On equivariant bijections relative to the defining characteristic

Abstract

This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in the defining characteristic case. Using a general argument based on the representation theory of connected reductive groups with disconnected center, we show that the inductive McKay condition holds if the Schur multiplier of the simple group has order 2. As a consequence, the simple groups P Ω 2 m + 1 ( p n ) and PSp 2 m ( p n ) are “good” for p > 2 and the simple groups E 7 ( p n ) are “good” for p > 3 in the sense of Isaacs, Malle and Navarro. We also describe the action of the diagonal and field automorphisms on the semisimple and the regular characters.