for G(4) and G(5)

\begin{matrix}C & d_{u} & ρ(C) & χ_{u} \\ 31 & 0 & ( \begin{matrix}0 \\ 3 \end{matrix} ) & (2, −) \\ 2^{2} & 1 & ( \begin{matrix}1 \\ 2 \end{matrix} ) & (1, 1) \\ 1^{4} & 2 & ( \begin{matrix}0 & 4 \\ 2 & 6 \end{matrix} ) & (1^{2}, −) \\ 1_{0}^{4} & 4 & ( \begin{matrix}1 & 5 \\ 1 & 5 \end{matrix} ) & (−, 1^{2}) \end{matrix}  \begin{matrix}C & d_{u} & ρ(C) & χ_{u} \\ 5 & 0 & ^{( 2 )} & (2, −) \\ 31^{2} & 1 & \binom{0 5}{2} & (1, 1) \\ 31_{0}^{2} & 2 & \binom{1 5}{1} & (1^{2}, −) \\ 2^{2}1 & 2 & \binom{0 4}{3} & (−, 2) \\ 1^{5} & 4 & \binom{0 4 8}{2 6} & (−, 1^{2}) \end{matrix}

Recall that an irreducible character χ ∈ Irr(W) of a Weyl group W is called special if its a- and b-invariants coincide (see [5, 9.4.8 and Remark 9.4.10] for the definition of the a-invariant). In the case of a Weyl group of type B_{m}, the special characters are precisely those indexed by a distinguished symbol S ∈ X_{1,m}^{1,0} (see [5, 6.4.4]). In the case of connected groups, all special characters occur as images under the Springer correspondence of pairs (u, 1), see [6, 13.1.1] for the case of good characteristic, and [7, Lemma 2.2 and 3.2] for bad characteristic, or also the remark after Thm. 2.1 in [4]. The analogous result remains true for disconnected groups of type G(n):

Table 3 Restricted Springer correspondence for G(6)

\begin{matrix}C & d_{u} & ρ(C) & χ_{u} \\ 51 & 0 & ( \begin{matrix}0 \\ 4 \end{matrix} ) & (3, −) \\ 3^{2} & 1 & ( \begin{matrix}1 \\ 3 \end{matrix} ) & (2, 1) \\ 31^{3} & 2 & ( \begin{matrix}0 & 4 \\ 2 & 7 \end{matrix} ) & (21, −) \\ 3_{0}^{2} & 2 & ( \begin{matrix}2 \\ 2 \end{matrix} ) & (1, 2) \\ 2^{2}1^{2} & 3 & ( \begin{matrix}0 & 5 \\ 2 & 6 \end{matrix} ) & (1^{2}, 1) \\ 2^{2}1_{0}^{2} & 4 & ( \begin{matrix}1 & 5 \\ 1 & 6 \end{matrix} ) & (1, 1^{2}) \\ 1^{6} & 6 & ( \begin{matrix}0 & 4 & 8 \\ 2 & 6 & 10 \end{matrix} ) & (1^{3}, −) \\ 1_{0}^{6} & 9 & ( \begin{matrix}1 & 5 & 9 \\ 1 & 5 & 9 \end{matrix} ) & (−, 1^{3}) \end{matrix}

Corollary 4. All special characters of W (B_{m}) occur as images under the Springer correspondence of an outer unipotent class of G(2m + 1) as well as of an outer unipotent class of G(2m).

Proof. Let S ∈ X_{1,m}^{1,0} be distinguished. Then S + ((0, 3, 6, …), (1, 4, 7, …)) is a distinguished symbol in D_{1,m}^{3,1}, so by Proposition 4(a) it parametrizes an outer unipotent class of G(2m + 1). By Theorem 7(a) its Springer correspondent is the character indexed by

S + ( \begin{matrix}0 & 3 & 6 & … \\ 1 & 4 & 7 & … \end{matrix} ) − ( \begin{matrix}0 & 4 & 8 & … \\ 1 & 5 & 9 & … \end{matrix} ) ∈ X_{1,m}^{0,0},

so by S ∈ X_{1,m}^{1,0}.

For G(2m), let again S ∈ X_{1,m}^{1,0} be distinguished with entries c_{1} ≤ c_{2} ≤ c_{3} ≤ . . .. Let

S′ := ( \begin{matrix}0 & c_{2} + 1 & c_{4} + 1 & … \\ c_{1} & c_{3} & c_{5} & … \end{matrix} ) ∈ X_{0,m}^{1,0}.

