  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.
