# Cuspidal characters and automorphisms

## TL;DR

This paper studies how outer automorphisms affect irreducible characters of finite groups of Lie type, providing results for cuspidal characters and verifying the inductive McKay condition for specific simple groups.

## Contribution

It offers new insights into the action of automorphisms on cuspidal characters and confirms the inductive McKay condition for additional infinite families of simple groups.

## Key findings

- Confirmed the behavior of automorphisms on cuspidal characters.
- Verified the inductive McKay condition for certain simple groups.
- Extended understanding of automorphisms in finite groups of Lie type.

## Abstract

We investigate the action of outer automorphisms of finite groups of Lie type on their irreducible characters. We obtain a definite result for cuspidal characters. As an application we verify the inductive McKay condition for some further infinite families of simple groups at certain primes.