Then S′+((0, 3, 6, …), (1, 4, 7, …)) is a distinguished symbol in D_{0,m}^{3,1}, hence by Proposition 4(a) it parametrizes an outer unipotent class of G(2m). By Theorem 7(b) its Springer correspondent is the character indexed by

[S′ + ( \begin{matrix}0 & 3 & 6 & … \\ 1 & 4 & 7 & … \end{matrix} ) − ( \begin{matrix}0 & 4 & 8 & … \\ 1 & 5 & 9 & … \end{matrix} )]^{tr} ∈ X_{0,m}^{0,0},

with entries ((c_{1}, c_{3} − 1, c_{5} − 2, …), (0, c_{2}, c_{4} − 1, …)), so by S ∈ X_{0,m}^{1,0}. ⊓ ⊔ 6 Restricted Springer correspondence for GO_{2n}

Let GO_{2n}, n ≥ 3, be the general orthogonal group over an algebraically closed field of characteristic 2, the extension of SO_{2n} by a non-trivial graph automorphism σ of order 2. In this section we determine the restricted Springer correspondence for the unipotent conjugacy classes in the coset GO_{2n} \ SO_{2n}.

Again, the classes and the correspondence are best described in terms of symbols.

6.1 Unipotent classes

The unipotent conjugacy classes of GO_{2n} not contained in SO_{2n} are parametrized by pairs (λ, ϵ) where λ = (λ_{1} ≤ λ_{2} ≤ …) ⊢ 2n is a partition of 2n into an odd number of non-zero parts such that any odd part occurs an even number of times, and ϵ : {λ_{i}} → {0, 1, ω} is a map from the set of parts of λ to {0, 1, ω} which is equal to ω on parts of odd length, different from ω on parts of even length, and equal to 1 on parts of even length which appear an odd number of times (see [13, I.2.6]).

Let (λ, ϵ) be the parameter of a unipotent class C in GO_{2n} \ SO_{2n}. We partition the sequence λ_{1} ≤ λ_{2} ≤ … into blocks of length 1 or 2 such that λ_{i} lies in a block of length 1 if and only if ϵ_{λ_{i}} = 1. We then set

c_{i} := (λ_{i} − 2)/2 + 2(i − 1)  if {λ_{i}} is a block,
\begin{matrix}c_{i} & := & (λ_{i} − 1)/2 + 2(i − 1), \\ c_{i+1} & := & (λ_{i} − 3)/2 + 2i = c_{i} + 1 \end{matrix} \} if \{λ_{i}, λ_{i+1}\} is a block, ϵ(λ_{i}) = ω,
\begin{matrix}c_{i} & := & λ_{i}/2 + 2(i − 1), \\ c_{i+1} & := & λ_{i}/2 + 2(i − 1) = c_{i} \end{matrix} \}  if \{λ_{i}, λ_{i+1}\} is a block, ϵ(λ_{i}) = 0.

Let ρ(C) denote the distinguished symbol ((c_{1}, c_{3}, …), (c_{2}, c_{4}, …)) ∈X_{1,n−1}^{4,0}. Note that it also defines an element of Y_{1,n−1}^{4}.

Proposition 5. (a) The map C ↦ ρ(C) is a bijection from the set of unipotent classes in GO_{2n} \ SO_{2n} to D_{1,n−1}^{4,0}.

(b) For any unipotent element u ∈ GO_{2n} \ SO_{2n} we have

d_{u} = g (ρ ([u])) .

The proof is entirely analogous to that of Proposition 4, using the results of Spaltenstein [13, Prop. II.6.3] on d_{u} for part (b).

6.2 Induced classes

We now describe Spaltenstein’s results on the induction of unipotent classes [13, II.7] in terms of the associated symbols.

Let L be a σ-stable Levi subgroup of GO_{2n}. Then L has the shape

L = GO_{2m} × GL_{n_{1}} × … × GL_{n_{s}}, n = m + n_{1} + … + n_{s},

where σ acts as the graph automorphism on the first factor and centralizes the factor GL_{n_{1}} × … × GL_{n_{s}}. Thus a unipotent class C in Lσ may be described by a tuple (C_{0}, C_{1} …, C_{s}) of unipotent classes C_{0} of GO_{2m}\SO_{2m}, respectively C_{i} of GL_{n_{i}}, 1 ≤ i ≤ s. As for the case of G(n) it is sufficient to describe induction in the case when s = 1, C_{i} = {1}.

Let (λ, ϵ) be the label of a unipotent class in GO_{2n} \ SO_{2n}. Write c_{i} for the number of parts in λ equal to i. Consider the following two types of operations:

(a_{1})’ if c_{i} ≥ 2, then decrease λ_{i}^{∗} by 2, (a_{2})’ if i is even, ϵ(i) = 1 and λ_{i−1}^{∗} = λ_{i}^{∗}, then decrease λ_{i}^{∗} and λ_{i−1}^{∗} by 1 each.

Denote a partition obtained by one of these two operations from λ by λ′. Moreover, define ϵ′ by ϵ′(j) = ϵ(j) if λ_{j}^{∗} = (λ′)_{j}^{∗}, and ϵ′(j) ≠ 1 if ϵ(j) = 0, ϵ′(j) ≠ 0 if ϵ(j) = 1.

