Unisingular representations of rank 1 finite simple groups of Lie type

A representation $\mathrm{\Phi }:G\to {\mathrm{G}\mathrm{L}}_n(\mathbb{F})$ of a finite group $G$ is called unisingular if the matrix $\mathrm{\Phi }(g)$ 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.