On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917097217458176 |
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| author | Takamatsu, Teppei |
| author_facet | Takamatsu, Teppei |
| contents | We prove that Lusztig's semi-infinite Deligne--Lusztig variety for $\mathrm{GSp}$ (and its inner form) is isomorphic, as a set with action, to an affine Deligne--Lusztig variety at infinite level, generalizing a result of Chan--Ivanov. Furthermore, we show that a component of some affine Deligne--Lusztig variety $X^0_{w_r}(b)_{\mathcal{L}}$ for $\mathrm{GSp}$ can be written, up to perfection, as a direct product of a classical Deligne--Lusztig variety with an affine space. We also study the varieties $X_h$ defined by Chan and Ivanov, and show that $X_h$ at infinite level can be realized as a subset of semi-infinite Deligne--Lusztig varieties defined using components of affine Deligne--Lusztig varieties such as $X^0_{w_r}(b)_{\mathcal{L}}$ above, even in the $\mathrm{GSp}$ case. This reinterprets previous constructions of representations from $X_h$ as instances of Lusztig's conjectural picture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17382 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$ Takamatsu, Teppei Algebraic Geometry Number Theory 11G25, 20G25 We prove that Lusztig's semi-infinite Deligne--Lusztig variety for $\mathrm{GSp}$ (and its inner form) is isomorphic, as a set with action, to an affine Deligne--Lusztig variety at infinite level, generalizing a result of Chan--Ivanov. Furthermore, we show that a component of some affine Deligne--Lusztig variety $X^0_{w_r}(b)_{\mathcal{L}}$ for $\mathrm{GSp}$ can be written, up to perfection, as a direct product of a classical Deligne--Lusztig variety with an affine space. We also study the varieties $X_h$ defined by Chan and Ivanov, and show that $X_h$ at infinite level can be realized as a subset of semi-infinite Deligne--Lusztig varieties defined using components of affine Deligne--Lusztig varieties such as $X^0_{w_r}(b)_{\mathcal{L}}$ above, even in the $\mathrm{GSp}$ case. This reinterprets previous constructions of representations from $X_h$ as instances of Lusztig's conjectural picture. |
| title | On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$ |
| topic | Algebraic Geometry Number Theory 11G25, 20G25 |
| url | https://arxiv.org/abs/2306.17382 |