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

Frenkel, Edward; Gaitsgory, Dennis

doi:10.1090/S1088-4165-09-00360-4

D-modules on the affine flag variety and representations of affine Kac-Moody algebras
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by Edward Frenkel and Dennis Gaitsgory

Represent. Theory 13 (2009), 470-608

DOI: https://doi.org/10.1090/S1088-4165-09-00360-4

Published electronically: November 2, 2009

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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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