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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.05092 |
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| _version_ | 1866915570370215936 |
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| author | Choudhury, Ashutosh Roy Deshpande, Tanmay |
| author_facet | Choudhury, Ashutosh Roy Deshpande, Tanmay |
| contents | Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$ and let $\ell$ be a prime number different from $p$. Let $U \subseteq G$ be a maximal unipotent subgroup, $T$ a maximal torus normalizing $U$ and $W$ the Weyl group of $G$. Let $\mathcal{L}$ be a non-degenerate multiplicative $\overline{\mathbb{Q}}_{\ell} $-local system on $U$. R. Bezrukavnikov and the second author have proved that the bi-Whittaker category, namely the triangulated monoidal category of $(U, \mathcal{L})$-biequivariant $\overline{\mathbb{Q}}_{\ell}$-complexes on $G$ is monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}_{W}^{\circ}(T) \subseteq \mathscr{D}_{W}(T)$ of "central sheaves" on the torus. In particular it has the structure of a symmetric monoidal category coming from the symmetric monoidal structure on $\mathscr{D}_W(T)$.
In this paper, we give another construction of a symmetric monoidal structure on the above category and prove that it agrees with the one coming from the above construction. For this, among other things, we generalize a proof by Gelfand for finite groups to the geometric setup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05092 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Construction of the Symmetric Monoidal Structure of the Geometric Whittaker Model Choudhury, Ashutosh Roy Deshpande, Tanmay Representation Theory Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$ and let $\ell$ be a prime number different from $p$. Let $U \subseteq G$ be a maximal unipotent subgroup, $T$ a maximal torus normalizing $U$ and $W$ the Weyl group of $G$. Let $\mathcal{L}$ be a non-degenerate multiplicative $\overline{\mathbb{Q}}_{\ell} $-local system on $U$. R. Bezrukavnikov and the second author have proved that the bi-Whittaker category, namely the triangulated monoidal category of $(U, \mathcal{L})$-biequivariant $\overline{\mathbb{Q}}_{\ell}$-complexes on $G$ is monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}_{W}^{\circ}(T) \subseteq \mathscr{D}_{W}(T)$ of "central sheaves" on the torus. In particular it has the structure of a symmetric monoidal category coming from the symmetric monoidal structure on $\mathscr{D}_W(T)$. In this paper, we give another construction of a symmetric monoidal structure on the above category and prove that it agrees with the one coming from the above construction. For this, among other things, we generalize a proof by Gelfand for finite groups to the geometric setup. |
| title | A Construction of the Symmetric Monoidal Structure of the Geometric Whittaker Model |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2412.05092 |