On fusion in unipotent blocks.

Marc Cabanes, Michel Enguehard

Introduction

The context of this note is as follows. One considers a connected reductive group G and a Frobenius endomorphism F: G → G defining G over a finite field of order q. One denotes by G^{F} the associated (finite) group of fixed points.

Let ℓ be a prime not dividing q. We are interested in the ℓ-blocks of the finite group G^{F}. Such a block is called unipotent if there is a unipotent character (see for instance [6, Definition 12.1]) among its representations in characteristic zero. Roughly speaking, it is believed that the study of arbitrary blocks of G^{F} might be reduced to unipotent blocks (see [2, Théorème 2.3], [5, Remark 3.6]). In view of certain conjectures about blocks (see for instance [9]), it would be interesting to further reduce the study of unipotent blocks to the one of principal blocks (blocks containing the trivial character). Our Theorem 7 is a step in that direction : we show that the local structure of any unipotent block of G^{F} is very close to the one of a principal block of a group of related type (notion of “control of fusion”, see [13, § 49]).

1. Centralizers of ℓ-subgroups.

Let G, F, q, G^{F} be as in the introduction. We abbreviate this by saying that (G, F) is “a connected reductive group defined over IF_{q}”.

We freely use the notions of “rationally irreducible components”, corresponding to the orbits of F on the irreducible components of a reductive group G with Frobenius endomorphism F, and of associated “rational type” (see [4, § 1.1]).

Notation 1. [4, Notation 2.3] Let ℓ be a prime not dividing q. Define G_{a} as the central product of Z^{∘}(G) and of the rationally irreducible components of [G, G] whose rational type is of the form (A_{n}, εq^{m}) with ℓ dividing q^{m} − ε. Let G_{b} be the central product of the components of [G, G] not included in G_{a}. Then G = G_{a}.G_{b} (central product).

In this paper, we have to deal with centralizers of nilpotent subgroups. Here are some elementary facts.

Lemma 2. Let G be a connected reductive group and let Y be a finite nilpotent group of semi-simple automorphisms (in the sense of [12, § 7]) of G. Then

(i) (G^{Y})^{∘} is reductive. (ii) if G = G_{a} relatively to a prime ℓ (Notation 1) and Y ⊆ G^{F} is a nilpotent group of semi-simple elements, then H := C_{G}^{∘}(Y) is reductive, F -stable, and H = H_{a}. Proof of Lemma 2. We prove (i) by induction on |Y |. When Y is cyclic, this is [7, Théorème 1.8.(i)]. If Y is not cyclic, let Z be a cyclic non-trivial subgroup of Z(Y). Then Y/Z induces a group of automorphisms of the reductive group K := (G^{Z})^{∘}. By the induction hypothesis (K^{Y})^{∘} = (K^{Y/Z})^{∘} is reductive. Since G^{Y} ⊂ G^{Z} and G^{Y} /K^{Y} is isomorphic to a subgroup of G^{Z}/K, then G^{Y} /(K^{Y})^{∘} is a finite group and (G^{Y})^{∘} = (K^{Y})^{∘}. So (G^{Y})^{∘} is reductive.

(ii) Point (i) applies to the group of inner automorphisms defined by Y, so H is reductive while it is obviously F -stable. If π: G → G_{ad} is the natural epimorphism, then π(C_{G}^{∘}(Y)) = π(C_{G}^{∘}(Z(G)Y)) = C_{G_{ad}}^{∘}(π(Y)) thanks to the standard description of connected centralizers in terms of roots (see [6, Theorem 3.5.3]), and the type of C_{G}^{∘}(Y) is determined by Y′ := π(Y) ⊆ G_{ad}^{F}. Now, let π′: G′ → G_{ad} where G′ is the direct product of general linear groups GL with same rational type as G and corresponding reduction modulo its (connected) center. Then C_{G_{ad}}^{∘}(Y′) = π′(C_{G′}^{∘}(Y′′)) where Y′′ = (π′)^{−1}(Y′)^{F} since (π′)^{−1}(Y′) = Y′′Z(G′) by connexity of Z(G′) and Lang’s theorem.

