EQUIVARIANCE AND EXTENDIBILITY IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER

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1 EQUIVARIANCE AND EXTENDIBILITY IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER MARC CABANES AND BRITTA SPÄTH Abstract. We show that several character correspondences for finite reductive groups G are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to G has connected center. The correspondences we consider are the socalled Jordan decomposition of characters introduced by Lusztig and the generalized HarishChandra theory of unipotent characters due to BrouéMalle Michel. In addition we consider a correspondence giving character extensions, due to the second author, in order to verify the inductive McKay condition from IsaacsMalleNavarro for the nonabelian finite simple groups of Lie types 3 D 4, E 8, F 4, 2 F 4, and G 2. Contents Introduction 1 1. Main notions and background results 2 2. Equivariance in groups with connected center 8 3. An equivariant bijection of l degree characters The inductive McKay condition for simple groups of types 3 D 4, E 8, F 4 and G The inductive McKay condition for simple groups of type 2 F The inductive AMcondition for blocks with maximal defect 17 References 22 Introduction In the last years, several longstanding conjectures about representations of finite groups were reduced to statements about simple groups, see [IMN07, NS12, NT11, Spä11a]. Recall that the McKay conjecture for a given finite group G and a prime number l asserts that, if M is the normalizer of a Sylow lsubgroup in G, then the numbers of irreducible characters with degree prime to l are the same in G and M, i.e. Irr l (G) = Irr l (M). The reduction theorem from IsaacsMalleNavarro [IMN07] states that the socalled inductive McKay condition (see [IMN07, 10]) on finite simple groups implies the conjecture for all finite groups. By earlier results and the classification of finite simple groups, there is now a special focus on these conditions for finite simple groups of Lie type (see [GLS98] for the terminology on simple groups). A key ingredient in this inductive condition on simple groups is the existence of bijections, that have to be equivariant with respect to the action of certain automorphisms. This brings forth the question of checking equivariance for several correspondences of characters occurring in the representation theory of quasisimple groups of Lie type. Jordan decomposition of characters 1
2 2 MARC CABANES AND BRITTA SPÄTH (Lusztig, see [Lus84]), dharishchandra theory of unipotent characters (BrouéMalleMichel, see [BMM93]), and to a lesser extent the socalled extension maps for normalizers of Levi subgroups (Späth, see [Spä09, Spä10b, Spä10c]) are such correspondences. In the present paper we will establish and use equivariance results about those correspondences to show the following. Theorem. The finite simple groups of types 3 D 4, E 8, F 4, 2 F 4, and G 2 satisfy the inductive McKay condition from [IMN07, 10] for all primes. This clears the way from all cases displaying, or resulting from, Ree or Suzuki automorphisms (types A 1, 2 B 2 and 2 G 2 are treated in [IMN07, 1517]). A very useful trait of those simple groups is that they can be defined as groups of rational points in a connected reductive group with connected center. This allows to take advantage of a representation theory which is in much better shape. Concerning other conjectures, in particular dealing with blocks, the more recent work [Spä11a] has adapted the result from [IMN07] to the blockwise versions of the McKay conjecture, also called the AlperinMcKay conjecture. So we also consider here how our equivariant bijections are compatible with the partitioning into blocks and the implications of our considerations towards the inductive AMcondition from Definition 7.2 of [Spä11a]. Note that in [NS12], it has been proven that the inductive AMcondition assumed for all nonabelian simple groups also implies the missing direction of Brauer s height zero conjecture (see [KM11] for the direction recently established). A similar approach is followed in [Spä11b] for the blockwise version of Alperin s weight conjecture, leading to a socalled inductive BAWcondition (see Definition 4.1 of [Spä11b]). Corollary 6.5 and Corollary 6.9 below give a checking of the inductive AM and BAWconditions in certain cases of the above theorem. This paper is organized as follows. The next section introduces most of the notations and relevant background results on finite reductive groups. In Section 2, we describe some more specific ingredients pertaining to representations of those groups in view of describing a bijection Irr l (G) Irr l (N) due to Malle (see [Mal07, Theorem 7.8]), where N can be thought of as the normalizer of some Sylow lsubgroup. In Section 3 we give our main results about equivariance leading to an equivariant version of Malle s bijection in our case of finite groups G F with connected Z(G). Sections 4 and 5 contain the applications to the specific simple groups mentioned above. Section 6 deals with the application to the AlperinMcKay conjecture and Alperin s weight conjecture. 1. Main notions and background results 1.1. Sets, groups, and characters. Let G be a group. A Gset X is any set endowed with an action of G by (g, x) g.x for g G and x X. The stabilizer of an element x X is denoted by G x, or I G (x) := {g G g.x = x} when the expression of x is more complicated. Many actions are induced by the conjugacy action of a group G on itself. The resulting equivalence relation is then denoted by G conj. If X, X are Gsets, one will say that a map f : X X is Gequivariant, or is a morphism of Gsets, if and only if f(gx) = g.f(x) for any g G, x X. We denote by Inn(G) Aut(G) the subgroup of inner automorphisms in the automorphism group of G, and Out(G) = Aut(G)/Inn(G) the outer automorphism group. If g G and σ
3 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER 3 Aut(G) we denote by gσ and σg the corresponding compositions of σ with the inner automorphism x gxg 1 associated to g. Assume G is a finite group. We denote by Irr(G) the set of irreducible (complex) characters of G. If l is a prime, we denote by Irr l (G) the set of χ Irr(G) with χ(1) prime to l. If H G is a subgroup, we denote induction to G of central functions on H by Ind G H and the restriction of central function on G to H by Res G H. If ν Irr(H), one denotes by Irr(G ν) the set of irreducible constituents of the induced character Ind G H (ν). The action of Aut(G), the group of automorphism of G, on Irr(G) is denoted by σ.χ := χ σ 1 for σ Aut(G) and χ Irr(G). If T is a subgroup of G, we denote W G (T ) := N G (T )/T. This notation extends to pairs (T, θ) where θ Irr(T ), then W G (T, θ) := G T,θ /T, also denoted by N G (T, θ)/t Inductive McKay condition. Let G 0 be a finite nonabelian simple group. The condition imposed by [IMN07, 10] on G 0 and a given prime l has been reformulated in [Spä10a, Propositions 2.3 and 2.8 and its proof] and is twofold. The whole is often called inductive McKay condition for G 0 and l. Let G be the universal covering group of G 0, i.e. the maximal perfect central extension of G 0 by its Schur multiplier, and Q a Sylow lsubgroup of G. Then G and Q have to satisfy the following: Equivariance ([IMN07], ]): there exists an Aut(G) Q stable subgroup N with N G (Q) N G, and an Aut(G) Q equivariant bijection Ω : Irr l (G) Irr l (N), where Ω(Irr l (G ν)) = Irr l (N ν) for every ν Irr(Z(G)). Cohomology ([IMN07, ]): for every character χ Irr l (G) there exists a group A(χ) with G := G/ ker(res G Z(G) (χ)) A(χ), with χ and Ω(χ) extending to A(χ) and N A(χ) (N) respectively, plus certain additional properties. When the outer automorphism group Out(G 0 ) is cyclic, this latter cohomological condition is then a consequence of the equivariance condition stated first (see [Mal08a, 2]), which in turn also simplifies when the Schur multiplier Z(G) of G 0 is trivial. This applies to the simple groups we treat (see Theorem 4.1 and Theorem 5.1) Connected reductive groups and duals. We refer to [Cr85, 4], [CE04, 8], and [DM91, 13]. In what follows, G will denote a connected reductive algebraic group over the algebraic closure F of the field with p elements, p a prime. One denotes by F : G G an algebraic endomorphism such that F δ (δ = 1 or 2) is a Frobenius endomorphism associated with a definition of G over the subfield of F with q elements (the case δ = 2 accounts for Ree and Suzuki automorphisms). We call F stable subgroups of G also rational. Levi subgroups of G are centralizers of tori. For a maximal torus T of G let X(T) = Hom(T, F ), Y (T) = Hom(F, T), and R(G, T), or simply R X(T) the set of Troots in G. Let x α : (F, +) G be the oneparameter (unipotent) subgroup t x α (t) associated with α R(G, T). Note that F induces a permutation of R(G, T) as soon as T is F stable. Choosing a Borel subgroup containing T gives a basis R(G, T), which in turn is in bijection with a generating set of W G (T). For a rational torus T, the G F classes of F stable Levi subgroups are parametrized by equivalence classes of pairs (I, w) ( type of the relevant Levi subgroups) with I and w W G (T) such that wf (I) = I, see [DM91, 4.3].
