Let $G$ be a reductive algebraic group defined over a field $k$ and $X$ an affine $G$-variety.

In the case $k$ is algebraically closed we have the following result:

Let $x\in X$ such that the orbit $G\cdot x$ is closed, then the stabilizer $G_x$ is reductive.

Is this result also valid for more general fields, as perfect fields?

`$G$`

and affine quotient`$G/H$`

here. The algebraically closed case in any characteristic is documented in papers by Richardson, Cline-Parshall-Scott, Borel, Haboush, and others. What is your own starting point? $\endgroup$ – Jim Humphreys Nov 5 '10 at 17:55