Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

TL;DR

This paper establishes a Galois-equivariant Jordan decomposition for characters of finite reductive groups when the centralizer is connected, and proves the uniqueness of this decomposition under certain conditions.

Contribution

It introduces a Galois-equivariant Jordan decomposition for finite reductive groups with connected centralizer, extending previous results to this specific case.

Findings

Jordan decomposition can be chosen to be Galois-equivariant when the centralizer is connected

Uniqueness of the Jordan decomposition is proven under the connected centralizer condition

The results generalize the known cases with connected center to connected centralizer scenarios.

Abstract

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne--Michel's unique Jordan decomposition in the connected center case.