Equivariant character correspondences and inductive McKay condition for type A

TL;DR

This paper verifies the inductive McKay condition for simple groups of Lie type A, using character theory and automorphism-equivariant Jordan decomposition, advancing the understanding of the McKay conjecture.

Contribution

It establishes the inductive McKay condition for type A groups, introducing new methods involving equivariant character correspondences and generalized Gelfand-Graev representations.

Findings

Verification of the inductive McKay condition for type A groups.

Development of automorphism-equivariant Jordan decomposition.

Potential applicability of methods to other Lie types.

Abstract

As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type $A_{n-1}$, split or twisted. Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$ related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.