4 4 MARC CABANES AND BRITTA SPÄTH For a pair (G, T), one defines the associated root datum (X(T), R, Y (T), R ). When moreover T is F stable, F induces endomorphisms X(F ) and Y (F ) of X(T) and Y (T), along with a permutation φ of R. When δ = 1, the restriction of X(F ) to ZR is φ multiplied by a constant power of p. We say another group (G, T, F ) defined over the same finite field is in duality with (G, T, F ) if and only if we are given an F stable maximal torus T G and isomorphisms X(T) = Y (T ), Y (T) = X(T ) exchanging the subsets of roots and coroots and sending X(F ) to the transpose of Y (F ). If T 0, resp. T 0, is a maximally split torus (i.e. contained in F stable Borel subgroups) of G, resp. G, then (G, T 0, F ) and (G, T 0, F ) are in duality. In order to simplify notations we write F as F. This generalizes to pairs of rational maximal tori (T, T ) whose types with regard to T 0 and T 0 are u W G(T 0 ), u 1 W G (T 0 ) = W G(T 0 ) opp. Two rational Levi subgroups L G and L G are said to be in duality if and only if there are rational maximal tori T L and T L and a duality between (G, T, F ) and (G, T, F ) such that R(L, T) X(T) and R(L, T ) X(T ) = Y (T) correspond by r r. This is possible if and only if they have transpose types (I, w), (I, w 1 ) with regard to maximally split tori. This clearly induces a bijection between the G F classes of F stable Levi subgroups of G and the G F classes of F stable Levi subgroups of G One denotes G ad := G/Z(G). Note that this quotient only depends on F and the root system R, its root datum being (ZR, R, Hom Z (ZR, Z), R ). One denotes by G sc the simplyconnected covering of [G, G], its root datum is (Hom Z (ZR, Z), R, ZR, R ) (see [Cr85, 1.11], [CE04, 8.1]). Recall the notation W G (L) := N G (L)/L. In the case of a Levi subgroup containing the maximal torus T, this is isomorphic to W WG (T)(W L (T)) Automorphisms. We keep (G, F ) as above. We refer to [Spr98, 9] for the notions of isogenies (always onto) and pmorphisms between root data (X(T), R, Y (T), R ). If σ : G G is an isogeny preserving T, recall X(σ): X(T) X(T) the induced morphism. We are interested in the cases where σf = F σ and X(σ) has cokernel a finite pgroup (this is a little more restrictive than inducing a pmorphism). This ensures that σ is a bijection, thus inducing a finite group automorphism G F G F (see proof of [Spr98, 9.6.5]). We denote by Aut F (G F ) the group of automorphisms of G F induced by the above σ s. The corresponding σ : G G is determined by its restriction to G F up to powers of F, a consequence of the uniqueness theorem of [Spr98, 9.6]. This allows to define the stabilizer Aut F (G F ) V for any F stable closed connected subgroup V G. Note also that Aut F (G F ) is indeed the abstract automorphism group of the finite group G F as soon as G = G sc and G F /Z(G) F is simple nonabelian (see [GLS98, 1.15]). But our purpose is here rather to describe field and graph automorphisms of groups G F with connected center Z(G) = Z (G), and relate them with automorphisms of the dual group G F. Elements of (G ad ) F (or simply of N T (G F )) induce elements of Aut F (G F ) by conjugation. They are called diagonal automorphisms, we have Inn(G F ) Inn((G ad ) F ), both normal in Aut F (G F ). Choose a maximally split torus T 0. One has Aut F (G F )/Inn((G ad ) F ) = Aut F (G F ) T0 /Inn((G ad ) F ) T0 and classes in the latter are determined by induced endomorphisms of X(T 0 ). Note also that the above process induces an isomorphism between Aut F (G F )/Inn((G ad ) F ) and Aut F (G F )/Inn((G ad )F ). Having the same image is equivalent to the following condition.
