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

TL;DR

This paper characterizes when certain pairs of parabolic subgroups produce dense orbits on flag varieties, linking this to Hermitian symmetric pairs, and classifies the closed orbits in these cases.

Contribution

It provides a complete classification of closed $K$-orbits on products of partial flag varieties for Hermitian symmetric pairs.

Findings

Dense $K$-orbits exist if and only if $(G,K)$ is Hermitian symmetric.

Characterization of pairs of parabolic subgroups with dense product.

Complete classification of closed $K$-orbits on flag varieties.

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