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D-modules on the affine flag variety and representations of affine Kac-Moody algebras


Authors: Edward Frenkel and Dennis Gaitsgory
Journal: Represent. Theory 13 (2009), 470-608
MSC (2010): Primary 17B67; Secondary 13N10
DOI: https://doi.org/10.1090/S1088-4165-09-00360-4
Published electronically: November 2, 2009
MathSciNet review: 2558786
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Abstract: The present paper studies the connection between the category of modules over the affine Kac-Moody Lie algebra at the critical level, and the category of D-modules on the affine flag scheme $ G((t))/I$, where $ I$ is the Iwahori subgroup. We prove a localization-type result, which establishes an equivalence between certain subcategories on both sides. We also establish an equivalence between a certain subcategory of Kac-Moody modules, and the category of quasi-coherent sheaves on the scheme of Miura opers for the Langlands dual group, thereby proving a conjecture of the authors in 2006.


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Additional Information

Edward Frenkel
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: frenkel@math.berkeley.edu

Dennis Gaitsgory
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: gaitsgde@math.harvard.edu

DOI: https://doi.org/10.1090/S1088-4165-09-00360-4
Received by editor(s): December 6, 2007
Received by editor(s) in revised form: July 6, 2009
Published electronically: November 2, 2009
Additional Notes: The first author was supported by DARPA and AFOSR through the grant FA9550-07-1-0543.
The second author was supported by NSF grant 0600903.
Article copyright: © Copyright 2009 Edward Frenkel and Dennis Gaitsgory

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