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)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.