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

We prove for finite reductive groups G of classical type, that every character χ∈Irr(L) extends to its inertia group in N, where L is an abelian centraliser of a Sylow d-torus S of G and N:=NG(S). This gives a precise description of the irreducible characters of N and proves |Irrℓ′(G)|=|Irrℓ′(N)| according to Malle (2007) [10] for all primes ℓ determined by S. Furthermore this enables us to verify the McKay conjecture in this situation for some primes.