Geometric Description of Epimorphic Subgroups

TL;DR

This paper characterizes observable and epimorphic subgroups of affine algebraic groups over algebraically closed fields of characteristic zero using geometric orbit closure properties.

Contribution

It provides a geometric criterion for identifying observable and epimorphic subgroups based on orbit closure conditions.

Findings

Observable subgroups correspond to orbits with 0 in their closure.

Epimorphic subgroups correspond to closed orbits.

Provides a geometric perspective on subgroup properties.

Abstract

Let $G$ be an affine algebraic group over an algrebraically closed field $\mathbb K$ of characteristic 0 and $H$ be a subgroup of $G$. The stabilizer of all the set of all vector-functions of $\mathbb K[G]^H$ with respect to the right action of $H$ is $\hat H$. $V^H=V^{\hat H}$ for a $G$-module $V$. The subgroup $H$ is called observable if $H=\hat H$ and epimorphic if $G=\hat H$. In this work I show that under some natural restrictions $H$ is observable if and only if some orbit of some group contains 0 in the closure and $H$ is epimorphic if and only the same orbit is closed.