Generic Stabilizers for Simple Algebraic Groups

Abstract

We prove a myriad of results related to the stabilizer in an algebraic group G of a generic vector in a representation V of G over an algebraically closed field k. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of G and the group $\mathit{G}(\mathit{k})$ of k-points. For G simple and V faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those G and V for which the stabilizer in general position is smooth, or $dim\mathit{V}/\mathit{G}<dim\mathit{G}$, or there is a $\mathit{v}\in \mathit{V}$ whose stabilizer in G is trivial.