Closed orbits on partial flag varieties and double flag variety of finite type

Abstract

Let $G$ be a connected reductive algebraic group over $ \C $. We denote by $K=(G^{\theta }{)}_0$ the identity component of the fixed points of an involutive automorphism $\theta$ 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) $P_1\cap P_2=Q$ 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) yield to conclude that the $K$-orbit through the origin $(e{P}_1,e{P}_2)$ of $G/P_1\times G/P_2$ 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 $G/P_1\times G/P_2$.

7 pages