Abstract: Let $G$ be a simple algebraic group, $U$ be a maximal unipotent subgroup of $G$, and $X$ be the affine $G$-variety corresponding to the algebra of regular functions on the homogeneous space $G/U$. We classify all subgroups $H$ of $G$ such that the restricted action $H:X$ is stable, and that $H$ is either a simple or a semisimple irreducible subgroup of $G$.
Full text of the article: