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) $P_{1} \cap P_{2} = Q$ 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 $G/P_{1} \times G/P_{2}$.

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