On semisimple classes and semisimple characters in finite reductive groups

Abstract

In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group G defined over a finite field with corresponding Frobenius map F and derive the number of F-stable semisimple classes of G with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup G F to their inertia group in the full automorphism group. As a consequence, we prove that “twisted” and “untwisted” simple groups of type E 6 are “good” in defining characteristic, which is a contribution to the general program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture in representation theory of finite groups.