Closed Orbits and Descents for Enhanced Standard Representations of Classical Groups

TL;DR

This paper classifies closed orbits in enhanced standard representations of classical groups over algebraically closed fields of characteristic zero, revealing their stability under MVW-extensions and detailing stabilizers and normal space actions.

Contribution

It provides a complete classification of closed orbits in these representations, including stability under extensions and explicit descriptions of stabilizers and actions.

Findings

Classified all closed orbits in the enhanced standard representation.

Proved all closed orbits are stable under MVW-extensions.

Explicitly described stabilizer groups and their actions on normal spaces.

Abstract

Let $G=\mathrm{GL}_n(\mathbb{F})$, $\mathrm{O}_n(\mathbb{F})$, or $\mathrm{Sp}_{2n}(\mathbb{F})$ be one of the classical groups over an algebraically closed field $\mathbb{F}$ of characteristic $0$, let $\breve{G}$ be the MVW-extension of $G$, and let $\mathfrak{g}$ be the Lie algebra of $G$. In this paper, we classify the closed orbits in the enhanced standard representation $\mathfrak{g}\times E$ of $G$, where $E$ is the natural representation if $G=\mathrm{O}_n(\mathbb{F})$ or $\mathrm{Sp}_{2n}(\mathbb{F})$, and is the direct sum of the natural representation and its dual if $G=\mathrm{GL}_n(\mathbb{F})$. Additionally, for every closed $G$-orbit in $\mathfrak{g}\times E$, we prove that it is $\breve{G}$-stable, and determine explicitly the corresponding stabilizer group as well as the action on the normal space.