Étale and affine representations of Lie algebras and algebraic groups

A rational representation $p : G \to \mathrm{Aff}(V )$ of an algebraic group $G$ for the affine group over a complex vector space $V$ is called an étale affine representation of $G$ if there exists a point $v \in V$ such that the orbit $\rho(G)v$ is Zariski-open in $V$ and the isotropy group $G_v$ at $v$ is a finite group.

For (complex) reductive groups, every étale affine representation is equivalent to a linear representation and we obtain a special case of a prehomogeneous representation of $G$. Such representations have been classified by Sato and Kimura in some cases. The induced representation on the Lie algebra level gives rise to a pre-Lie algebra structure on the Lie algebra ${g}$ of $G$. For a Lie group $G$, pre-Lie algebra structures on ${g}$ correspond to left-invariant affine structures on $G$. This refers to a well-known question by John Milnor from 1977 on the existence of complete left-invariant affine structures on solvable Lie groups. We present results on the existence of étale affine representations of reductive groups, and discuss a related conjecture of V. Popov concerning attenable groups and linearizable subgroups of the affine Cremona group. Furthermore we give some results on pre-Lie algebra structures on Lie algebras. For modular Lie algebras these results are based on the article First Cohomology Groups for Classical Lie Algebras by Jens Carsten Jantzen.