Counting p ′-characters in finite reductive groups

Abstract

This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field 𝔽q of characteristic p > 0 with corresponding Frobenius map F. We prove that, if p is a good prime for G and if the group of F‐coinvariants of the component group of the centre of G has prime order, then the relative McKay conjecture holds for GF at the prime p. In particular, this conjecture is true for GF in the defining characteristic for a simple simply connected group G of type Bn, Cn, E6 or E7. Our main tools are the theory of Gelfand–Graev characters for connected reductive groups with disconnected centre developed by Digne, Lehrer and Michel and the theory of cuspidal Levi subgroups. We also explicitly compute the number of semisimple classes of GF for any simple algebraic group G.

On equivariant bijections relative to the defining characteristic

“…2 Remark 5.12. In the proof of Theorem 5.11, we need the assumption that p is a good prime for G only in order to apply the results of [3]. Proof.…”

On semisimple classes and semisimple characters in finite reductive groups

“…1Z ,3 is in bijection with the D-orbits of size 3 of Irr s (B σ i F j , 1 Z ). As above,[1, 2.17] and[5, Lemma 5.4] implies that |O′ σ i F j D,1Z ,3 | = p 2n/d . This discussion proves that, if ν = 1 Z , then for every H ≤ A Now, if ν = ε m with m = ±1, then A ν = σF and for every H = σ i F n/d ≤ A ν , the same argument shows that | = p 2n/d = |O ′H D,ε m ,3 |.…”

“…In [5], we explicitly computed the number of semisimple classes of G * F * when G * is simple and p is a good prime for G * . For that, we used the theory of Gelfand-Graev characters for connected reductive groups with disconnected center, developed by Digne-Lehrer-Michel in [12] and [13].…”

“…This method gives a lot of information on the F * -stable semisimple G * -classes of G * with disconnected centralizer (by the centralizer of an F * -stable class, we mean the centralizer of a fixed F *stable representative), which allows us to prove the inductive McKay condition in defining characteristic for simple groups coming from simple algebraic groups with fundamental group of order 2; see [7,Theorem 1.1]. However, we cannot derive from [5] all information that we need, especially phenomena appearing only in the algebraic group, as for example the description of the F * -stable semisimple classes with disconnected centralizer such that the fixed-point subgroup A G * (s) F * is trivial (s is any F * -stable representative); see Remark 2.14. In this work, we will consider the Brauer complex, initially introduced by J. Humphreys in [18] for describing p-modular representation theory of G F . When G is a simple simply-connected group, Deriziotis proved in [11] that the F -stable semisimple classes of G (and thus, the semisimple classes of G F ) are parametrized by the faces of the Brauer complex of maximal dimension.…”

The McKay conjecture and Brauer's induction theorem

“…Theorem 5.2 is a generalisation of a result due to Brunat ([5], Theorem 1.1). We use many of the same ingredients as the proof in [5], but avoid explicit computations.…”