Galois group action and Jordan decomposition of characters of finite reductive groups with connected center

TL;DR

This paper investigates how Galois automorphisms affect the Jordan decomposition of irreducible characters of finite reductive groups with connected center, providing explicit descriptions of the transformed characters.

Contribution

It establishes the behavior of Jordan decompositions of characters under Galois automorphisms for connected reductive groups with connected center.

Findings

Explicit description of Galois action on Jordan decompositions

Extension of character theory to Galois conjugates

Insights into the structure of finite reductive groups

Abstract

Let $\mathbf{G}$ be a connected reductive group with connected center defined over $\mathbb{F}_q$, with Frobenius morphism F. Given an irreducible complex character $\chi$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galois automorphism $\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we give the Jordan decomposition of the image ${^\sigma \chi}$ of $\chi$ under the action of $\sigma$ on its character values.