Jordan decomposition and real-valued characters of finite reductive groups with connected center

TL;DR

This paper characterizes all real-valued irreducible complex characters of finite reductive groups with connected center over finite fields using Jordan decomposition, providing a comprehensive parameterization.

Contribution

It introduces a complete parameterization of real-valued characters of finite reductive groups with connected center via Jordan decomposition, advancing understanding of their representation theory.

Findings

All real-valued irreducible characters are explicitly parameterized.

The parameterization is achieved through Jordan decomposition.

Provides a framework for analyzing characters of finite reductive groups.

Abstract

Let $\bf{G}$ be a connected reductive group with connected center defined over $\mathbb{F}_q$, with Frobenius morphism $F$. We parameterize all of the real-valued irreducible complex characters of ${\bf G}^F$ using the Jordan decomposition of characters.