Action of automorphisms on irreducible characters of groups of type \textsf{A}

TL;DR

This paper explicitly describes how automorphisms act on irreducible characters of groups of type A, using rational classes of unipotent support, and applies this to the McKay conjecture.

Contribution

It provides an explicit description of automorphism actions on characters of groups of type A and introduces a criterion for constituents of Gelfand-Graev characters.

Findings

Automorphisms act on characters via rational classes of unipotent support.

Criteria for identifying constituents of Gelfand-Graev characters.

A short proof of the global Sp{"a}th's criterion for the McKay condition.

Abstract

Let $G$ be a finite group isomorphic to $SL_n(q)$ or $SU_n(q)$ for some prime power $q$. In this paper, we give an explicit description of the action of automorphisms of $G$ on the set of its irreducible complex characters. This is done by showing that irreducible constituents of restrictions of irreducible characters of $GL_n(q)$ (resp. $GU_n(q)$) to $SL_n(q)$ (resp. $SU_n(q)$) can be distinguished by the rational classes of their unipotent support which are equivariant under the action of automorphisms. Meanwhile, we give a criterion to explicitly determine whether an irreducible character is a constituent of a given generalized Gelfand-Graev character of $G$. As as application, we give a short proof of the global side of Sp{\" a}th's criterion for the inductive McKay condition for the irreducible characters of $G$.