Sur les caract\`eres des groupes r\'eductifs finis : applications aux groupes sp\'eciaux lin\'eaires et unitaires

TL;DR

This paper surveys the theory of ordinary characters of finite reductive groups with non-connected centers and proves Lusztig's conjecture on characteristic functions of character sheaves for all finite reductive groups of type A.

Contribution

It provides a comprehensive survey of character theory and confirms Lusztig's conjecture for type A finite reductive groups, including non-split cases.

Findings

Confirmed Lusztig's conjecture for all finite reductive groups of type A

Provided a detailed survey on characters of finite reductive groups with non-connected centers

Extended results to split and non-split groups of type A

Abstract

In the first chapters, this paper contains a survey on the theory of ordinary characters of finite reductive groups with non-connected centre. The last chapters are devoted to the proof of Lusztig's conjecture on characteristic functions of character sheaves for all finite reductive groups of type A, split or not.