CLOSED ORBITS ON PARTIAL FLAG VARIETIES AND DOUBLE FLAG VARIETY OF FINITE TYPE

Let G be a connected reductive algebraic group over C. We denote by K = (G^θ)_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 P_1, P_2 of G such that (i) P_1 ∩ P_2 = Q and (ii) P_1 P_2 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 (eP_1, eP_2) of G/P_1 × G/P_2 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/P_1 × G/P_2