The following can be proved just as Lemma 10:

Lemma 11. Assume that (λ′, ϵ′) is obtained from (λ, ϵ) by application of (a_{1})’ or (a_{2})’ above for some index i. Then the symbol ρ((λ′, ϵ′)) is obtained by decreasing exactly one of the entries of the symbol ρ((λ, ϵ)) by 1.

Similarly as in Theorem 6 we obtain, using [13, 6.7 and 6.28]:

Theorem 8. Let C be a unipotent class in GO_{2n} \ SO_{2n}. Then the symbol ρ(C̃) of the induced class

C̃ := Ind_{GO_{2m}×GL_{n−m}σ}^{GO_{2n}}(C × {1})

is obtained from the symbol ρ(C) by increasing each of the largest n − m entries by 1.

This allows to give an explicit description of rigid unipotent classes of GO_{2n} (compare with [13, II.7.9]):

Corollary 5. A unipotent class C in GO_{2n} \SO_{2n} is rigid if and only if the associated symbol ρ(C) ∈ D_{1}^{4,0} has the following property:

– for all 1 ≤ j ≤ |ρ(C)|, either c_{j} = c_{j−1} or c_{j} −c_{j−2} ≤ 4 or c_{j+1} −c_{j−1} ≤ 4,

where we have set c_{−1} = c_{0} := −4 (that is, reducing all entries c_{j}, c_{j+1}, … by 1 does not lead to a symbol in D_{1}^{4,0}).

6.3 The restricted Springer correspondence

We now have all ingredients to prove the analogue of Theorem 7:

Theorem 9. The map

D_{1,n−1}^{4,0} ⟶ X_{1,n−1}^{0,0}, S ↦ S − ( \begin{matrix}0 & 4 & … \\ 0 & 4 & … \end{matrix} ) ,

represents the restricted Springer correspondence for the outer unipotent classes of GO_{2n} in characteristic 2.

Proof. This is entirely analogous to the proof of Theorem 7, using the description of induced classes in Theorem 8. ⊓ ⊔

Example 3. We give tables for the restricted Springer correspondence in GO_{2n} \ SO_{2n}, n = 6, 8. Since GO_{6} ≅ G(4), the results for GO_{6} reproduce those for G(4) in Example 2. But note that the Weyl group of G(4)^{σ} has to be interpreted as W (C_{2}), while that of GO_{6} ≅ G(4) has type W (B_{2}), so the two correspondences differ by the non-trivial automorphism of W (B_{2}), which interchanges the characters indexed by (−, 2) and (1^{2}, −).

Table 4 Restricted Springer correspondence for GO_{6}

\begin{matrix}C & d_{u} & ρ(C) & χ_{u} \\ 6 & 0 & ^{( 2 )} & (2, −) \\ 41^{2} & 1 & \binom{0 5}{1} & (1, 1) \\ 2^{3} & 2 & \binom{0 4}{2} & (−, 2) \\ 21^{4} & 4 & \binom{0 4 8}{1 5} & (−, 1^{2}) \end{matrix}

As for G(n), all special characters lie in the image of the Springer correspondence (compare with Corollary 4):

Corollary 6. All special characters of W (B_{n−1}) occur as images under the Springer correspondence of a unipotent class in GO_{2n} \ SO_{2n}.

Proof. Let S ∈ X_{1,n−1}^{1,0} be distinguished. Then S +((0, 3, 6, …), (0, 3, 6, …)) is a distinguished symbol in D_{1,n−1}^{4,0}, so by Proposition 5(a) it parametrizes an outer unipotent class of GO_{2n}. By Theorem 9 its Springer correspondent is the character indexed by

S + ( \begin{matrix}0 & 3 & 6 & … \\ 0 & 3 & 6 & … \end{matrix} ) − ( \begin{matrix}0 & 4 & 8 & … \\ 0 & 4 & 8 & … \end{matrix} ) ∈ X_{1,n−1}^{0,0},

so by S ∈ X_{1,n−1}^{1,0}. ⊓ ⊔

Table 5 Restricted Springer correspondence for GO_{8}

\begin{matrix}C & d_{u} & ρ(C) & χ_{u} \\ 8 & 0 & ^{( 3 )} & (3, −) \\ 61^{2} & 1 & \binom{0 6}{1} & (2, 1) \\ 42^{2} & 2 & \binom{0 5}{2} & (1, 2) \\ 42_{0}^{2} & 3 & \binom{1 5}{1} & (1^{2}, 1) \\ 3^{2}2 & 3 & \binom{0 4}{3} & (−, 3) \\ 41^{4} & 4 & \binom{0 4 9}{1 5} & (1, 1^{2}) \\ 2^{3}1^{2} & 5 & \binom{0 4 8}{1 6} & (−, 21) \\ 21^{6} & 9 & \binom{0 4 8 12}{1 5 9} & (−, 1^{3}) \end{matrix}

Acknowledgements We thank the referee for several helpful comments and for communicating Theorem 5 to us.

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