2025/06/14 by Kyo Nishiyama, Nishiyama, Kyo, Taito Tauchi +1
Mathematics · Physics and Astronomy · #11E72 #22E15 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #primary 14M15 #secondary 05E14
paper · pdf · doi:10.48550/arxiv.2506.12663
openalex publication_date 2025/06/14 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28
Let G be a connected reductive algebraic group over ℝ , and H its symmetric subgroup. For parabolic subgroups PG ⊂ G and PH ⊂ H , the product of flag varieties \mathfrakX = H/PH × G/PG is called a double flag variety, on which H acts diagonally. Now let G be either U(n,n) or Sp2n(ℝ). We classify the H-orbits on \mathfrakX 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.