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We prove for finite reductive groups $G$ of classical type, that everyirreducible character of $L$ extends to its inertia group in $N$, where $L$ isan abelian centraliser of a Sylow $d$-torus $\mathbf S$ of $G$ and$N:=N_G(\mathbf S)$. This gives a precise description of the irreduciblecharacters of $N$. Furthermore it enables us to verify the McKay conjecture inthis situation for $G$ and some primes.

Regular Sylow $d$-Tori of classical groups and the McKay conjecture