Upper bounds for eigenvalue multiplicities of almost cyclic elements in irreducible representations of simple algebraic groups

TL;DR

This paper investigates the eigenvalue multiplicities in irreducible representations of simple algebraic groups, establishing bounds based on the group's rank for elements with limited eigenvalue multiplicities.

Contribution

It provides new bounds on eigenvalue multiplicities for certain elements in irreducible representations of simple algebraic groups, linking eigenvalue multiplicity to group rank.

Findings

Bound on eigenvalue multiplicity in terms of group rank

Characterization of elements with limited eigenvalue multiplicities

Extension of previous eigenvalue multiplicity results

Abstract

We study the irreducible representations of simple algebraic groups in which some non-central semisimple element has at most one eigenvalue of multiplicity greater than 1. We bound the multiplicity of this eigenvalue in terms of the rank of the group.