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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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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... –  YCor Aug 25 at 10:41
    
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 $\mathbb R$. –  Venkataramana Aug 25 at 10:52

1 Answer 1

up vote 2 down vote accepted

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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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. –  Vít Tuček Aug 26 at 7:53
    
Vit Tucek's answer is a precise and relevant answer to my confusing question! Thanks a lot. –  Lucien from IHP Aug 26 at 13:25
    
In that case you can accept my answer by clicking on the tick sign. ;) –  Vít Tuček Aug 26 at 13:44
    
A question concerning the terminology: to what refers the "R" in "R-space"? –  Lucien from IHP Aug 28 at 13:00
    
I have no idea. And it's pretty hard to google. :) –  Vít Tuček Aug 28 at 14:19

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