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

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 $left (mathbb{C}^{ times } Gright )_{x}$ is conjugate to $left (mathbb{C}^{ times } Gright )_{y}$. 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 $left (mathbb{C}^{ times } Gright )_{x}$ contains a conjugate of $left (mathbb{C}^{ times } Gright )_{y}$. The proof involves using the Kempf-Ness theorem to reduce the result to the principal orbit type theorem for compact Lie groups.