Etale representations for reductive algebraic groups with one-dimensional center

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.

original version of this paper has been split up; the current version contains an extension of the structural part, see arxiv:1610.07005 for the classification results second update: fixed some mistakes in the introduction; Theorem B from the previous version now attributed to a previous paper by Baues; additional references

References in corpus (1)

• Tables of prehomogeneous modules and étale modules of reductive algebraic groups

Cited by in corpus (1)

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