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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.08264 |
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| _version_ | 1866908822354788352 |
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| author | Popkovich, Aleksandr |
| author_facet | Popkovich, Aleksandr |
| contents | A theorem of R. Travkin and R. Yang, initially conjectured by D. Gaiotto, states that for a generic (not a root of unity) $q$ the category of $q$-twisted D-modules on the affine Grassmannian $Gr_{GL_N}$ which are equivariant with respect to a certain subgroup (defined by a choice of $0 \le M <N$) of $GL_N$ is equivalent to the category of representations of the quantum supergroup $U_q(\mathfrak{gl}(M|N))$. We aim to see whether this equivalence should hold when $q$ is a root of unity. We begin by asking if there is a natural bijection between the sets of irreducible objects. In this note we make an observation that suggests this should be the case: we show that there is a natural bijection between irreducible objects in the Gaiotto category and in the category of representations of a supergroup $GL(M|N)$ in positive characteristic. The proof is based on the version of the Serganova's algorithm formulated by J. Brundan and J. Kujawa in arXiv:math/0210108. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08264 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Irreducible objects in the Gaiotto category at roots of unity Popkovich, Aleksandr Representation Theory Algebraic Geometry A theorem of R. Travkin and R. Yang, initially conjectured by D. Gaiotto, states that for a generic (not a root of unity) $q$ the category of $q$-twisted D-modules on the affine Grassmannian $Gr_{GL_N}$ which are equivariant with respect to a certain subgroup (defined by a choice of $0 \le M <N$) of $GL_N$ is equivalent to the category of representations of the quantum supergroup $U_q(\mathfrak{gl}(M|N))$. We aim to see whether this equivalence should hold when $q$ is a root of unity. We begin by asking if there is a natural bijection between the sets of irreducible objects. In this note we make an observation that suggests this should be the case: we show that there is a natural bijection between irreducible objects in the Gaiotto category and in the category of representations of a supergroup $GL(M|N)$ in positive characteristic. The proof is based on the version of the Serganova's algorithm formulated by J. Brundan and J. Kujawa in arXiv:math/0210108. |
| title | Irreducible objects in the Gaiotto category at roots of unity |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2602.08264 |