Orbit structures on real double flag varieties for symmetric pairs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913893982404608 |
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| author | Nishiyama, Kyo Tauchi, Taito |
| author_facet | Nishiyama, Kyo Tauchi, Taito |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12663 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orbit structures on real double flag varieties for symmetric pairs Nishiyama, Kyo Tauchi, Taito Representation Theory primary 14M15, secondary 05E14, 11E72, 22E15 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. |
| title | Orbit structures on real double flag varieties for symmetric pairs |
| topic | Representation Theory primary 14M15, secondary 05E14, 11E72, 22E15 |
| url | https://arxiv.org/abs/2506.12663 |