Etale representations for reductive algebraic groups with one-dimensional center

TL;DR

This paper investigates etale modules for reductive algebraic groups with one-dimensional centers, establishing constraints on their structure and proving non-existence results for certain group configurations.

Contribution

It characterizes etale modules for such groups, showing irreducible submodules are non-regular and deriving conditions that exclude their existence in specific cases.

Findings

Irreducible submodules of etale modules are non-regular.

Constraints on ranks of simple factors in non-regular prehomogeneous modules.

No etale modules exist for groups of the form GL_1×S×...×S with S simple.

Abstract

A complex vector space $V$ is a prehomogeneous $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit. The module is called etale if $\dim V=\dim G$. We study etale modules for reductive algebraic groups $G$ with one-dimensional center. For such $G$, even though every etale module is a regular prehomogeneous module, its irreducible submodules have to be non-regular. For these non-regular prehomogeneous modules, we determine some strong constraints on the ranks of their simple factors. This allows us to show that there do not exist etale modules for $G=\mathrm{GL}_1\times S\times\cdots\times S$, with $S$ simple.