  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 $G/P_{1} \times G/P_{2}$.

Representation Theory;Algebraic Geometry