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

TL;DR

This paper proves that for classical finite reductive groups, irreducible characters of certain abelian centralizers extend to their inertia groups, aiding in verifying the McKay conjecture for these groups and specific primes.

Contribution

It establishes the extension property of irreducible characters for regular Sylow d-tori in classical groups, providing a detailed description of their irreducible characters and verifying the McKay conjecture in this context.

Findings

Irreducible characters of abelian centralizers extend to inertia groups.

Provides a detailed description of irreducible characters of normalizers.

Verifies the McKay conjecture for classical groups and some primes.

Abstract

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