5 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER 5 Definition 1.5. One says σ Aut F (G F ) and σ Aut F (G F ) are of the same type, denoted by σ σ if and only if they are in classes of σ 0 stabilizing T 0 up to inner automorphisms, resp. σ0 stabilizing T 0 and such that X(σ 0) is the transpose of X(σ0 ) by the isomorphisms of duality, up to an element of W G (T 0 ) F. X(F ) in Aut(X(T 0 ) Z C). This is because two elements σ, σ Aut F (G F ) can be supposed to both preserve T 0, then our claim amounts to showing that if σ, σ Aut F (G F ) T0, they are equal mod diagonal automorphisms if and only if X(σ) and X(σ ) are equal mod W G (T 0 ) F. This is clear from the isomorphism theorem ([Spr98, 9.6.2]) since diagonal automorphisms are given by T 0 conjugations commuting with F, and inner automorphisms are represented by elements of W G (T 0 ) F. Proposition 1.6. If L and L are in duality, then for every σ Aut F (G F ) L there is σ Aut F (G F ) L such that σ σ. A field automorphism of G F is an automorphism, which, up to inner automorphisms, is the restriction to G F of some endomorphism F : G G associated to a definition of G over a finite subfield of F (and commuting with F ). Up to inner automorphisms of G F it stabilizes T 0 and is the restriction to G F of an isogeny of G of the form x α (t) x α (t q ) on [G, G] for q a power of p. Proposition 1.7. If L is an F stable closed subgroup of G containing a (F stable) maximal torus of G and σ Aut F (G F ) is a field automorphism such that σ(l) = L, then σ acts by inner automorphisms on W G (L) F. When G is of irreducible type (i.e. irreducible root system R), a graph automorphism of G F is an automorphism which, up to inner automorphisms, is the restriction to G F of some endomorphism γ : G G stabilizing T 0 and of the form x α (t) x φ(α) (±t qα ) on [G, G] for φ a graph automorphism permuting an F stable basis R of R, and q α a power of p which is 1 when α is long and the sign ± is + when α R R. Note that when δ = 2 or when F itself induces a non trivial permutations of R, then there is no non trivial graph automorphism. Proof of Propositions 1.6 and 1.7. For Proposition 1.6, since L and L are in duality, we may assume L T and L T with R(L, T) = R(L, T ). Let X(σ): X(T) X(T) the induced automorphism. The existence of σ : G G commuting with F and inducing the transpose of X(σ) on Y (T ) = X(T) is ensured by [Spr98, 9.6.5] and Lang s theorem (we first have tσ F = F σ for some t T 0, but then Lang s theorem ensures the existence of t 0 T 0 such that t 0 σ commutes with F ). Let s prove now Proposition 1.7. If T is an F stable maximal torus of L, then conjugacy of maximal tori tells us that W G (L) = W WG (T)(W L (T)) with compatible action of F. As a first consequence, one may assume G = G ad = [G, G]. By a standard application of Lang s theorem, let g G such that g T 0 = T, so that n := g 1 F (g) N G (T 0 ) and (W WG (T)(W L (T)), F ) is sent by g to (W WG (T 0 )(W L g(t 0 )), nf ). Then let F p be the endomorphism of G sending x α (t) to x α (t p ). It commutes with F and fixes all elements of W G (T 0 ). So it acts trivially on any subgroup of W WG (T 0 )(W L g(t 0 )). It clearly generates the field automorphisms up to inner automorphisms. There remains to see that it comes from an automorphism of G F in the same class mod inner automorphisms. Through conjugacy by g, F p corresponds with g 1 F p(g)f p, so it suffices to show that we can arrange g 1 F p(g) G F. Since F p acts trivially on W G (T 0 ), F p(n) nt 0. By Lang s theorem applied to F p and nf in T 0, there are t, t T 0 such that nt is fixed by F p and
6 6 MARC CABANES AND BRITTA SPÄTH nt = (gt) 1 F (gt). Then replacing g by gt, one sees that one might assume that n was F pfixed in the first place. Then it is easily deduced that x := g 1 F p(g) is fixed by F, using F p(n) = n and F F p = F pf. Now xf p generates field automorphisms mod inner automorphisms while acting trivially on W G (L) F, so σ acts by an inner automorphism of G F which has to normalize L Sylow Φ d tori. Keep (G, F ) as before and assume δ = 1. Let d 1 be an integer and Φ d Z[x] the associated cyclotomic polynomial. Using the notion of polynomial orders of pairs (G, F ) (G an algebraic group defined over F q, F the associated Frobenius endomorphism, see [BM92] [CE04, 13.1]), Φ d tori, and dsplit (F stable) Levi subgroup of G can be defined (see [CE04, 13.3]). Sylow (i.e. maximal) Φ d tori of (G, F ) are G F conjugate (see [BM92] and [CE04, 13.18]). The dual of a dsplit Levi subgroup is also dsplit ([CE04, 13.9] ). Let l be a prime not dividing q. We shall investigate Irr l (G F ). The equivariance condition from [IMN07, 10] (see 1.2 above) requires the existence of a subgroup N G F such that any automorphism of G F stabilizing a given Sylow lsubgroup also stabilizes N. Let us take d the (multiplicative) order of q mod l (resp. mod 4) when l is odd (resp. l = 2). Let us define the following Condition 1.9. Let N be the normalizer in G F of a Sylow Φ d torus S of G. We assume that N G F (Q) N < G F for some Sylow lsubgroup Q of G F. This occurs apart from a few exceptions when l = 2 and G F is of type C n with q 3, 5 mod 8; or l = 3 and G F is of type A 2, 2 A 2, G 2 with q in certain classes mod 9 (see [Mal07, 5.14, 5.19]). Note that [Mal07] assumes G to be simple as an algebraic group, but one reduces easily to that case by factoring out Z (G). In order to check equivariance properties we need to generalize the above condition from inner to all automorphisms. Proposition Assume Condition 1.9 is satisfied. Through the identification of Aut F (G F ) with a subquotient of the group of abstract automorphisms of G (see 1.4), one has Aut F (G F ) Q Aut F (G F ) S. Proof. Let σ 0 Aut F (G F ) Q. We assume it is the restriction of σ : G G a bijective algebraic endomorphism commuting with F. Then σ(s) is an F stable closed subgroup of G (see for instance [CE04, A.2.4]). Using the definition of polynomial order given for instance in [Cb94, 4.3], it is easy to see that σ(s) has same polynomial order as S, hence is a Sylow Φ d torus of G. So x (σ(s)) = S for some x G F by conjugacy of Sylow Φ d tori. Moreover Q and x Q = x (σ(q)) are Sylow lsubgroups of N G F (S), since Q N G F (Q) N G F (S) by Condition 1.9. So there is y N G F (S) such that yx Q = Q, and still yx σ(s) = S. But then N G F (Q) N G F (S) implies S yx = S and therefore σ(s) = S DeligneLusztig characters and geometric series. (G, F ) is as in 1.3. We denote by G ss the set of semisimple elements of G. For T an F stable maximal torus and θ Irr(T F ), we denote by R G T θ ZIrr(GF ) the Deligne Lusztig generalized character (see [DM91, 11.14];[CE04, 8.3]). From a duality between (G, T 0, F ) and (G, T 0, F ) (see 1.3), one gets a bijection between the four sets ([DM91, 13.13]; [CE04, 8.21]):
