Restrictions of irreducible characters of finite groups of Lie type to derived subgroups and regular semisimple elements

TL;DR

This paper investigates how most irreducible characters of finite groups of Lie type behave when restricted to derived subgroups, revealing that they generally remain irreducible, supported by asymptotic and semisimple element analysis.

Contribution

It introduces the observation that nearly all irreducible characters stay irreducible upon restriction to derived subgroups, using asymptotic and regular semisimple element results.

Findings

Most irreducible characters remain irreducible on derived subgroups

Asymptotic results support the irreducibility behavior

Analysis of strongly regular semisimple elements underpins the findings

Abstract

In this note, we formulate an observation that "almost all" irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.