Jordan Decomposition for the Alperin-McKay Conjecture

TL;DR

This paper advances the understanding of the Alperin-McKay conjecture by leveraging Jordan decomposition and block equivalences to reduce the verification process to quasi-isolated blocks in finite groups of Lie type.

Contribution

It introduces a method to reduce the inductive Alperin-McKay condition verification to quasi-isolated blocks using Jordan decomposition and automorphism lifting techniques.

Findings

Reduction of the inductive condition verification to quasi-isolated blocks.

Extension of Bonnafé-Rouquier equivalence to include automorphisms.

Application of Jordan decomposition in the context of the conjecture.

Abstract

Sp\"ath showed that the Alperin-McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin-McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnaf\'e-Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.