Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory)
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| Format: | Preprint |
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2018
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| _version_ | 1866911975351517184 |
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| author | Finkelberg, Michael Krylov, Vasily Mirković, Ivan |
| author_facet | Finkelberg, Michael Krylov, Vasily Mirković, Ivan |
| contents | Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_07721 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory) Finkelberg, Michael Krylov, Vasily Mirković, Ivan Algebraic Geometry Representation Theory Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory. |
| title | Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory) |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/1805.07721 |