The McKay-conjecture for exceptional groups and odd primes

Abstract

Let $\mathbf{G}$ be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map $F:\mathbf{G}→\mathbf{G}$ and $\mathbf{G}:=\mathbf{G}^F$, such that the root system is of exceptional type or $\mathbf{G}$ is a Suzuki-group or Steinberg’s triality group. We show that all irreducible characters of $C_G(\mathbf{S})$, the centraliser of $\mathbf{S}$ in $G$, extend to their inertia group in $N_G(\mathbf{S})$, where $\mathbf{S}$ is any $F$-stable Sylow torus of $(\mathbf{G},F)$. Together with the work in [17] this implies that the McKay-conjecture is true for $G$ and odd primes $\ell$ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [15].