Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem

Abstract

The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(\rho ,V)$ be a regular representation of $G$. Let $X$ be an irreducible ${\mathbb{C}}^{\times }G$ invariant Zariski closed subset such that $G$ has a closed orbit that has maximal dimension among all orbits (this is equivalent to: generic orbits are closed). Then there exists an open subset, $W$,of $X$ in the metric topology which is dense with complement of measure $0$ such that if $x,y\in W$ then ${\left({\mathbb{C}}^{\times }G\right)}_x$ is conjugate to ${\left({\mathbb{C}}^{\times }G\right)}_y$. Furthermore, if $Gx$ is a closed orbit of maximal dimension and if $x$ is a smooth point of $X$ then there exists $y\in W$ such that ${\left({\mathbb{C}}^{\times }G\right)}_x$ contains a conjugate of ${\left({\mathbb{C}}^{\times }G\right)}_y$. The proof involves using the Kempf-Ness theorem to reduce the result to the principal orbit type theorem for compact Lie groups.