Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
Author:
Ryan Cohen Reich
Journal:
Represent. Theory 16 (2012), 345449
MSC (2010):
Primary 22E57
Posted:
August 3, 2012
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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 BeilinsonDrinfeld 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 MirkovićVilonen equivalence. The second generalization, due recently to FinkelbergLysenko, 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 MirkovićVilonen 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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 [BBD82]
 A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5171 (French). MR 751966 (86g:32015)
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 T. Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209216. MR 1996415 (2004f:14037)
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 A. Braverman and D. Gaitsgory, Geometric Eisenstein series, Invent. Math. 150 (2002), no. 2, 287384. MR 1933587 (2003k:11109)
 [BFGM02]
 A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld's compactifications, Selecta Math. (N.S.) 8 (2002), no. 3, 381418. MR 1931170 (2003h:14060)
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 V. Drinfeld, On a conjecture of Kashiwara, Math. Res. Lett. 8 (2001), no. 56, 713728.
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 P. Deligne and J. Milne, Tannakian categories, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer, 1981.
 [FGI$^+$05]
 B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure, and A. Vistoli, Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, Amer. Math. Soc., 2005. MR 2222646 (2007f:14001)
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 M. Finkelberg and S. Lysenko, Twisted geometric Satake equivalence, J. Inst. Math. Jussieu 9 (2010), no. 4, 719739. MR 2684259 (2011i:22020)
 [FGV01]
 E. Frenkel, D. Gaitsgory, and K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. of Math. (2) 153 (2001), no. 3, 699748. MR 1836286 (2002e:11156)
 [Gai07]
 D. Gaitsgory, On de Jong's conjecture, Israel J. Math. 157 (2007), 155191.
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 V. Ginzburg, Perverse sheaves on a Loop group and Langlands' duality (1995), available at http://arxiv.org/abs/alggeom/9511007.
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 J. Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, SpringerVerlag, 1971 (French). MR 0344253 (49:8992)
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 G. Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston Inc., Boston, MA, 1993. MR 1227098 (94m:17016)
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 G. Lusztig, Singularities, character formulas, and a analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983. MR 737932 (85m:17005)
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 I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95143. MR 2342692 (2008m:22027)
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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 481091043
Email:
ryanr@math.ucla.edu
DOI:
http://dx.doi.org/10.1090/S108841652012004204
PII:
S 10884165(2012)004204
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
Posted:
August 3, 2012
Article copyright:
© Copyright 2012 Ryan Cohen Reich
