Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

TL;DR

This paper investigates the values of irreducible characters on specific elements related to regular unipotents in extended finite general linear and unitary groups, revealing new insights into their real-valued characters and conjugacy classes.

Contribution

It introduces a novel analysis of character values on elements linked to regular unipotents in extended groups, expanding understanding of their representation theory.

Findings

Identification of elements squaring to regular unipotents in extended groups

Evaluation of irreducible character values on these elements

Results on real conjugacy classes and real-valued characters

Abstract

When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.