7 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER 7 (a) G F classes of pairs (T, θ) where T is an F stable maximal torus of G and θ Irr(T F ), (b) W G (T 0 )classes of pairs (wf, θ), for w W G (T 0 ) and θ Irr(T wf 0 ), (a ) G F classes of pairs (T, s) where T is an F stable maximal torus of G and s T F, (b ) W G (T 0 )classes of pairs (wf, s), for w W G (T 0 ) and s T 0 wf. The identifications between (a) and (b) and between (a ) and (b ) is by use of the types of F stable tori (see 1.3). The identifications between (b) and (b ) are made through the exact sequences and wf 1 1 X(T 0 ) X(T 0 ) Irr(T Res wf 0 ) 1 wf 1 1 X(T 0 ) X(T 0 ) = Y (T 0) T Nw 0 wf 1, where N w is a norm map as defined in [CE04, 8.12]. Assume the center of G is connected, so that Inn(G F ) = Inn(G F ad ). Proposition Assume (T, θ) and (T, s) are related as above. If σ Aut F (G F ) and σ Aut F (G F ) satisfy σ σ, then (σ(t), σ.θ) and (σ (T ), σ (s)) are also related. Proof. By conjugacy of maximally split tori, one may assume that σ(t 0 ) = T 0 and σ (T 0 ) = T 0 up to inner automorphisms. The induced endomorphisms of the groups X(T 0 ) and X(T 0 ) may even be assumed to be transposed to each other up to powers of X(F ). Then our claim easily results from taking the images of the exact sequences above and using the commutation of F with σ (seen as endomorphism of G) and σ. When (T, θ) and (T, s) are related one writes R G T s := RG T θ. One defines E(G F, s) as the set of irreducible components of all possible R G T s with F stable T containing s. The elements of E(G F, 1) are called unipotent characters. For the following Jordan decomposition of characters, due to Lusztig [Lus84, 4.23], see for instance [DM91, 13.23]. Let us recall that when the center of G is connected, then C G (s) is connected for any semisimple s G (see [DM91, (ii)]). Theorem Recall that we assume that the center of G is connected. partition Then we have a Irr(G F ) = s E(G F, s) where s ranges over (G ss) F and where E(G F, s) = E(G F, s ) if and only if s and s are G F  conjugate. Moreover, for any s (G ss) F there is a bijection E(C G (s) F, 1) E(G F, s), denoted ζ χ G s,ζ, where C G (s) is a connected reductive group, and we have χ G s,ζ, RG T s G F = ζ, RC G (s) T for any F stable maximal torus T of G containing s. Furthermore the degrees satisfy (1.14) χ G s,ζ (1) = G F : C G (s) F p ζ(1). 1 CG (s) F
8 8 MARC CABANES AND BRITTA SPÄTH Generalized HarishChandra theory. Keep (G, F ) as in 1.3, assuming now δ = 1. For a parabolic subgroup P G admitting a Levi decomposition with F stable Levi subgroup L (though P needs not be F stable itself), one has the DeligneLusztig induction functor R G L P : ZIrr(LF ) ZIrr(G F ) (see [DM91, 11.1]) and its adjoint R G L P for the scalar product of characters. When taken at unipotent characters, those functors are independent of the parabolic chosen for L, (see [BMM93, 1.33]), so we write simply R G L : ZIrr(LF ) ZIrr(G F ) and R G L. Let d 1 be an integer. A (unipotent) dcuspidal pair of G is any pair (L, λ) where L is a dsplit Levi subgroup of G and λ E(L F, 1) is such that R L M (λ) = 0 for any dsplit M L, M L. Denote by E(G F, L, λ) the set of components of the induced character R G L (λ). In the case of d = 1, this corresponds to HarishChandra theory of cusp forms. The general case introduced by BrouéMalleMichel gives rise to a very similar parametrization. Theorem ([BMM93], [Mal07, 3.2]) There is a bijection I G L,λ : Irr(W G(L, λ) F ) E(G F, L, λ) η I G L,λ (η). We will be interested only in the cases where L is a minimal dsplit Levi subgroup, that is the centralizer of a Sylow Φ d torus. Then any unipotent character of L is of course dcuspidal Maximal extendibility. In order to deal with Irr l (M) for M the normalizer of a Sylow lsubgroup of a finite group or the group called N in Condition 1.9, we describe a situation where Clifford theory is particularly simple. Definition Let L M be finite groups. We say that maximal extendibility holds with respect to L M if any χ Irr(L) extends as an irreducible character χ Irr(I M (χ)). Accordingly there exists a map Λ : Irr(L) L I M Irr(I) with χ χ. We call such a map an extension map for L M. For a subgroup A of Aut(M) L, we call Λ Aequivariant if a Λ(χ) = Λ( a χ) for any a A, χ Irr(L). Clifford theory gives at once Proposition Given an extension map Λ for L M and χ Irr(L), there exists a bijection Irr(I M (χ)/l) Irr(M χ) defined by η Ind M I M (χ) (Λ(χ)η). We use later a strengthened version of the following result. Theorem 1.20 ([Spä09, Theorem A], [Spä10c, Theorem 1.1]). Assume (G, F ) is simple, simplyconnected and defined over F q. Let d 1 be an integer and let C be a minimal dsplit Levi subgroup in G. Then maximal extendibility holds with respect to C F N G (C) F. 2. Equivariance in groups with connected center We keep (G, F ) as in 1.3, so that F δ is a Frobenius map defining G over F q. Take (G, F ) in duality with (G, F ). We assume that G has connected center. This implies that all its Levi subgroups also have connected center and that C G (s) is (reductive) connected for any s (G ) ss (see [DM91, 13.14, 13.15]).
9 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER Jordan decomposition and automorphisms. We analyse equivariance in Theorem Theorem 2.2. Let G be a connected reductive group with connected center defined over a finite field F q. Let F : G G be an isogeny such that a power F δ (δ = 1 or 2) of F is a Frobenius endomorphism associated with the definition of G over F q. Then there is a Jordan decomposition ( ) Irr(G F ) E(C G (s) F, 1) /G F conj s (G ss) F such that χ G s,λ (s, λ) χ G s,λ = σ.χg σ (s),σ λ for any σ Aut F (G F ), σ Aut F (G F ) with σ σ (see Definition 1.5). Proof. Assume first that F is a standard Frobenius endomorphism coming from the definition of G over a finite field (that is δ = 1). Then our statement is essentially contained in the uniqueness results given in [DM90, 7.1.(vi)]. Note that Hypothesis 3.3 of [DM90] is satisfied (see [CE04, 15.11], [Mal08b, 2.4]). We explain below how to rephrase the approach given there in our setting (see also Remark 2.3 below). Note first that since the center of G is connected, and therefore all centralizers of semisimple elements of G are connected, the G F classes of G ss F are in bijection with F stable G conjugacy classes of G ss, which in turn also coincide with F stable W G (T 0 )classes in T 0 (see [DM91, 13.12] and [CE04, 8.4]. For such a t T 0 (and associated n N G (T 0 ) such that F (t) = n 1 tn), one denotes E(G F, t) the associated Lusztig series (see 1.11). In [DM90, 7.1], one gets a bijection π t : E(G F, t) E(C G (t) nf, 1) with several properties, among which the one of Lusztig s Theorem Another property is associated to any pair of adjoint bijective algebraic morphisms ϕ, ϕ commuting with F and preserving T 0 and T 0. In this situation, [DM90, 7.1.(vi)] tells us that ϕ 1.χ E(G F, ϕ (t)) for all χ E(G F, t) and that (E) π ϕ (t)(ϕ 1.χ) = ϕ.π t (χ). Now, for s G ss F corresponding to t, that is s = t g where g G is such that gf (g 1 ) = n, one defines χ G s,λ := π 1 t ( g λ) for λ (E(C G (t) nf, 1)) g = E(C G (s) F, 1). This does not depend on the choice of g since another choice would differ by an element of C G (s) F. Our property will then be immediate from (E) above for the cases where σ preserves T 0 and X(σ) = X(σ ). We know that we can reduce to this up to diagonal automorphisms on each side. On the G side, there are none since Z(G) is connected. On the G side, they act on s by sending it to a G F conjugate (an easy consequence of connectedness of C G (s)) and diagonal automorphisms preserve unipotent characters. So we get the general case. For non standard Frobenius endomorphisms (that is δ = 2) defining finite groups of type 2 B 2 and 2 G 2, proper reductive subgroups are of classical type, so the uniqueness in terms of scalar products with DeligneLusztig characters (see Remark 2.3 below) applies for noncentral s, while the case of a central s is trivial. In the case of 2 F 4, the types appearing as centralizers of non central semisimple elements of G are either classical or 2 B 2. For type 2 B 2, an argument on eigenvalues of the Frobenius endomorphism F 2 on cohomology spaces of DeligneLusztig varieties of type a Coxeter element is then possible as in [DM90, 7.1.(ii) and (iv)], since 2 B 2 is a Levi subgroup and, by [Lus84,
