Action of Hecke algebra on the double flag variety of type AIII
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929420789350400 |
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| author | Fresse, Lucas Nishiyama, Kyo |
| author_facet | Fresse, Lucas Nishiyama, Kyo |
| contents | Consider a connected reductive algebraic group $ G $ and a symmetric subgroup $ K $. Let $ \mathfrak{X} = K/B_K \times G/P $ be a double flag variety of finite type, where $ B_K $ is a Borel subgroup of $ K $, and $ P $ a parabolic subgroup of $ G $. A general argument shows that the orbit space $ \mathbb{C}\,\mathfrak{X}/K $ inherits a natural action of the Hecke algebra $ \mathscr{H} = \mathscr{H}(K, B_K) $ of double cosets via convolutions. However, to find out the explicit structure of the Hecke module is a quite different problem.
In this paper, we determine the explicit action of $ \mathscr{H} $ on $ \mathbb{C}\,\mathfrak{X}/K $ in a combinatorial way using graphs for the double flag variety of type AIII. As a by-product, we also get the description of the representation of the Weyl group on $ \mathbb{C}\,\mathfrak{X}/K $ as a direct sum of induced representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_10476 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Action of Hecke algebra on the double flag variety of type AIII Fresse, Lucas Nishiyama, Kyo Representation Theory Primary 20C08, Secondary 14M17, 14M15, 20G05 Consider a connected reductive algebraic group $ G $ and a symmetric subgroup $ K $. Let $ \mathfrak{X} = K/B_K \times G/P $ be a double flag variety of finite type, where $ B_K $ is a Borel subgroup of $ K $, and $ P $ a parabolic subgroup of $ G $. A general argument shows that the orbit space $ \mathbb{C}\,\mathfrak{X}/K $ inherits a natural action of the Hecke algebra $ \mathscr{H} = \mathscr{H}(K, B_K) $ of double cosets via convolutions. However, to find out the explicit structure of the Hecke module is a quite different problem. In this paper, we determine the explicit action of $ \mathscr{H} $ on $ \mathbb{C}\,\mathfrak{X}/K $ in a combinatorial way using graphs for the double flag variety of type AIII. As a by-product, we also get the description of the representation of the Weyl group on $ \mathbb{C}\,\mathfrak{X}/K $ as a direct sum of induced representations. |
| title | Action of Hecke algebra on the double flag variety of type AIII |
| topic | Representation Theory Primary 20C08, Secondary 14M17, 14M15, 20G05 |
| url | https://arxiv.org/abs/2206.10476 |