2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ @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 Nov 5 '10 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 Nov 5 '10 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 Nov 9 '10 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 Feb 14 '17 at 21:32
  • 1
    $\begingroup$ See Jim's comment to this question. The assertion is contained in his book, but I cannot find it now.... $\endgroup$ – Mikhail Borovoi Feb 15 '17 at 11:47

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.