So, to prove the last assertion, assume G = G_{a} is a direct product of general linear groups. Then C_{G}^{∘}(Y) is the direct product of the centralizers of the projections of Y. We assume G ≅ GL(n)^{c} has irreducible rational type (A_{n}, εq^{c}) with εq^{c} ≡ 1 (mod ℓ). Using the fact that Y has a semi-simple representation in the underlying nc-dimensional space and F permutes the isotypic components, one sees that C_{G}^{∘}(Y) is of type ×_{i}(A_{m_{i}}, (εq^{c})^{c_{i}}) where i ranges over the orbits of F^{c} acting on the simple representations of Y involved, (m_{i} + 1) is the multiplicity of the corresponding simple representation and c_{i} the cardinality of the F^{c}-orbit. It is then clear that C_{G}^{∘}(Y) = C_{G}^{∘}(Y)_{a}.

2. Blocks and Brauer’s third main theorem.

Let G be a finite group, let ℓ be a prime number and O a local complete finite extension of ZZ_{ℓ}. Let K denote the fraction field of O. Assume that K is algebraically closed, so that the elements of Irr(G) correspond with simple KGmodules. All ℓ-blocks of G are now considered as primitive idempotents in the center Z(O[G]) of the group algebra O[G]. They induce a partition of Irr(G), and we denote by b_{G}(χ) the block associated with χ ∈ Irr(G). Let k = O/J(O) and a ↦ \overline{a} be the reduction map modulo J(O) from O[G] to k[G]. Then Z(k[G]) is the image of Z(O[G]) and primitive idempotents are in bijection. We recall the Brauer morphism associated with an ℓ-subgroup Q of G (see [13, § 40]) :

Br_{Q} : (O[G])^{Q} → k[C_{G}(Q)],

∑_{g∈G} λ_{g}g ↦ \overline{∑_{g∈C_{G}(Q)} λ_{g}g}.

The definitions and main properties of subpairs (Q, b) (also called “Brauer pairs”), inclusion of subpairs, defect groups, self-centralizing subpairs and their canonical character are taken from [13, § 40, § 41].

The following is a generalization of R. Brauer’s third main theorem (where H = {1}, ρ = 1_{G}).

Proposition 3. Let ρ ∈ Irr(G) and H be a subgroup of G such that Res_{H}^{G}ρ ∈ Irr(b_{H}) where b_{H} is an ℓ-block of H with central defect group (in H). Let Q ⊆ Q′ two ℓ-subgroups of C_{G}(H). Then (Q, b_{C_{G}(Q)}(Res_{C_{G}(Q)}^{G}ρ)) ⊆ (Q′, b_{C_{G}(Q′)}(Res_{C_{G}(Q′)}^{G}ρ)) in G. Proof of the Proposition. For all subgroup H′ such that H ⊆ H′ ⊆ G, denote σ_{H′} = ∑_{g∈H′} ρ(g^{−1})g ∈ Z(O[H′]). One has Res_{H′}^{G}ρ ∈ Irr(H′), and the ℓ-block b_{H′} of H′ corresponding to Res_{H′}^{G}ρ satisfies the condition

(E)  \overline{b_{H′}} \overline{σ_{H′}} = \overline{σ_{H′}}

(this last fact because ∑_{g∈H′} ρ(g^{−1})g is proportional to the primitive idempotent of Z(K[H′]) associated with Res_{H′}^{G}ρ, see [11, Proposition 3.6.22]). Blocks with central defect groups are easy to express from one of their irreducible characters (combine [11, Proposition 3.6.22, 5.8.14]), this yields b_{H} = \frac{ρ(1)}{|H:Z(H)_{ℓ}|} ∑_{g∈H_{ℓ′}} ρ(g^{−1})g, and \frac{ρ(1)}{|H:Z(H)_{ℓ}|} is a unit in O. Then there exists some g ∈ H such that \overline{ρ(g)} ≠ 0. Therefore \overline{σ_{H′}} ≠ 0 in k[G], and (E) above characterizes b_{H′} thanks to the orthogonality of distinct blocks.

