2026-09-19T06:28:05Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/836992022-01-13T07:54:04Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Roman Bezrukavnikov.
Singh, Bhairav
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2014-01-09T19:46:13Z
2014-01-09T19:46:13Z
2013
http://hdl.handle.net/1721.1/83699
864159424
Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 39-40).
One of the fundamental results in geometric representation theory is the geometric Satake equivalence, between the category of spherical perverse sheaves on the affine Grassmannian of a reductive group G and the category of representations of its Langlands dual group. The category of spherical perverse sheaves sits naturally in an equivariant derived category, and this larger category was described in terms of the dual group by Bezrukavnikov-Finkelberg. Recently, Finkelberg-Lysenko proved a "twisted" version of the geometric Satake equivalence, which involves perverse sheaves associated to twisted local systems on a line bundle over the affine Grassmannian. In this thesis we extend the Bezrukavnikov-Finkelberg description of the equivariant derived category to the twisted setting. Our method builds on theirs, but some additional subtleties arise. In particular, we cannot use Ginzburg's results on equivariant cohomology. We get around this by using localization techniques in equivariant cohomology in a more detailed way, allowing as to reduce certain computations to those of Ginzburg and Bezrukavnikov-Finkelberg. We also use show how our methods can be extended to explain an equivalence between Iwahori-equivariant peverse sheaves and twisted Iwahori-equivariant perverse sheaves on dual affine Grassmannians. This equivalence was observed earlier by Arkhipov-Bezrukavnikov-Ginzburg by combining several deep results, and they posed the problem of finding a more direct explanation. Finally, we explain how our results fit into the (quantum) geometric Langlands program.
by Bhairav Singh.
Ph.D.
40 pages
eng
Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Some results related to the quantum geometric Langlands program
Thesis
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<Abstract>One of the fundamental results in geometric representation theory is the geometric Satake equivalence, between the category of spherical perverse sheaves on the affine Grassmannian of a reductive group G and the category of representations of its Langlands dual group. The category of spherical perverse sheaves sits naturally in an equivariant derived category, and this larger category was described in terms of the dual group by Bezrukavnikov-Finkelberg. Recently, Finkelberg-Lysenko proved a "twisted" version of the geometric Satake equivalence, which involves perverse sheaves associated to twisted local systems on a line bundle over the affine Grassmannian. In this thesis we extend the Bezrukavnikov-Finkelberg description of the equivariant derived category to the twisted setting. Our method builds on theirs, but some additional subtleties arise. In particular, we cannot use Ginzburg's results on equivariant cohomology. We get around this by using localization techniques in equivariant cohomology in a more detailed way, allowing as to reduce certain computations to those of Ginzburg and Bezrukavnikov-Finkelberg. We also use show how our methods can be extended to explain an equivalence between Iwahori-equivariant peverse sheaves and twisted Iwahori-equivariant perverse sheaves on dual affine Grassmannians. This equivalence was observed earlier by Arkhipov-Bezrukavnikov-Ginzburg by combining several deep results, and they posed the problem of finding a more direct explanation. Finally, we explain how our results fit into the (quantum) geometric Langlands program.</Abstract>
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