Orbit structures on real double flag varieties for the Siegel parabolic subgroups

Abstract

Let $ G $ be a connected reductive algebraic group over $ \mathbb{R} $, and $ H $ its symmetric subgroup. For parabolic subgroups $ P_{G} \subset G $ and $ P_{H} \subset H $, the product of flag varieties $ \mathfrak{X} = H/P_H \times G/P_G $ is called a double flag variety, on which $ H $ acts diagonally. Now let $G$ be either $\mathrm{U}(n,n)$ or $\mathrm{Sp}_{2n}(\mathbb{R})$. We classify the $H$-orbits on $ \mathfrak{X} $ in both cases and show that they admit exactly the same parametrization. Concretely, each orbit corresponds to a signed partial involution, which can be encoded by simple combinatorial graphs. The orbit structure reduces to several families of smaller flag varieties, and we find an intimate relation of the orbit decomposition to Matsuki duality and Matsuki-Oshima's notion of clans. We also compute the Galois cohomology of each orbit, which exhibits another classification of the orbits by explicit matrix representatives.