Counting p′-characters in finite reductive groups

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 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 G(F) at the prime p. In particular, this conjecture is true for G(F) in the defining characteristic for a simple simply connected group G of type B-n, C-n, E-6 or E-7. 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 G(F) for any simple algebraic group G.