A note on \'etale representations from nilpotent orbits

TL;DR

This paper explores classical constructions related to nilpotent orbits in semisimple Lie algebras to identify new examples of étale representations for reductive algebraic groups, clarifying their interrelations.

Contribution

It describes two classical constructions for nilpotent orbit classification and determines which reductive groups and étale representations emerge from them.

Findings

Identifies new étale representations arising from classical nilpotent orbit constructions.

Clarifies the relationship between Vinberg's and Bala-Carter's constructions.

Provides criteria for when reductive groups admit étale representations in these frameworks.

Abstract

A linear \'etale representation of a complex algebraic group $G$ is given by a complex algebraic $G$-module $V$ such that $G$ has a Zariski-open orbit on $V$ and $\dim G=\dim V$. A current line of research investigates which \'etale representations can occur for reductive algebraic groups. Since a complete classification seems out of reach, it is of interest to find new examples of \'etale representations for such groups. The aim of this note is to describe two classical constructions of Vinberg and of Bala & Carter for nilpotent orbit classifications in semisimple Lie algebras, and to determine which reductive groups and \'etale representations arise in these constructions. We also explain in detail the relation between these two~constructions.