10 10 MARC CABANES AND BRITTA SPÄTH p.373] the four elements of E( 2 B 2 (q), 1) (the trivial and Steinberg characters plus two cuspidal characters) have distinct degrees except the two cuspidal characters, but their eigenvalues of F 2 are then 1± 1 2 q by [Lus77, 7.4.(a)]. Remark 2.3. In groups with connected center and classical type, any element χ Irr(G F ) is characterized by the scalar products χ, R G T (θ) G F of χ with DeligneLusztig characters (see [Lus88, 8.1.(a)]). Then the equivariance of Theorem 2.2 is more elementarily a consequence of Theorem 1.13 again and Proposition 1.12, along with the property of DeligneLusztig characters σ.r G T θ = RG σtσ.θ (see [DM90, 9.2]). Corollary 2.4. Let G, F be as above, L, L F stable Levi subgroups in duality. Let s (L ) F ss, λ E(C L (s) F, 1) and let χ L s,λ E(LF, s) obtained by an equivariant Jordan decomposition as above. Then there is a collection of isomorphisms i s,λ : I G F (L, χl s,λ )/LF W CG (s )(C L (s ), λ ) F for any (s, λ ) in the L F conj class of (s, λ) such that, for any σ σ with σ Aut F (G F ) L, σ Aut F (G F ) L (see Proposition 1.6), i σ (s ),σ.λ σ 1 σ i s,λ up to inner automorphisms of I G F (L, χ L s,λ )/L F. Proof. See also [CE04, 1.9.(ii)]). Assume the duality between L and L is given with respect to the rational maximal tori T and T. Let us start with the case s = 1, λ = 1 CL (s) F, so that I G F (L, 1 L F )/L F = W G (L) F. An isomorphism i 1,1 : W G (L) W G (L ) is given by the isomorphism W G (T) W G (T ) opp of duality, which restricts as W L (T) W L (T ) opp while W G (L) = W WG (T)(W L (T)) as recalled in 1.3. Then we get i 1,1 : W G (L) F W G (L ) F by restricting i 1,1. It satisfies i 1,1 σ 1 = σ i 1,1 for any σ Aut F (G F ) L,T, σ Aut F (G F ) L,T inducing adjoint endomorphisms of X(T). This gives our claim for pairs σ σ by the definition of this relation. The rest of the corollary is a matter of comparing both sides of the sought map as subgroups of the above. For s (L ) F ss, an L conjugacy brings it to an element s T so that W CG (s )(C L (s )) is actually a subgroup of W G (L ). Then the equivariance of Theorem 2.2 tells us that i 1,1 maps I G F (L, χ L s,λ ) into W CG (s )(C L (s )) with image the one of W CG (s)(c L (s)) F by the L conjugacy used in the first place (see also the proof of [CE04, 1.9.(iii)]). This gives an isomorphism i s,λ with the claimed equivariance property. Having done that for (s, λ) allows to define it for any L F conjugate since the left hand side does not change and the right hand side is fixed by C L (s) F Generalized HarishChandra theory and automorphisms. We now turn to the question of equivariance in Theorem We now assume δ = 1 (see 1.3). Theorem 2.6. Let (G, F ) be a connected reductive group defined over a finite field as in 1.3 with δ = 1, let d 1 be an integer, L a minimal dsplit Levi subgroup of G, and λ E(L F, 1). Then there is a bijection I G L,λ : Irr(W G(L, λ) F ) E(G F, L, λ) which is Aut F (G F ) (L,λ) equivariant.
11 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER 11 Proof. Let us first reduce to a simple G = G ad. Unipotent characters are the same for the three finite groups G F ( G) F (G ad ) F (for G G a regular embedding, see [DM91], 13.20]) and N G (L, λ) F /L F is the same too. This also implies that N G F ad (L, λ) acts trivially on E(G F, 1) and Irr(W G (L, λ)) F. Now G = i G i a product of simple algebraic groups and F G i = G ϕ(i) for some permutation ϕ. One may assume that ϕ is trivial by replacing G with a product of representatives of orbits and F on this product by its power on the representative (see also [BMM93, 1.37]). Then (G, F ) = i I (G i, F i ). Let σ Aut F (G F ) (L,λ) be the restriction of a bijective endomorphism of G denoted the same. Arguing that [G i, g i ] generates a dense subgroup of G i for each g i G i \{1}, and since σ(g i ) = [G, σ(g i )] is a closed subgroup of G (see [CE04, A2.4]), one sees that for any i there is = I(i) I such that σ(g i ) = I(i) G i while I = ii(i) by surjectivity. Taking dimensions and summing over i I, one sees that each I(i) has to be a single element. So σ induces a permutation of the i s and Aut F (G F ) has a wreath product structure on each isotypic component of the product i I (G i, F i ). It is then clear that it suffices to find an Aut F (G F 1 1 ) (L 1,λ 1 )equivariant bijection between components of R G 1 L 1 (λ 1 ) and Irr(N G1 (L 1, λ 1 ) F 1 /L F 1 1 ) to get our claim. We now assume that G is simple of adjoint type, defined over F q with F : G G the associated Frobenius endomorphism. The group Out(G F ) is generated by the classes of graph and field automorphisms, the latter generating a normal subgroup (see 1.4). This normal subgroup acts trivially on unipotent characters (see [Mal08b, 2.5]) and field automorphisms stabilizing (L, λ) also act trivially on the sets Irr(W G (L, λ) F ) thanks to Proposition 1.7. One is now left to compare the action of graph automorphisms (including Suzuki and Ree automorphisms) on E(G F, 1) and Irr(W G F (L, λ)). In [Mal07, 4.3], the actions on both sets are described and compared (with a slightly different goal) in all cases where the action is non trivial (types D 2n, B 2 (2 2m+1 ), G 2 (3 2m+1 ), F 4 (2 2m+1 )) with some orbits of order 2 and 3. They are checked there to be the same on the E(G F, L, λ) and the Irr(W G (L, λ) F ) sides (see [Mal07, p. 202] ). Remark 2.7. If (H, F ) is a connected reductive group defined over a finite field as in 1.3 with δ = 1, let us consider the set T of all triples (G, L, λ) where G is an F stable connected reductive subgroup of maximal rank, L is a minimal dsplit Levi subgroup of G and λ E(L F, 1). Then Aut F (H F ) acts on T. It is easy to see that the above Theorem 2.6 implies that we have maps I G L,λ : Irr(W G(L, λ) F ) E(G F, L, λ) such that σ.il,λ G (η) = IσG σl,σ.λ (σ.η) for any (G, L, λ) T, η Irr(W G (L, λ) F ), and σ Aut F (H F ). This can be seen as a natural transformation between the functors (G, L, λ) E(G F, L, λ) and (G, L, λ) Irr(N H (L, λ) F /L F ) and Theorem 2.6 as the isomorphism property on automorphism groups of the category T Equivariant maximal extensions. Here is an equivariant version of Theorem 1.20 adapted to our needs.
