Closed orbits and reductive groups

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?

-
@Ana The answer seems to be (trivially) yes, if the question is stated correctly. Please write in detail, what result you want to be valid over more general fields. – Mikhail Borovoi Nov 5 '10 at 17:25
I'd also need to see a more precise formulation: are the groups in question assumed to be connected? any restriction on the characteristic of the field? Basically you are looking at a reductive group $G$ 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? – Jim Humphreys Nov 5 '10 at 17:55
I know that the result is true for $k$ algebraically closed, what i want to know is if it remains valid for more general fields. – Ana Nov 9 '10 at 15:36