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

Published electronically:
November 2, 2009

MathSciNet review:
2558786

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Abstract | References | Similar Articles | Additional Information

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 , where 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:
http://dx.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