12 12 MARC CABANES AND BRITTA SPÄTH Theorem 2.9. Let (G, F ) be defined over F q with G = G sc, δ = 1 (see 1.3) and an exceptional root system or G F = 3 D 4 (q). Let A Aut F (G F ) be the subgroup generated by inner automorphisms of G F, along with graph automorphisms and field endomorphisms of G commuting with F (see 1.4 above). Let d 1 be an integer and C a minimal dsplit Levi subgroup of G. Then there exists an A C equivariant extension map (see Definition 1.18) for the inclusion C F N G (C) F. Proof. Assume first that C is a torus. Then we can apply Lemma 8.1 (for inner and field automorphisms) and Lemma 8.2 (for graph automorphisms) of [Spä09]. This gives us the desired extension map. Assume now that C is not a torus (non regular cases in the terminology of [Spä09]). Then [Spä09, 9.2] allows to associate with any χ Irr(C F ) an extension χ Irr(N G (C) F χ ) which is A C,χ fixed. Choosing a representative system of Irr(C F ) under A C action, it is then easy to define an A C equivariant extension map χ χ. The following gives a couple of extra cases where equivariant maximal extendibility will be used later. Lemma Assume (G, F ) is such that G is adjoint and G F is of type 2 A n (q) or B n (2 m ) with n 2, m 1, and q a prime power such that (n + 1, q + 1) = 1. Let d 1 be an integer and C a minimal dsplit Levi subgroup of G. Then there exists an Aut F (G F ) C equivariant extension map for C F N G (C) F. Proof. Note first that we can take G = GL n (F), respectively Sp 2n (F) with 2F = 0, since with our hypotheses we have G F = G F ad Z(G)F and our claim about G F ad is clearly equivalent to the same for G F. Now in G = GL n (F) and Sp 2n (F), it is elementary to describe a minimal dsplit Levi subgroup C and one has N G (C) F = N 1 C with an abelian C, both factors are Aut F (G F ) C stable and C N 1 N 1 represents the same situation as C F N G (C) F in a unitary or symplectic group of smaller rank. One may then reduce to the case where C = 1 and the proof of Lemma 8.1 of [Spä09] applies. 3. An equivariant bijection of l degree characters We now recall the principle of Malle s bijection from [Mal07, 7.8] and prove a version in our case. While Malle deals with the case where G is simple, we assume here that Z(G) is connected, which brings many simplifications. Then we show that the equivariance results of the preceding part allow to choose such a bijection to be equivariant for relevant automorphisms. We keep G a connected reductive group defined over the finite field F q with corresponding Frobenius endomorphism F : G G (that is δ = 1 in the setup of 1.3), and connected center Z(G) = Z (G) Malle s parametrization. Assume l is a prime not dividing q, and d is the multiplicative order of q in Z/lZ, if l is odd, and in Z/4Z, if l = 2. Let S 0 be a Sylow Φ d torus of G, C := C G (S 0 ), and N := N G (S 0 ) and N := N F. Assume C G in duality with C. Let S 0 be the Sylow Φ dtorus of G contained in the center of C (so that C = C G (S 0 )). Definition 3.2. Let M G be the set of quadruples (S, s, λ, η) where
13 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER 13 S is a Sylow Φ d torus of G, s C G (S ) F ss is such that G F : C G (s) F l = 1, λ E(C G (s, S ) F, 1) Irr l (C G (s, S ) F ) and η Irr l (W CG (s)(c G (s, S ), λ) F ). Let M N be the set of triples (s, λ, η) where s (C ) F ss is such that N G (S 0 )F : C G (s, S 0 )F l = 1, λ E(C G (s, S 0 )F, 1) Irr l (C G (s, S 0 )F ), and η Irr l (W CG (s)(c G (s, S 0 ), λ)f ) There exists a natural action of G F on M G, as well as an action of N G (S 0 )F on M N. Assume now Condition 1.9 and that maximal extendibility holds with respect to C F N F, see Definition Accordingly there exists an Nequivariant extension map Λ : Irr(C F ) C F I N F Irr(I). We associate to elements of M G and M N characters of G F, respectively N. Let us define the map ψ G : M G Irr(G F ) by M G (S, s, λ, η) χ G s,ζ with ζ = IC G (s) C G (s,s ),λ (η), and we denote by ψ G (S,s,λ,η) the image of (S, s, λ, η). This character is constant on G F orbits in M G. Similarly, let ψ N : M N Irr(N) be the map defined by M N (s, λ, η) Ind N I N (χ C s,λ )(Λ(χC s,λ ).η i s,λ), where we use the notations of Theorems 2.2 and 2.6 for Jordan decomposition and generalized HarishChandra parametrization, while i s,λ is as in Corollary 2.4. We denote by ψ(s,λ,η) N the image of (s, λ, η), and N G (S 0 )F conjugate elements of M N determine the same character in Irr(N), as Λ is Nequivariant. The following bijection was introduced by Malle (see [Mal07]) in a slightly different situation. Proposition 3.3. There exists a bijection Ω : Irr l (G F ) Irr l (N), with ψ G S 0,s,λ,η ψn (s,λ,η) for any (s, λ, η) M N. This bijection has the following additional property. Remark 3.4. For (s, λ, η) M N we have ψ N (s,λ,η) Irr(N ν) and ψg (S 0,s,λ,η) Irr(GF ν) for the same character ν Irr(Z(G F )), see [Mal07, 2.2]. An essential step for the proof of Proposition 3.3 is the following statement about unipotent characters. Lemma 3.5. [Mal07, Corollary 6.6] A character ζ E(H F, 1) satisfies ζ Irr l (H F ) if and only if ζ = I H L,λ (η) for a minimal dsplit Levi subgroup L, λ E(LF, 1) Irr l (L F ) and η Irr l (W H (L, λ) F ). Proof of Proposition 3.3. The statement is the one of Theorem 7.8 of [Mal07]. While Malle assumes G to be simplyconnected we assume that G has a connected center, which simplifies the proof. We nevertheless briefly recall the main steps. About ψ G. According to Theorem 1.13 a character κ Irr l (G F ) can be written as χ G s,ζ for some element s (G ss) F and a character ζ E(C G (s) F, 1). By the degree formula (1.14)
