Unisingular representations of rank 1 finite simple groups of Lie type

Abstract

A representation $\Phi :G\to {\mathrm{GL}}_n(\mathbb{F})$ of a finite group $G$ is called unisingular if the matrix $\Phi (g)$ admits $1$ as an eigenvalue for any $g\in G$. In this paper, we determine all the complex irreducible unisingular representations of the finite simple groups of Lie type of rank $1$ and of the almost simple sporadic groups.