Geometric Description of Epimorphic Subgroups

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.