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!



Rational homogenous spaces and symmetric spaces
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](http://www.sfbtr71.de/publications/pedit_isothermic.pdf) (pdf).


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


