Real double flag variety for the symmetric pair $(U(p,p),GL_{p}(\mathbb{C}))$ and Galois cohomology

Abstract: Let $G$ be the indefinite unitary group $U(p,p)$, $H\simeq GL_{p}(\mathbb{C})$ its symmetric subgroup, $P_{S}$ the Siegel parabolic subgroup of $G$, and $B_{H}$ a Borel subgroup of $H$. In this article, we give a classification of the orbit decomposition $H\backslash (H/B_{H}\times G/P_{S})$ of the real double flag variety by using the Galois cohomology in the case where $p=2$.