Almost cyclic regular elements in irreducible representations of simple algebraic groups

TL;DR

This paper investigates the eigenvalue multiplicities of elements in simple algebraic groups' representations, showing that near-regular eigenvalue conditions imply strong regularity of the element, extending previous results on regularity.

Contribution

It extends prior work by proving that if most eigenvalues of a group element's representation are simple, then the element is strongly regular, with some exceptions.

Findings

Most eigenvalues being simple implies strong regularity of the element.

The result generalizes earlier findings on regularity and weight multiplicities.

Exceptions to the rule are explicitly characterized.

Abstract

Let $G$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p\geq 0$ and let $\phi$ be a $p$-restricted irreducible representation of $G$. Let $T$ be a maximal torus of $G$ and $s\in T$. We say that $s$ is strongly regular if $\alpha(s)\ne\beta(s)$ for all distinct $T$-roots $\alpha$ and $\beta$ of $G$. Our main result states that if all but one of the eigenvalues of $\phi(s)$ are of multiplicity 1 then, with a few specified exceptions, $s$ is strongly regular. This can be viewed as an extension of our earlier result saying that under the same hypotheses, $s$ must be regular and all non-zero weights of $\phi$ are of multiplicity 1.