14 14 MARC CABANES AND BRITTA SPÄTH the fact that l κ(1) implies that l ζ(1) and C G (s) F contains a Sylow lsubgroup of G F. This last condition implies that C G (s) has to contain a Sylow Φ d torus of G (see (1) in the proof of Theorem 5.9 in [Mal07]). So we may assume S 0 C G (s). This condition may also be rephrased as saying that N CG (s)(s 0 )F contains a Sylow lsubgroup of N G (S 0 )F, since G F : N G (S 0 )F l = 1 (apply [BM92, Theorem 3.4(4)] in G ). By [BM92, 3.4] all Sylow Φ d tori of G are G F conjugate. Hence via (s, λ, η) (S 0, s, λ, η) the N G (S 0 )orbits in M N are in bijection with G F orbits in M G. By Lemma 3.5 the unipotent character ζ satisfies l ζ(1) if and only if ζ can be written as I C G (s) C G (s,s ),λ(η) for η Irr l (W C 0 G (s)(c G (s, S 0 ), λ)f ). Consequently, ψ G and therefore Ω are welldefined. About ψ N. We use the normal subgroup C F N with maximal extendibility. The set Irr l (N) is in bijection with Nconjugacy classes of pairs (ξ, η) where ξ Irr l (C F ) is such that N : I N (ξ) l = 1 and η Irr l (I N (ξ)/c F ). Applying the above Lemma to ξ allows to write it as χ C s,λ with s C such that C F : C C (s) F l = 1 and λ E(C C (s) F, 1) Irr l (C C (s) F ). The isomorphism of Corollary 2.4 allows to identify η with an element of Irr l (W CG (s)(c G (s, S 0 ), λ)f ). Comparing cardinalities on applying Corollary 2.4 to (s, λ) and (1, 1) shows that N : I N (χ C s,λ ) = G F : N CG (s)(s 0 )F. Applying again [BM92, 3.4.(4)] in G then tells us that N : I N (ξ) l = 1 if and only if N G (S 0 )F : N G (S 0 )F C G (s) F l = 1, which matches the definition of M N. On the other hand, the map is a bijection thanks to Proposition 1.19 and what has been said about M N /N G (S 0 )F conj M G /G F conj The main theorem. The bijection of Proposition 3.3 relies on the choice of three correspondences of characters. Using now the choices described in Theorems 2.2, 2.6 and 2.9, we show that this bijection can be made equivariant. Theorem 3.7. Let G be a connected reductive group defined over a finite field of order q with associated Frobenius endomorphism F : G G. Assume that Z(G) is connected. Let l be a prime not dividing q, d the integer determined by q and l as in 1.8 and C be a minimal dsplit Levi subgroup for G. Assume the following hypothesis (H) Condition 1.9 is satisfied and the inclusion C F N G (C) F satisfies maximal extendibility (see Definition 1.18) with an Aut F (G F ) C equivariant map Λ (see Theorem 2.9). Then one may choose the Jordan decompositions λ χ G s,λ and the dharishchandra parametrizations η I?,?? (η) such that the bijection of Proposition 3.3 is Aut F (G F ) C equivariant. Irr l (G F ) Irr l (N G (C) F ) Proof. Denote G := G F. Note that C can be taken as the one denoted so in 3.1 above and N G (C) F = N. Let σ Aut F (G F ) C. Thanks to Proposition 3.3, it suffices to show that if (s, λ, η) M N, then there is some triple (s, λ, η ) M N such that σ 1.ψ N (s,λ,η) = ψn (s,λ,η ) and σ 1.ψ G (S 0,s,λ,η) = ψg (S 0,s,λ,η ). Let σ Aut F (G F ) C with σ σ (see Proposition 1.6). We are going to show the above with (s, λ, η ) = (σ (s), σ λ, σ η).
15 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER 15 Concerning Jordan decomposition in G and C, applying Theorem 2.2 we have σ 1.χ G s,ζ = χ G σ (s),σ ζ for any ζ E(C G (s), 1), and similarly σ 1.χ C s,λ = χc σ (s),σ λ for any λ E(C C (s), 1) for a suitable choice of Jordan decomposition that we suppose made in both G and C. For the characters of N, this is enough for our purpose since by our hypothesis on the extension map Λ from C F to N, we can write σ 1.ψ(s,λ,η) N = σ 1.Ind N I N (χ C s,λ )(Λ(χC s,λ )η i s,λ) = Ind N I N (σ 1.χ C s,λ )(σ 1.Λ(χ C s,λ )η i s,λ σ) = Ind N I N (σ 1.χ C s,λ )(Λ(σ 1.χ C s,λ )η i s,λ σ) = Ind N I N (χ C σ (s),σ λ )(Λ(σ 1.χ C s,λ )(σ.η) i σ (s),σ λ) by Corollary 2.4 = Ind N I N (χ C σ (s),σ λ )(Λ(χC σ (s),σ.λ )(σ.η) i σ (s),σ λ) = ψ(σ N (s),σ.λ,σ.η). For the characters of G, we had σ 1.ψ(S G 0,s,λ,η) = σ 1.χ G s,ζ = χg σ (s),σ ζ for ζ = IC G (s) C (η). So C (s),λ we get our claim as soon as we can write σ (I C G (s) C C (s),λ (η)) = Iσ (C G (s)) σ (C C (s)),σ.λ (σ η). This in turn is a consequence of Theorem 2.6 along with Remark 2.7 for H = G and the triples (C G (s), C C (s), λ). This finishes our proof. 4. The inductive McKay condition for simple groups of types 3 D 4, E 8, F 4 and G 2. The inductive McKay condition from [IMN07, 10] has been recalled already in 1.2 above. Theorem 4.1. The finite simple groups of Lie types 3 D 4, E 8, F 4, and G 2 satisfy the inductive McKay condition of [IMN07, 10] for every prime l. Proof. The simple groups with exceptional Schur multipliers (see [GLS98, 6.1]) have been treated in [Mal08a, 4.1]. This accounts for simple groups of types F 4 (2), G 2 (3), G(4). We consider the simple group of type G(2) as being of type 2 A 2 (3). So, for the types considered in our theorem, we may assume that the Schur multiplier of our simple group is trivial and that the outer automorphism group is cyclic generated by field automorphisms, see [GLS98, 2.5]. Let G be the adjoint group of type listed above (except for G 2 (2) where we take G = PGL 3 (F) with 3F = 0) and defined over the field with q elements thanks to a Frobenius endomorphism F : G G (taking q = 9 and the twisted F for PGL 3 ). Note that G F is then our simple group, with trivial Schur multiplier and cyclic outer automorphism group, so we can use the formulation of the inductive McKay condition in 1.2 above. Hence our claim actually reduces to showing that there exists an equivariant bijection for the group G 0 = G F. Let l be a prime not dividing q, then q and l determine d as in 1.8. Let S be a Sylow Φ d torus, C = C G (S) be the corresponding minimal dsplit Levi subgroup in G and N = N G (C). We assume that N contains N G F (Q) for some Sylow lsubgroup Q of G F. The only possible exceptions have been determined in Theorem 5.14 and Lemma 5.18 of [Mal07], and the inductive McKay condition has been verified in these cases in Section 3.3 of [Mal08b]. Thanks to Proposition 1.10, the group N F is also Aut F (G F ) Q stable.
