On Unipotent Supports of Reductive Groups with a Disconnected Centre

TL;DR

This paper extends Lusztig's relationship between characters and unipotent supports to cases where the center of a simple algebraic group is disconnected, and explores applications to the group's character theory over finite fields.

Contribution

It generalizes Lusztig's numerical relationship to disconnected centers and applies this to deepen understanding of the character theory of finite groups of Lie type.

Findings

Extended Lusztig's relationship to disconnected centers

Derived new formulas relating characters and unipotent supports

Provided applications to character theory of finite groups

Abstract

Let $G$ be a simple algebraic group defined over a finite field of good characteristic, with associated Frobenius endomorphism $F$. In this article we extend an observation of Lusztig, (which gives a numerical relationship between an ordinary character of $G^F$ and its unipotent support), to the case where $Z(G)$ is disconnected. We then use this observation in some applications to the ordinary character theory of $G^F$.