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 , 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.

**[AB]**S. Arkhipov, R. Bezrukavnikov,*Perverse sheaves on affine flags and Langlands duality*, Israel J. of Math.**170**(2009), 135-185.**[ABBGM]**S. Arkhipov, R. Bezrukavnikov, A. Braverman, D. Gaitsgory and I. Mirković,*Modules over the small quantum group and semi-infinite flag manifold*, Transformation Groups**10**(2005), 279-362. MR**2183116 (2008a:14027)****[ABG]**S. Arkhipov, R. Bezrukavnikov, V. Ginzburg,*Quantum Groups, the loop Grassmannian, and the Springer resolution*, JAMS**17**(2004), 595-678. MR**2053952 (2005g:16055)****[BB]**A. Beilinson and J. Bernstein,*Localisation de -modules*, C. R. Acad. Sci. Paris Ser. I Math.**292**(1981), 15-18. MR**610137 (82k:14015)****[BD]**A. Beilinson and V. Drinfeld,*Quantization of Hitchin's integrable system and Hecke eigensheaves*, available at http://www.math.uchicago.edu/arinkin/langlands/**[BV]**A. Beilinson and V. Vologodsky,*A DG guide to Voevodsky's motives*, GAFA**17**(2008), 1709-1787. MR**2399083 (2009d:14018)****[Bez]**R. Bezrukavnikov,*Non-commutative counterparts of the Springer resolution*, MR**2275638 (2009d:17026)**ICM talk, arXiv:math/0604445.**[CHA]**A. Beilinson and V. Drinfeld,*Chiral algebras*, AMS Colloquium Publications**51**, AMS, 2004. MR**2058353 (2005d:17007)****[Dr]**V. Drinfeld,*DG quotients of DG categories*, J. of Algebra**272**(2004), 643-691. MR**2028075 (2006e:18018)****[FF]**B. Feigin and E. Frenkel,*Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras*, in*Infinite Analysis*, eds. A. Tsuchiya, T. Eguchi, M. Jimbo, Adv. Ser. in Math. Phys.**16**, 197-215, Singapore: World Scientific, 1992. MR**1187549 (93j:17049)****[FG1]**E. Frenkel and D. Gaitsgory,*D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras*, Duke Math. J.**125**(2004), 279-327. MR**2096675 (2005h:17040)****[FG2]**E. Frenkel and D. Gaitsgory,*Local geometric Langlands correspondence and affine Kac-Moody algebras*, in: Algebraic Geometry and Number Theory, Progr. Math.**253**(2006), 69-260, Birkhäuser Boston. MR**2263193 (2008e:17023)****[FG3]**E. Frenkel and D. Gaitsgory,*Fusion and convolution: applications to affine Kac-Moody algebras at the critical level*, Pure Appl. Math. Quart.**2**(2006), 1255-1312. MR**2282421 (2008d:17036)****[FG4]**E. Frenkel and D. Gaitsgory,*Localization of -modules on the affine Grassmannian*, math.RT/0512562, to appear in Annals of Math.**[FG5]**E. Frenkel and D. Gaitsgory,*Geometric realizations of Wakimoto modules at the critical level*, Duke Math. J.**143**(2008), 117-203. MR**2414746 (2009d:17034)****[Ga]**D. Gaitsgory,*Construction of central elements in the affine Hecke algebra via nearby cycles*, Invent. Math.**144**(2001), 253-280. MR**1826370 (2002d:14072)****[Ga]**D. Gaitsgory,*Appendix: Braiding compatibilities*, in: Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math.**40**(2004), 91-100, Math. Soc. Japan, Tokyo. MR**2074590 (2006f:14015)****[Ga1]**D. Gaitsgory,*The notion of category over an algebraic stack*, math.AG/0507192.**[Kr]**H. Krause,*The stable derived of a noetherian scheme*, Comp. Math.**141**(2005) 1128-1162. MR**2157133 (2006e:18019)****[Lu]**J. Lurie,*Derived Algebraic Geometry II: Non-commutative algebra*, available at J. Lurie's homepage.**[MV]**I. Mirković and K. Vilonen,*Geometric Langlands duality and representations of algebraic groups over commutative rings*, Annals of Math.**166**(2007), 95-143. MR**2342692 (2008m:22027)**

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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