Now, to check the proposition, it suffices to check the case when Q ◁ Q′. Then Res_{C_{G}(Q)}^{G}ρ is fixed by Q′ since ρ is a central function on G, thus σ_{C_{G}(Q)} and b_{C_{G}(Q)} are fixed by Q′. One has Br_{Q′}(σ_{C_{G}(Q)}) = \overline{σ_{C_{G}(Q′)}}. In order to check the inclusion it suffices to check Br_{Q′}(b_{C_{G}(Q)})\overline{σ_{C_{G}(Q′)}} = \overline{σ_{C_{G}(Q′)}}. Using the fact that Br_{Q′} induces an algebra morphism on k[G]^{Q′}, we have Br_{Q′}(b_{C_{G}(Q)})\overline{σ_{C_{G}(Q′)}} = Br_{Q′}(b_{C_{G}(Q)})Br_{Q′}(σ_{C_{G}(Q)}) = Br_{Q′}(b_{C_{G}(Q)}σ_{C_{G}(Q)}) = Br_{Q′}(σ_{C_{G}(Q)}) = \overline{σ_{C_{G}(Q′)}} as claimed.

3. On the local structure of unipotent blocks.

Let (G, F) be a connected reductive group defined over the field IF_{q} of order q. Let ℓ be an odd prime, good for G, not dividing q and ≠ 3 if (G, F) involves type 3D_{4}. Let e be the order of q mod ℓ. Let (L, λ) be a unipotent e-cuspidal pair of G and let b_{G^{F}} (L, λ) be the ℓ-block it defines (see [4, Theorem 4.4]). Then there is a maximal subpair (D, b_{D}) such that D is a Sylow ℓ-subgroup of C_{G}^{∘}([L, L])^{F}, b_{D} is the block defined by the character Res_{C_{G^{F}} (D)}^{L^{F}}λ and (Z^{∘}(L)_{ℓ}^{F} , b_{L^{F}} (λ)) ◁ (D, b_{D}), [4, Lemma 4.5].

The following proposition is about self-centralizing subpairs of (D, b_{D}).

Proposition 4. Let (Y, b_{Y}) be a self-centralizing subpair of G^{F} with canonical character χ ∈ Irr(b_{Y}) and assume (Y, b_{Y}) ⊆ (D, b_{D}). Then C_{G}^{∘}(Y)_{b} = [L, L] and Res_{[L,L]^{F}}^{C_{G}(Y)^{F}} χ = Res_{[L,L]^{F}}^{L^{F}} λ.

Proof of the Proposition. Note first that C_{G}^{∘}(Y) is reductive by Lemma 2.(i), so C_{G}^{∘}(Y)_{b} makes sense.

Let us prove the proposition by induction on dim G. If G is a torus, then D = Y = G_{ℓ}^{F} and our claim is clear. In the general case, note that, since (Y, b_{Y}) and (Z^{∘}(L)_{ℓ}^{F} , b_{L^{F}} (λ)) are self-centralizing and included in (D, b_{D}) [4, Lemma 4.5], then Z(Y) ∩ Z^{∘}(L)_{ℓ}^{F} ⊇ Z(D) [13, Proposition 41.3.b].

Assume that Z(D) ⊈ G_{a}. Let z ∈ Z(D) \ G_{a} and let H := G_{a}C_{G}^{∘}(z) = G_{a}C_{G_{b}}^{∘}(z). Then H is a Levi subgroup of G [4, Proposition 2.1.(ii)] and H ≠ G since Z(G_{b})^{F} is ℓ′. Moreover z ∈ Z^{∘}(L)_{ℓ}^{F}, so H ⊇ L [4, Proposition 3.3]. In order to apply the induction hypothesis, it suffices to check that the inclusion (Y, b_{Y}) ⊆ (D, b_{D}) actually takes place in H^{F}, or equivalently that D, Y, C_{G^{F}} (D), C_{G^{F}} (Y) are all included in H. To see that, it clearly suffices to check C_{G^{F}} (z) ⊆ H. The quotient (G_{a}C_{G_{b}}(z))^{F} /H^{F} equals (G_{a}C_{G_{b}}(z)/G_{a}C_{G_{b}}^{∘}(z))^{F} by Lang theorem and is in turn an isomorphic image of (C_{G_{b}}(z)/C_{G_{b}}^{∘}(z))^{F}. But z clearly acts on G_{b} as a rational ℓ-element of G_{b}, so (C_{G_{b}}(z)/C_{G_{b}}^{∘}(z))^{F} = 1 by [4, Proposition 2.1.(iii)]. Then (G_{a}C_{G_{b}}(z))^{F} ⊆ H^{F} and therefore C_{G^{F}} (z) ⊆ H. The induction hypothesis now gives our claim.

