Rational elements in representations of simple algebraic groups, I

TL;DR

This paper classifies rational elements of odd order in simple algebraic groups of types A, B, and C, and determines when these elements have eigenvalue 1 in irreducible representations.

Contribution

It explicitly determines all triples of simple algebraic groups, rational semisimple elements, and irreducible representations where the element's image has eigenvalue 1.

Findings

Classification of rational elements of odd order in simple algebraic groups.

Identification of conditions for eigenvalue 1 in representations.

Results hold over algebraically closed fields of any characteristic.

Abstract

A finite order element $g$ of a group $G$ is called rational if $g$ is conjugate to $g^i$ for every integer $i$ coprime to the order $g$. We determine all triples $(G,g,\phi)$, where $G$ is a simple algebraic group of type $A_n,B_n$ or $C_n$ over an algebraically closed field of characteristic $p\geq 0$, $g\in G$ is a rational odd order semisimple element and $\phi$ is an irreducible representation of $G$ such that $\phi(g)$ has eigenvalue 1.