Generic stabilizers for simple algebraic groups

TL;DR

This paper investigates the stabilizers of generic vectors in representations of simple algebraic groups, providing detailed results on their structure, smoothness, and dimensions at the group scheme level.

Contribution

It establishes the existence of stabilizers in general position for simple algebraic groups and characterizes when these stabilizers are smooth or trivial.

Findings

Existence of stabilizers in general position for simple groups.

Conditions for stabilizers to be smooth or trivial.

Classification of representations with specific stabilizer properties.

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 $G(k)$ 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 $\dim V/G < \dim G$, or there is a $v \in V$ whose stabilizer in $G$ is trivial.