Assume that Z(D) ⊆ G_{a}. Then D ⊆ G_{a} by [4, Notation 2.3.(ii)]. Thus Z^{∘}(L)_{ℓ}^{F} ⊆ G_{a} and therefore L ⊇ G_{b} by [4, Proposition 3.3.(ii)]. Then L = T_{a}G_{b} where T_{a} is a maximal torus of G_{a} [4, Proposition 3.3.(i)]. Since Y ⊆ D, then C_{G}^{∘}(Y) = C_{G_{a}}^{∘}(Y)G_{b}. Lemma 2.(ii) then implies C_{G}^{∘}(Y)_{b} = G_{b} = [L, L]. By [4, Proposition 3.1], λ ∈ E(L^{F} , 1) is the unipotent character whose restriction to G_{b}^{F} is a certain e-cuspidal λ_{b} ∈ E(G_{b}^{F} , 1). Note that λ_{b} is thus in a block of G_{b}^{F} with central defect [4, Proposition 4.3]. Note also that λ_{b}, considered as a character of G_{b}^{F} /(G_{a} ∩ G_{b})^{F} extends into a unipotent character λ̃_{b} of (G_{b}/(G_{a} ∩ G_{b}))^{F}. Now, (G_{b}/(G_{a} ∩ G_{b}))^{F} = (G/G_{a})^{F} = G^{F} /G_{a}^{F} by connexity of G_{a} and Lang Theorem, so λ̃_{b} defines a unique λ̃ ∈ Irr(G^{F}) having G_{a}^{F} in its kernel. Applied to L instead of G, this construction clearly defines an element of E(L^{F} , 1) which coincides with λ_{b} on G_{b}^{F}, so this is λ. Thus λ = Res_{L^{F}}^{G^{F}} λ̃ with λ̃ ∈ Irr(G^{F}).

We now apply Proposition 3 with G = G^{F}, H = G_{b}^{F}, ρ = λ̃. One obtains (Y, b_{C_{G^{F}} (Y)}(Res_{C_{G^{F}} (Y)}^{G^{F}}λ̃)) ⊆ (D, b_{C_{G^{F}} (D)}(Res_{C_{G^{F}} (D)}^{G^{F}}λ̃)) = (D, b_{D}), hence, by the

standard properties of subpairs [13, Corollary 40.9.a], b_{Y} = b_{C_{G^{F}} (Y)}(Res_{C_{G^{F}} (Y)}^{G^{F}}λ̃).

But λ̃ has Z(Y) in its kernel since Y ⊆ G_{a}^{F}. Then Res_{C_{G^{F}} (Y)}^{G^{F}}λ̃ is the canonical

character of b_{Y}, so χ = Res_{C_{G^{F}} (Y)}^{G^{F}}λ̃. Restricting further to [L, L]^{F} completes our proof.

For the notion of control subgroups, we refer to [13, § 49]. The hypotheses are the same as above.

Proposition 5. Let H be a subgroup of G^{F} such that C_{G}^{∘}([L, L])^{F} ⊆ H and H covers the quotient N_{G^{F}} ([L, L], Res_{[L,L]^{F}}^{L^{F}} λ)/[L, L]^{F}. Then H is a b_{G^{F}} (L, λ)-control subgroup of G^{F}.

Proof of the Proposition. Let us take a maximal subpair (D, b_{D}) as in Proposition 4, then D is a (Sylow) ℓ-subgroup of C_{G}^{∘}([L, L])^{F} and b_{D} = b_{C_{G^{F}} (D)}(Res_{C_{G^{F}} (D)}^{L^{F}}λ). Let Y = {(Y, b_{Y}) ⊆ (D, b_{D}) ; N_{G^{F}} (Y, b_{Y}) ⊆ H.C_{G^{F}} (Y)}. It is well-known that H is a b_{G^{F}} (L, λ)-control subgroup of G^{F} if and only if Y contains any self-centralizing (Y, b_{Y}) included in (D, b_{D}) (see [13, Proposition 48.6.b, 49.5]).

