On eigenvectors for semisimple elements in actions of algebraic groups

Abstract

Let $G$ be a simple simply connected algebraic group defined over an algebraically closed field $K$ and $V$ an irreducible module defined over $K$ on which $G$ acts. Let $E$ denote the set of vectors in $V$ which are eigenvectors for some non-central semisimple element of $G$ and some eigenvalue in $K^∗$. We prove, with a short list of possible exceptions, that the dimension of $\overline{E}$ is strictly less than the dimension of $V$ provided $\dim V > \dim G + 2$ and that there is equality otherwise. In particular, by considering only the eigenvalue $1$, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of $V$ provided $\dim V > \dim G + 2$, with a short list of possible exceptions. In the majority of cases we consider modules for which $\dim V > \dim G + 2$ where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds $\dim G$. In more difficult cases, when $\dim V$ is only slightly larger than $\dim G + 2$, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying $\dim V \leq \dim G + 2$, an immediate observation yields the result for $\dim V < \dim B$ where $B$ is a Borel subgroup of $G$, while in other cases we argue directly.