Classification of nondegenerate $G$-categories (with an appendix written jointly with Germán Stefanich)
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911587051241472 |
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| author | Gannon, Tom |
| author_facet | Gannon, Tom |
| contents | We classify a "dense open" subset of categories with an action of a reductive group, which we call nondegenerate categories, entirely in terms of the root datum of the group. As an application of our methods, we also:
(1) Upgrade an equivalence of Ginzburg and Lonergan, which identifies the category of bi-Whittaker $\mathcal{D}$-modules on a reductive group with the category of $\tilde{W}$-equivariant sheaves on a dual Cartan subalgebra $\mathfrak{t}^*$ which descend to the coarse quotient $\mathfrak{t}^*//\tilde{W}$, to a monoidal equivalence (where $\tilde{W}$ denotes the extended affine Weyl group) and
(2) Show the parabolic restriction of a very central sheaf acquires a Weyl group equivariant structure such that the associated equivariant sheaf descends to the coarse quotient $\mathfrak{t}^*//\tilde{W}$, providing evidence for a conjecture of Ben-Zvi-Gunningham on parabolic restriction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_11247 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Classification of nondegenerate $G$-categories (with an appendix written jointly with Germán Stefanich) Gannon, Tom Representation Theory Algebraic Geometry Category Theory We classify a "dense open" subset of categories with an action of a reductive group, which we call nondegenerate categories, entirely in terms of the root datum of the group. As an application of our methods, we also: (1) Upgrade an equivalence of Ginzburg and Lonergan, which identifies the category of bi-Whittaker $\mathcal{D}$-modules on a reductive group with the category of $\tilde{W}$-equivariant sheaves on a dual Cartan subalgebra $\mathfrak{t}^*$ which descend to the coarse quotient $\mathfrak{t}^*//\tilde{W}$, to a monoidal equivalence (where $\tilde{W}$ denotes the extended affine Weyl group) and (2) Show the parabolic restriction of a very central sheaf acquires a Weyl group equivariant structure such that the associated equivariant sheaf descends to the coarse quotient $\mathfrak{t}^*//\tilde{W}$, providing evidence for a conjecture of Ben-Zvi-Gunningham on parabolic restriction. |
| title | Classification of nondegenerate $G$-categories (with an appendix written jointly with Germán Stefanich) |
| topic | Representation Theory Algebraic Geometry Category Theory |
| url | https://arxiv.org/abs/2206.11247 |