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 | References | Similar Articles | Additional Information

Abstract: The *geometric Satake equivalence* of Ginzburg and Mirković-

Vilonen, for a complex reductive group , is a realization of the tensor category of representations of its Langlands dual group as a category of ``spherical'' perverse sheaves on the *affine grassmannian* . 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 is a collection of spaces over the powers whose general fiber is isomorphic to 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 . 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

Email:
ryanr@math.ucla.edu

DOI:
https://doi.org/10.1090/S1088-4165-2012-00420-4

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