Let G be a connected reductive algebraic group over C. We denote by K = (Gθ)0the identity component of the fixed points of an involutive automorphism θ of G. The pair (G, K) is called a symmetric pair. Let Q be a parabolic subgroup of K. We want to find a pair of parabolic subgroups P 1, P 2of G such that (i) P 1∩ P 2= Q and (ii) P 1 P 2is dense in G. The main result of this article states that, for a simple group G, we can find such a pair if and only if (G, K) is a Hermitian symmetric pair. The conditions (i) and (ii) imply that the K -orbit through the origin (eP 1, eP 2) of G/P 1× G/P 2is closed and it generates an open dense G -orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed K -orbits on G/P 1× G/P 2.
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