Take (Y, b_{Y}) self-centralizing and included in (D, b_{D}). By Proposition 4, C_{G}^{∘}(Y)_{b} = [L, L] and the restriction to [L, L]^{F} of the canonical character of b_{Y} equals Res_{[L,L]^{F}}^{L^{F}} λ. Then N_{G^{F}} (Y, b_{Y}), which acts by algebraic automorphisms of C_{G}^{∘}(Y) commuting with F, normalizes the pair ([L, L], Res_{[L,L]^{F}}^{L^{F}} λ). So N_{G^{F}} (Y, b_{Y}) ⊆ H.[L, L]^{F} ⊆ H.C_{G^{F}} (Y) as claimed.

Keep the same hypotheses as above concerning G, F, q, ℓ, L, λ. Proposition 6. The quotient N_{G}([L, L])/C_{G}^{∘}([L, L]).[L, L] is a finite group whose order is prime to ℓ. Proof. One knows that Z^{∘}(L) is a maximal torus in C_{G}^{∘}([L, L]) (see [4, Proposition 1.7.(i)]), so, by conjugacy of maximal tori, one has N_{G}([L, L])C_{G}^{∘}([L, L]) ⊆ N_{G}(Z^{∘}(L))C_{G}^{∘}([L, L]). Therefore N_{G}([L, L])C_{G}^{∘}([L, L]) = N_{G}(L)C_{G}^{∘}([L, L]) and N_{G}([L, L])/C_{G}^{∘}([L, L]).[L, L] = N_{G}(L)/C_{G}^{∘}([L, L]).L. Let now T be a maximal torus of L and denote the associated root systems (resp. Weyl groups) by the letter R (resp. W). The quotient to be checked is then isomorphic with N_{W (G)}(W (L))/W (TC_{G}^{∘}([L, L])W (L). We may assume that (G, F) is rationally irreducible.

The quotient N_{W (G)}(W (L))/W (L) corresponds with the subgroup C_{L} ⊆ W (G) consisting of the elements of W (G) normalizing a basis Π_{L} of the root system R_{L} of L (see [6, Lemma 9.2.1]), while W (TC_{G}^{∘}[L, L]) corresponds with the roots α such that the corresponding root subgroup U_{α} commutes with all the U_{β} for β ∈ R_{L}. The latter implies α ∈ (R_{L})^{⊥}, but, when all the roots of G have same length, this is an equivalence by the commutator formula. Now the group generated by the reflections associated with the elements of R_{G} ∩ (R_{L})^{⊥} is the subgroup of W (G) fixing all the elements of Π_{L} (see [1, V.3. Proposition 2]), so, when all the roots of G have same length, our quotient is then isomorphic with a group of diagram automorphisms of R_{L}. This rules out at once all the exceptional types since either no good prime divides the order of W (G), or G is of type some E_{i} and ℓ = 5, 7 but then, no proper subdiagram has a diagram automorphism of order ℓ.

In type A, B, C and D, the existence of an e-cuspidal unipotent character implies that [L, L] is of the same rational type with smaller rank than G on q (see [10]), so that N_{W (G)}(W (L)) = W (TC_{G}^{∘}([L, L]))W (L) short of type D where N_{W (G)}(W (L))/W (TC_{G}^{∘}([L, L]))W (L) is of order 2 and induces a diagram automorphism on [L, L].

