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 F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the F-coinvariants of the component group of the center of G has prime order and if p is a good prime for G, then the relative McKay conjecture holds for G at the prime p. In particular, this conjecture is true for G^F in defining characteristic for G a simple and simply-connected group of type B_n, C_n, E_6 and E_7. Our main tools are the theory of Gelfand-Graev characters for connected reductive groups with disconnected center developed by Digne-Lehrer-Michel and the theory of cuspidal Levi subgroups. We also explicitly compute the number of semisimple classes of G^F for any simple algebraic group G.