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

TL;DR

This paper investigates the real double flag variety associated with the symmetric pair $(U(p,p),GL_p(\mathbb{C}))$ in the special case $p=2$. By translating the problem to the complex setting and employing Galois cohomology, the authors reduce the orbit classification to analyzing $H^{1}(\mathbb{R},-)$ for stabilizers of complex orbits, and they obtain a complete list of real orbits corresponding to 17 $B$-orbits on $G/P$. The main result provides explicit representatives for the orbit decomposition $H\backslash(H/B_H\times G/P)$, accomplished via a detailed tau/sigma fixed-point analysis and a careful selection of orbit representatives from the complex classification (twisted by a Weyl-group element). The approach demonstrates the effectiveness of Galois cohomology in real double flag varieties and yields a concrete classification for $p=2$, with broader implications hinted for $p\ge 3$ via Matsuki duality.

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 $p=2$.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof