Etale representations for reductive algebraic groups with factors $S{p}_n$ or $S{O}_n$

Abstract

A complex vector space $V$ is an étale $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit and $\dim G=\dim V$. Such a module is called super-étale if the stabilizer of a point in the open orbit is trivial. Popov proved that reductive algebraic groups admitting super-étale modules are special algebraic groups. He further conjectured that a reductive group admitting a super-étale module is always isomorphic to a product of general linear groups. In light of previously available examples, one can conjecture more generally that in such a group all simple factors are either $S{L}_n$ for some $n$ or $S{p}_2$. We show that this is not the case by constructing a family of super-étale modules for groups with a factor $S{p}_n$ for arbitrary $n\ge 1$. A similar construction provides a family of étale modules for groups with a factor $S{O}_n$, which shows that groups with étale modules with non-trivial stabilizer are not necessarily special. Both families of examples are somewhat surprising in light of the previously known examples of étale and super-étale modules for reductive groups. Finally, we show that the exceptional groups $F_4$ and $E_8$ cannot appear as simple factors in the maximal semisimple subgroup of an arbitrary Lie group with a linear étale representation.