16 16 MARC CABANES AND BRITTA SPÄTH Theorem 2.9 implies that for C F N F there exists an N F equivariant extension map Λ as G F is isomorphic to (G sc ) F for the simply connected group G sc of same type (recall that Z(G sc ) F = {1} in types 3 D 4 (q), E 8 (q), F 4 (q), and G 2 (q), see [GLS98, Table 2.2]). According to Theorem 2.9, and Lemma 2.10 for 2 A 2 (3), we can choose Λ to be Aut F (G F ) C equivariant. We can now apply Theorem 3.7 and get a bijection Irr l (G F ) Irr l (N F ) which is Aut F (G F ) C  equivariant. Thus our claim. The above arguments may also apply to other types where the universal covering group of the simple group can also be related to a connected reductive group with connected center, namely finite simple groups of Lie type with trivial Schur multiplier. We make no attempt at completeness (see also Table 1 of [Bru09]). Proposition 4.2. The simple groups of Lie types B n (2 m ), E 7 (2 m ) and 2 E 6 (3 m ) (m 1, n 2) satisfy the inductive McKay condition of [IMN07, 10] for any prime. Proof. Types B n (2) are treated in [Cb11, 5] for n 3, while type B 2 (2) corresponds to an alternating group for which we have [Mal08a, 3.1]. The remaining simple groups have trivial Schur multipliers (see [GLS98, Theorem 2.5.1, 6.8]), and they write G = (G sc ) F = (G ad ) F. They satisfy Condition 1.9 thanks to [Mal07, 5.14, 5.19]. Their outer automorphism group is cyclic, generated by field automorphisms. For exceptional types E 7 (2 m ) and 2 E 6 (3 m ), we then get our claim by reasoning as in the proof of Theorem 4.1 above. For types B n (2 m ), the only missing argument is to show maximal extendibility as in Theorem 2.9. This is obtained from Lemma The inductive McKay condition for simple groups of type 2 F 4. Theorem 5.1. The finite simple groups of Lie type 2 F 4 all satisfy the inductive McKay condition of [IMN07, 10] for every prime. Proof. Recall that our type is some 2 F 4 (2 2m+1 ) for m 0. The case of m = 0 is [Mal08b, Proposition 3.17]. We assume m 1. Then the simple group is of the form G F where G is a simplyconnected group of type F 4 over an algebraically closed field of characteristic two, and F = F0 2m+1 where F 0 : G G is the Ree bijective endomorphism of [Cr72, ]. The group G F has trivial Schur multiplier, see Theorem of [GLS98] and Out(G F ) is cyclic generated by F 0, see Theorem of [GLS98]. We can use the formulation of the inductive McKay condition given in 1.2 above. For l = 2 we get the socalled defining characteristic case and here, the theorem has been checked in Proposition 6.3 of [HH09] (see also [Bru09] p.412). In the case of l = 3 and 2 2m+1 2, 5 mod 9 the statement is Proposition 3.16 of [Mal08b]. For all other primes l dividing G F and a given Sylow lsubgroup Q of G F there exists an F stable torus S with N G F (Q) N G F (S) by [Mal07, Theorem 8.4], where S is a socalled Sylow Φ (l) torus. By the proof of Lemma 6.5 of [Spä09] the group C G (S) is a maximal torus T. As F 0 (T) is G F conjugate to T we get Aut(G F ) Q Aut F (G F ) T. Malle s parametrization applies to this case by [Mal07, Theorem 8.5], where the Sylow Φ d torus is replaced by the Φ (l) torus S. Our claim is now that this bijection (depending on choices of a Jordan decomposition, d HarishChandra correspondence and equivariant extensions) can be chosen to be equivariant for automorphisms of G F stabilizing Q. We follow the steps of the proof of Theorem 3.7.
17 EQUIVARIANCE IN FINITE REDUCTIVE GROUPS WITH CONNECTED CENTER 17 The first point is to ensure that an equivariant Jordan decomposition can be chosen in both G and T. This is trivial for T, while for G it is included in our statement of Theorem 2.2. Secondly we describe an equivariant generalized HarishChandra theory as in Theorem 2.6. We have to give equivariant bijections Irr(W H (T ) F ) E(H F, T, 1) for any H = C G (s) with s (T ) F ss. The type of (H, F ) then has to be 2 F 4, 2 B 2, 2 A 2 or A 1 (see for instance [DL85]). For types 2 F 4 and 2 B 2, the bijection is given by [Mal07, Theorem 8.5]. For types 2 A 2 and A 1, all maximal F stable tori are minimal dsplit Levi subgroups for some d {1, 2, 6} and the bijection is a special case of Theorem The action of automorphisms of those groups on unipotent characters is trivial by [Mal08b, Theorem 2.5]. The same is true for the relative Weyl groups by Proposition 1.7 since all automorphisms are field automorphisms. Lemma 4.4 and the proof of Lemma 8.2 of [Spä09] show that there exists an Aut F (G F ) T  equivariant extension map Λ for T F N G F (T). This is an analogue of Theorem The inductive AMcondition for blocks with maximal defect We consider in this section lblocks, and prove in a kind of main case for groups with connected center that the bijection constructed in the preceding sections is also compatible with blocks. The aim is to verify the inductive AMcondition from Definition 7.2 of [Spä11a] for the considered groups, most primes l and maximal defect groups. This should be seen as a first step towards a general verification of the inductive AMcondition. We also show briefly how this can relate with the blockwise version of Alperin s weight conjecture (see [Spä11b]) The inductive AMcondition for G 0, l and defect group D. The inductive AMcondition for a simple group G 0 and a prime l stated in Definition 7.2 of [Spä11a] concerns the universal covering group G of G 0 and is a requirement that has to be satisfied for all lsubgroups occurring as defect groups of lblocks of G. For notions and basic results about lblocks (Brauer induction, defect groups, covering blocks), we refer to [N98]. Now, we consider instead of the sets Irr l (X) for finite groups X, the sets Irr 0 (X D) of all ordinary irreducible characters of X that lie in lblocks of X with defect group D and have height zero. We state here the condition that has to be satisfied for a specific lsubgroup D of G such that Irr 0 (G D). In this context, G and D have to satisfy the following: Equivariance: there exists an Aut(G) D stable subgroup N with N G (D) N G, and an Aut(G) D equivariant bijection Ω D : Irr 0 (G D) Irr 0 (N D), Block compatibility: for every χ Irr 0 (G D), if b is the lblock of N containing Ω D (χ), then b G (Brauer induction) is the lblock of G containing χ. Cohomology: for every χ Irr 0 (G D) and Ω D (χ) there exist projective representations with additional properties. As in the case of the inductive McKay condition (see 1.2 above), when the group of outer automorphisms Out(G) is cyclic, this third condition is then a consequence of the equivariance and block compatibility conditions stated first (see last paragraph of the proof of [Spä11a, 8.1]) The inductive AMcondition for groups with connected center. The framework is now the one of groups with connected center as in Theorem 3.7 above, with a prime l satisfying the conditions used in [CE99].
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