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Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian

Author: Ryan Cohen Reich
Journal: Represent. Theory 16 (2012), 345-449
MSC (2010): Primary 22E57
Published electronically: August 3, 2012
MathSciNet review: 2956088
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Abstract: The geometric Satake equivalence of Ginzburg and Mirković-
Vilonen, for a complex reductive group $ G$, is a realization of the tensor category of representations of its Langlands dual group $ {}^L G$ as a category of ``spherical'' perverse sheaves on the affine grassmannian $ \operatorname {Gr}_G = G(\mathbb{C}(\mspace {-3.5mu}(t)\mspace {-3.5mu})/G(\mathbb{C}[\mspace {-2mu}[t]\mspace {-2mu}])$. Since its original statement it has been generalized in two directions: first, by Gaitsgory, to the Beilinson-Drinfeld or factorizable grassmannian, which for a smooth complex curve $ X$ is a collection of spaces over the powers $ X^n$ whose general fiber is isomorphic to $ \operatorname {Gr}_G^n$ but with the factors ``fusing'' as they approach points with equal coordinates, allowing a more natural description of the structures and properties even of the Mirković-Vilonen equivalence. The second generalization, due recently to Finkelberg-Lysenko, considers perverse sheaves twisted in a suitable sense by a root of unity, and obtains the category of representations of a group other than the Langlands dual. This latter result can be considered as part of ``Langlands duality for quantum groups''.

In this work we obtain a result simultaneously generalizing all of the above. We consider the general notion of twisting by a gerbe and define the natural class of ``factorizable'' gerbes by which one can twist in the context of the Satake equivalence. These gerbes are almost entirely described by the quadratic forms on the weight lattice of $ G$. We show that a suitable formalism exists such that the methods of Mirković-Vilonen can be applied directly in this general context virtually without change and obtain a Satake equivalence for twisted perverse sheaves. In addition, we present new proofs of the properties of their structure as an abelian tensor category.

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Additional Information

Ryan Cohen Reich
Affiliation: UCLA Mathematics Department, 520 Portola Plaza, Los Angeles, California 90095
Address at time of publication: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043

Received by editor(s): October 17, 2010
Received by editor(s) in revised form: December 28, 2010, October 9, 2011, March 26, 2012, and March 30, 2012
Published electronically: August 3, 2012
Article copyright: © Copyright 2012 Ryan Cohen Reich