Orbit structures on real double flag varieties for symmetric pairs

TL;DR

This paper classifies the orbits of a symmetric subgroup acting on double flag varieties for specific real reductive groups, revealing combinatorial parametrizations and connections to duality theories.

Contribution

It provides a complete classification of $H$-orbits on double flag varieties for $ ext{U}(n,n)$ and $ ext{Sp}_{2n}( ext{R})$, introducing combinatorial graph encodings and linking to duality concepts.

Findings

Orbits correspond to signed partial involutions.

Orbit structure relates to Matsuki duality and clans.

Galois cohomology classifies orbits via explicit matrices.

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.