Abstract
Let G be a connected reductive algebraic group over C. We denote by K = (Gθ)0 the 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 P1, P2 of G such that (i) P1 ∩ P2 = Q and (ii) P1 P2 is 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 (eP1, eP2) of G/P1 × G/P2 is 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/P1 × G/P2.
