Relative Langlands duality and Koszul duality
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911704029331456 |
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| author | Braverman, Alexander Finkelberg, Michael Travkin, Roman |
| author_facet | Braverman, Alexander Finkelberg, Michael Travkin, Roman |
| contents | Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_ψ(Y)$ is polarized, so that $Q(X)=D_ψ(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then using a variant of the $S^1$-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_ψ(Y)\operatorname{-mod}^B)^{{\mathbb Z}/2}$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{B^\vee,\operatorname{mon}})^{{\mathbb Z}/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14085 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative Langlands duality and Koszul duality Braverman, Alexander Finkelberg, Michael Travkin, Roman Algebraic Geometry Representation Theory Symplectic Geometry Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_ψ(Y)$ is polarized, so that $Q(X)=D_ψ(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then using a variant of the $S^1$-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_ψ(Y)\operatorname{-mod}^B)^{{\mathbb Z}/2}$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{B^\vee,\operatorname{mon}})^{{\mathbb Z}/2}$. |
| title | Relative Langlands duality and Koszul duality |
| topic | Algebraic Geometry Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2604.14085 |