Theorem 7. Let (G, F) be a connected reductive group defined over IF_{q}, ℓ an odd prime, good for G, not dividing q and ≠ 3 if (G, F) involves rational type ^{3}D_{4}. Let e be the order of q mod ℓ. Let (L, λ) be a unipotent e-cuspidal pair of G, thus defining an ℓ-block b_{G^{F}} (L, λ) (see [4, Theorem 4.4]) and let T be a maximally split torus of [L, L]C_{G}^{∘}([L, L]). Let D be a Sylow ℓ-subgroup of C_{G}^{∘}([L, L])^{F}, hence a defect group of the b_{G^{F}} (L, λ) (see [4, Theorem 4.4.(ii)]). Then, there exists a subgroup H of N_{G^{F}} ([L, L], Res_{[L,L]^{F}}^{L^{F}} λ) and a ℓ′-subgroup T of T^{F} such that

(i) C_{G}^{∘}([L, L])^{F} ⊆ H ∩ [L, L]C_{G}^{∘}([L, L]) ⊆ T C_{G}^{∘}([L, L])^{F}, H[L, L]^{F} = N_{G^{F}} ([L, L], Res_{[L,L]^{F}}^{L^{F}} λ) ; (ii) D is a Sylow ℓ-subgroup of H ; (iii) H is a b_{G^{F}} (L, λ)-control subgroup of G^{F}.

Proof. Assuming point (i), Proposition 5 implies (iii) and Proposition 6 implies (ii).

Let us check (i). Write M = N_{G^{F}} ([L, L], Res_{[L,L]^{F}}^{L^{F}} λ), K = [L, L]C_{G}^{∘}([L, L]). As λ is unipotent T^{F} ⊆ M hence K^{F} ⊆ M. Let Π_{K} be an F -stable basis of the root system of K with respect to T. The letter W denotes the Weyl groups with respect to T.

Let V be the stabilizer of Π_{K} in N_{M}(T)/T^{F} considered as a subgroup N_{M}(T)T/T of W (G)^{F}. By [3, Proposition 4.2] there exists a subgroup N of N_{G}(T) such that N ∩ T = T_{2} and NT = N_{G}(T). Then, define N_{V} as the inverse image of V in N.

One has [L, L] ∩ C_{G}^{∘}([L, L]) ⊆ Z(L) ∩ [L, L] ⊆ Z([L, L]). As [L, L]_{b} = [L, L] (Proposition 4), Z([L, L])^{F} is of order prime to ℓ. It follows that T^{F} /(T ∩ [L, L]^{F} C_{G}^{∘}([L, L])^{F}) is of order prime to ℓ. Hence there exists a subgroup S′ of T_{ℓ′}^{F} such that T^{F} = S′(T ∩ [L, L]^{F} C_{G}^{∘}([L, L])^{F}). Then S = S′[S′, V] satisfies the same relation and is normalized by V. Put H = (N_{V})^{F} SC_{G}^{∘}([L, L])^{F}.

As (1 − F)T_{2} = T_{2} and V ⊆ W (G)^{F} one has (N_{V})^{F} T_{2} = N_{V}. The group [L, L]^{F} C_{G}^{∘}([L, L])^{F} is transitive on maximally split tori of K and nTn^{−1} is such a torus for any n ∈ M, hence M = N_{M}(T)[L, L]^{F} C_{G}^{∘}([L, L])^{F}. Now N_{M}(T) acts on the set of basis of the root system of K and W (K) ≅ N_{K}(T)/T acts regularly on this set. As Π_{K} is F -stable, its stabilizer C in W (G) is F -stable. This gives a semi-direct product N_{W (G)}(W (K)) = W (K).C hence N_{M}(T)/T^{F} = W (K)^{F} .V. Thus (N_{V})^{F} N_{K^{F}} (T) = N_{M}(T) and N_{V} ∩ K = N_{V} ∩ T = T_{2}. It follows that H[L, L]^{F} = (N_{V})^{F} SC_{G}^{∘}([L, L])^{F} [L, L]^{F} = N_{M}(T)C_{G}^{∘}([L, L])^{F} [L, L]^{F} = M and H ⊆ T_{2}^{F} SC_{G}^{∘}([L, L])^{F}.

Remark 8. The above Theorem implies that the “Brauer category” [13, § 47] of b_{G^{F}} (L, λ) is very close to the one of the principal block of a finite reductive group suitably extended by a group of diagram automorphisms. Certain examples suggest that the corresponding module categories of the block should be in some sense equivalent (see [8] for the existence of “perfect isometries” in linear and unitary groups).

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Université Paris 7 - Denis Diderot

UFR de Mathématiques 2, place Jussieu F-75005 PARIS

e-mail : cabanes@math.jussieu.fr