Generically free representations, I: Large representations

Abstract

This paper concerns a faithful representation $V$ of a simple linear algebraic group $G$. Under mild assumptions, we show that if $V$ is large enough, then the Lie algebra of $G$ acts generically freely on $V$. That is, the stabilizer in $\text{ Lie}\left(G\right)$ of a generic vector in $V$ is zero. The bound on $\text{ dim }V$ grows like ${\left(\text{ rank }G\right)}^2$ and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position.