Abstract

After a brief historical overview of the study of eigenvalue multiplicities in representations of simple algebraicgroups and the finite groups of Lie type, we report on two recent articles with A. Zalesski, in which we studysemisimple elements having “almost simple” spectrum in an irreducible representation of a simple algebraicgroup. More precisely, we show that if such an element acts with at most one eigenvalue of multiplicitygreater than 1 in some irreducible representation of a simple algebraic group, then all nonzero weights ofthe representation have multiplicity one and with very few exceptions the semisimple element is regular.We go on to study the behavior of regular semisimple elements acting on the representations whose nonzeroweights have multiplicity 1 and introduce the notion of a “strongly regular” semisimple element and showthat, with specified exceptions, a semisimple element acting with almost simple spectrum on some irreduciblerepresentation of a simple algebraic group is strongly regular.