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 $G x$ 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.