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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?

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  • $\begingroup$ @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. $\endgroup$
    – Mikhail Borovoi
    Commented Nov 5, 2010 at 17:25
  • $\begingroup$ 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? $\endgroup$
    – Jim Humphreys
    Commented Nov 5, 2010 at 17:55
  • $\begingroup$ 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. $\endgroup$
    – Ana
    Commented Nov 9, 2010 at 15:36
  • $\begingroup$ @MikhailBorovoi: Why the answer is yes? I can not see a reason for this even for algebraically closed fields. $\endgroup$
    – m07kl
    Commented Feb 14, 2017 at 21:32
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    $\begingroup$ See Jim's comment to this question. The assertion is contained in his book, but I cannot find it now.... $\endgroup$
    – Mikhail Borovoi
    Commented Feb 15, 2017 at 11:47

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