Darstellungstheorietage and Nikolaus Conference 2021

Speaker: Ana-Maria Retegan (Lausanne)

Title: Eigenvalue Multiplicities of Group Elements in Irreducible Representations of Simple Linear Algebraic Groups

Abstract:

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.