Generically free representations I: large representations

TL;DR

This paper proves that large enough representations of simple algebraic groups have a generically free Lie algebra action, with bounds depending on the group's rank, and explores implications for invariants and stabilizers.

Contribution

It establishes explicit bounds on the dimension of representations ensuring generic freeness of the Lie algebra action for simple algebraic groups.

Findings

Lie algebra acts generically freely on large representations

Bound on dimension grows quadratically with group rank

Results apply across different field characteristics

Abstract

For a simple linear algebraic group $G$ acting faithfully on a vector space $V$ and under mild assumptions, we show: if $V$ is large enough, then the Lie algebra of $G$ acts generically freely on $V$. That is, the stabilizer in the Lie algebra of $G$ of a generic vector in $V$ is zero. The bound on $\dim V$ grows like $(\mathrm{rank} G)^2$ 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 as well as the determination of the invariants of the representation.