Sylow d-tori of classical groups and the McKay conjecture, II

The aim of this paper is to describe the irreducible ordinary characters of the normalisers of Sylow tori of a finite reductive group G of classical type. We prove that every character χ∈Irr(L) extends as an irreducible character to its inertia group in N, where L is the centraliser and N the normaliser of a Sylow torus in G. Using this result Malle has established in Malle (2007) [8], a bijection between Irrℓ′(G) and Irrℓ′(N), where N is the normaliser of a suitable Sylow torus in G. This is a first step in proving the inductive McKay condition from Isaacs et al. (2007) [5] for the classical groups of Lie type. Furthermore the bijection enables us to prove the McKay conjecture for primes ℓ>3, different from the defining characteristic of G.