Darstellungstheorietage and Nikolaus Conference 2021

Let \(k\) be an algebraically 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 \(kG\)-module and let \(V_{g}(\mu)\) denote the eigenspace corresponding to the eigenvalue \(\mu\in k^{*}\) of \(g \in G\) on \(V\). We set \(\nu_{G}(V)=\min\{\dim(V)-\dim(V_{g}(\mu))\mid g \in G, \mu \in k^{*}\) with \(\mu g \neq 1\}\). 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 behavior.