Spectra of non-regular elements in irreducible representations of simple algebraic groups

TL;DR

This paper characterizes the natural representations of classical simple algebraic groups by analyzing the spectra of non-regular semisimple elements, revealing minimal dimension conditions for almost simple spectra.

Contribution

It provides a classification of irreducible representations where non-regular semisimple elements have almost simple spectra, especially highlighting classical types and minimal dimensions.

Findings

Classical groups are characterized by spectra of non-regular semisimple elements.

Minimal dimension representations exhibit almost simple spectra for certain elements.

Most such representations correspond to natural representations up to Frobenius twists.

Abstract

We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if G is a simply connected simple linear algebraic group and f is a non-trivial irreducible representation of G in some GL(V) for which there exists a non-regular non-central semisimple element s in G such that f(s) has almost simple spectrum, then, with few exceptions, G is of classical type and dim V is minimal possible. Here the spectrum of a diagonalizable matrix is called simple if all eigenvalues are of multiplicity 1, and almost simple if at most one eigenvalue is of multiplicity greater than 1. This yields a kind of characterization of the natural representation (up to their Frobenius twists) of classical algebraic groups in terms of the behavior of semisimple elements.