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What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian symmetric space?

This is certainly well-known by the experts, but I'm not one of them...

Thanks for any help!

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  • $\begingroup$ Could you clarify: what do mean by "is"? $(G/P)(\mathbf{R})$ is a set, a topological space, a real analytic manifold... A compact riemannian symmetric space is a metric space, a Riemannian manifold, a topological space, a real analytic manifold... $\endgroup$
    – YCor
    Commented Aug 25, 2014 at 10:41
  • $\begingroup$ My confusion is what are "real points"? There is no ${\mathbb R}$-structure on $G/P$, a priori, unless $G$ and $P$ are defined over $\mathbb R$. $\endgroup$
    – Venkataramana
    Commented Aug 25, 2014 at 10:52

1 Answer 1

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Compact Riemannian symmetric spaces admitting a Lie group of diffeomorphisms $G$ properly containing the isometry group are essentialy (up to covers) the symmetric $R$-spaces, which are of the form $G/P$. This is a celebrated theorem of Nagano [Nag, Theorem 3.1].

The list of these spaces is e.g. in the appendix of Isothermic submanifolds of symmetric $R$-spaces (pdf).

[Nag] Tadashi Nagano, Transformation groups on compact symmetric spaces, Trans. Amer. Math. Soc. 118 (1965), 428–453. MR0182937

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  • $\begingroup$ Why the downvote? It's not entirely clear to me what the OP means by $(G/P)(\mathbb{R})$, but the Nagano's theorem should bring him closer to answer in any case. $\endgroup$
    – Vít Tuček
    Commented Aug 26, 2014 at 7:53
  • $\begingroup$ Vit Tucek's answer is a precise and relevant answer to my confusing question! Thanks a lot. $\endgroup$
    – Lucien
    Commented Aug 26, 2014 at 13:25
  • $\begingroup$ A question concerning the terminology: to what refers the "R" in "R-space"? $\endgroup$
    – Lucien
    Commented Aug 28, 2014 at 13:00
  • $\begingroup$ I have no idea. And it's pretty hard to google. :) $\endgroup$
    – Vít Tuček
    Commented Aug 28, 2014 at 14:19
  • $\begingroup$ R is for "racine". Terminology coined by Jacques Tits, I believe. $\endgroup$
    – Fran Burstall
    Commented Nov 23, 2014 at 0:53

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