Is there a nice reference for reductive groups over local fields, which for example contains discussion of things such as: Given a semisimple element in $G(F)$, its $G(F)$-conjugacy class is closed in the analytic topology of $G(F)$, and related things?
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1$\begingroup$ I don't know such a reference. However, I would suggest to ask a reference for the following assertion: the $G(\overline F)$-conjugacy class of any semisimple element in $G(\overline F)$ is Zariski-closed. The tags in such a question should include invariant-theory. $\endgroup$– Mikhail BorovoiCommented Feb 15, 2020 at 19:58
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1$\begingroup$ Now it remains to show that if $F$ is a local field and $X$ is a homogeneous space of $G$, then the orbits of $G(F)$ in $X(F)$ are closed in the analytical topology. Proof: the number of orbits is finite (see Serre, Galois Cohomology), and they are open, hence the complement of each orbit is open, hence each orbit is closed. $\endgroup$– Mikhail BorovoiCommented Feb 15, 2020 at 20:06
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1$\begingroup$ Serre (GC, III.4.4, Thm. 5) assumes that $F$ is perfect when proving that there are finitely many $G(F)$-orbits in $X(F)$. $\endgroup$– Mikhail BorovoiCommented Feb 16, 2020 at 12:51
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1$\begingroup$ In char 0, since $X$ is homogeneous, for any $x\in X(F)$ the map $$\phi_x\colon G\to X, \quad g\mapsto g\cdot x$$ is smooth, and hence the map on $F$-points $G(F)\to X(F)$ is open by the implicit function theorem, and hence the $G(F)$-orbit of $x$ is open. $\endgroup$– Mikhail BorovoiCommented Feb 16, 2020 at 12:57
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1$\begingroup$ However, I think that all will work also in char p, if the (scheme-theoretical) stabilizer of $x$ in $G$ is a smooth connected reductive group. $\endgroup$– Mikhail BorovoiCommented Feb 16, 2020 at 13:03
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