On semisimple classes and semisimple characters in finite reductive groups

TL;DR

This paper investigates semisimple classes and characters in finite reductive groups, focusing on elements with disconnected centralizers, their stability, and implications for the McKay Conjecture in representation theory.

Contribution

It provides new results on the number of F-stable semisimple classes with disconnected centralizers and proves certain groups are 'good' in defining characteristic, advancing the McKay Conjecture.

Findings

Count of F-stable semisimple classes with disconnected centralizer derived

Extendibility of semisimple characters to inertia groups established

Twisted and untwisted E_6 groups shown to be 'good' in defining characteristic

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 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.