Etale representations for reductive algebraic groups with factors $Sp_n$ or $SO_n$

Abstract

A complex vector space $V$ is an \'etale $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit and $\dim G=\dim V$. Such a module is called super-\'etale if the stabilizer of a point in the open orbit is trivial. Popov proved that reductive algebraic groups admitting super-\'etale modules are special algebraic groups. He further conjectured that a reductive group admitting a super-\'etale 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 $SL_n$ for some $n$ or $Sp_2$. We show that this is not the case by constructing a family of super-\'etale modules for groups with a factor $Sp_n$ for arbitrary $n\geq1$. A similar construction provides a family of \'etale modules for groups with a factor $SO_n$, which shows that groups with \'etale modules with non-trivial stabilizer are not necessarily special. Both families of examples are somewhat surprising in light of the previously known examples of \'etale and super-\'etale 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 \'etale representation.