Almost cyclic regular elements in irreducible representations of simple algebraic groups

2022/03/06 by Testerman, Donna M., Zalesski, Alexandre
#20G05 (primary) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2203.02900

Abstract

Let G be a simple linear algebraic group defined over an algebraically closed field of characteristic p≥ 0 and let ϕ be a p-restricted irreducible representation of G. Let T be a maximal torus of G and s∈ T. We say that s is strongly regular if α(s)≠β(s) for all distinct T-roots α and β of G. Our main result states that if all but one of the eigenvalues of ϕ(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 ϕ are of multiplicity 1.

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