Action of Automorphisms on Irreducible Characters of Symplectic Groups

TL;DR

This paper describes how automorphisms act on irreducible characters of finite symplectic groups, utilizing Deligne--Lusztig induction, and applies this to verify a key condition in the inductive McKay conjecture.

Contribution

It provides a detailed description of automorphism actions on characters of symplectic groups and establishes the inductive McKay condition for these groups.

Findings

Automorphism action on irreducible characters is explicitly characterized.

A version of Deligne--Lusztig induction equivariance under automorphisms is formulated.

The inductive McKay condition is verified for symplectic groups.

Abstract

Assume $G$ is a finite symplectic group $\mathrm{Sp}_{2n}(q)$ over a finite field $\mathbb{F}_q$ of odd characteristic. We describe the action of the automorphism group $\mathrm{Aut}(G)$ on the set $\mathrm{Irr}(G)$ of ordinary irreducible characters of $G$. This description relies on the equivariance of Deligne--Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Sp\"ath's criterion for the inductive McKay condition holds for the irreducible characters of $\mathrm{Sp}_{2n}(q)$.