Towards the inductive McKay--Navarro Condition for groups of Lie type

TL;DR

This paper develops tools to verify the inductive McKay--Navarro condition for groups of Lie type, establishing a key bijection and verifying conditions for unipotent characters, advancing the understanding of local-global conjectures in representation theory.

Contribution

It introduces new methods for proving the inductive McKay--Navarro condition for Lie type groups and confirms the condition for specific cases involving type A and unipotent characters.

Findings

Established a bijection for quasisimple groups of Lie type A

Proved the inductive conditions for unipotent characters

Provided tools for verifying the McKay--Navarro condition in Lie type groups

Abstract

We gather tools for proving the inductive McKay--Navarro (or Galois--McKay) condition for groups of Lie type and odd primes. We use this to establish a bijection in the case of quasisimple groups of Lie type A satisfying the equivariance properties needed for the condition. We also prove the inductive conditions for the subset of unipotent characters.