Eigenvalue multiplicities of group elements in irreducible representations of simple linear algebraic groups

Let $k$ be an algebraicallly closed field of characteristic $p\geq 0$ and let $G$ be a linear algebraic group of rank $\ell\geq 1$ over $k$. Let $V$ be a rational finite-dimensional $kG$-module and let $V_g(\mu)$ denote the eigenspace corresponding to the eigenvalue $\mu\in k^*$ of $g \in G$ on $V$. We set $\nu_G(V)=\min\{\dim(V)-\dim(V_g(\mu))| g \in G \setminus Z(G), \mu \in k^*\}$. In this talk we will identify pairs $(G,V)$ of simple simply connected linear algebraic groups and of rational irreducible tensor-indecomposable $kG$-modules with the property that $\nu_G(V)\leq \sqrt{\dim(V)}$. This problem is an extension of the classification result obtained by Guralnick and Saxl for $\nu_G(V)\leq \max\{2,\frac{\sqrt{\dim(V)}}{2}\}$. One motivation for studying such problems is to identify subgroups of linear algebraic groups based on element behaviour.