Matrices of simple spectrum in irreducible representations of cyclic extensions of simple algebraic groups

TL;DR

This paper classifies certain irreducible projective representations of algebraic groups with cyclic extensions, focusing on those with simple spectrum for specific elements, extending previous results for the case when the group is equal to its connected component.

Contribution

It extends the classification of irreducible projective representations with simple spectrum to groups with cyclic extensions of simple algebraic groups.

Findings

Identifies conditions for irreducible projective representations with simple spectrum.

Provides a classification extending known results for connected simple algebraic groups.

Characterizes representations where a specific element induces a simple spectrum.

Abstract

Let $H$ be a linear algebraic group whose connected component $G\neq 1$ is simple and $H/G$ is cyclic. We determine the irreducible projective representations $\phi$ of $H$ such that $\phi(G)$ is irreducible and $\phi(h)$ has simple spectrum for some $h\in H$. The latter means that all irreducible constituents of the group $\phi( \langle h\rangle)$ are of multiplicity 1. (Here $\langle h\rangle$ is the subgroup of $H$ generated by $h$.) This extends an earlier known result for $H=G$.