Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



On Koszul duality for Kac-Moody groups

Authors: Roman Bezrukavnikov and Zhiwei Yun
Journal: Represent. Theory 17 (2013), 1-98
MSC (2010): Primary 20G44, 14M15, 14F05
Published electronically: January 2, 2013
MathSciNet review: 3003920
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For any Kac-Moody group $ G$ with Borel $ B$, we give a monoidal equivalence between the derived category of $ B$-equivariant mixed complexes on the flag variety $ G/B$ and (a certain completion of) the derived category of $ G^\vee $-monodromic mixed complexes on the enhanced flag variety $ G^\vee /U^\vee $, here $ G^\vee $ is the Langlands dual of $ G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $ U$-equivariant mixed complexes on the partial flag variety $ G/P$ and a certain ``Whittaker model'' category of mixed complexes on $ G^\vee /B^\vee $. In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in [BGS96].

References [Enhancements On Off] (What's this?)

  • [ABG04] Arkhipov, S; Bezrukavnikov, R; Ginzburg, V. Quantum groups, the loop Grassmannian, and the Springer resolution. J. Amer. Math. Soc. 17 (2004), no. 3, 595-678. MR 2053952 (2005g:16055)
  • [B87] Beilinson, A. On the derived category of perverse sheaves. In $ K$-theory, arithmetic and geometry (Moscow, 1984-1986), 27-41, Lecture Notes in Math., 1289, Springer, Berlin, 1987. MR 923133 (89b:14027)
  • [BB93] Beilinson, A.; Bernstein, J. A proof of Jantzen conjectures. In I. M. Gel'fand Seminar, 1-50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993. MR 1237825 (95a:22022)
  • [BBD82] Beilinson, A.; Bernstein, J.; Deligne, P. Faisceaux pervers. In Analysis and topology on singular spaces I (Luminy, 1981), 5-171, Astérisque 100, Soc. Math. France, Paris, 1982. MR 751966 (86g:32015)
  • [BBM04a] Beilinson, A; Bezrukavnikov, R; Mirković, I. Tilting exercises. Mosc. Math. J.4(2004), no. 3, 547-557, 782. MR 2119139 (2006a:14022)
  • [BG99] Beilinson, A.; Ginzburg, V. Wall-crossing functors and D-modules. Represent. Theory 3 (1999), 1-31. MR 1659527 (2000d:17007)
  • [BGS88] Beilinson, A.; Ginsburg, V.; Schechtman, V. Koszul duality, J. Geom. Phys. 5 (1988), no. 3, 317-350. MR 1048505 (91c:18011)
  • [BGS96] Beilinson, A; Ginzburg, V; Soergel, W. Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010)
  • [BL94] Bernstein, J.; Lunts, V. Equivariant sheaves and functors. Lecture Notes in Mathematics, 1578. Springer-Verlag, Berlin, 1994. iv+139 pp. MR 1299527 (95k:55012)
  • [B06] Bezrukavnikov, R. Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves. Invent. Math. 166 (2006), no. 2, 327-357. MR 2249802 (2008e:14018)
  • [BBM04b] Bezrukavnikov, R; Braverman, A; Mirković, I. Some results about the geometric Whittaker model. Adv. Math. 186 (2004), no. 1, 143-152. MR 2065510 (2005e:20068)
  • [BF08] Bezrukavnikov, R.; Finkelberg, M. Equivariant Satake category and Kostant-Whittaker reduction. Mosc. Math. J. 8 (2008), no.1, 39-72, 183. MR 2422266 (2009d:19008)
  • [D80] Deligne, P. La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. No. 52 (1980), 137-252. MR 601520 (83c:14017)
  • [G91] Ginzburg, V. Perverse sheaves and $ \mathbb{C}^*$-actions. J. Amer. Math. Soc. 4 (1991), no. 3, 483-490. MR 1091465 (92d:14013)
  • [GKM98] Goresky, M.; Kottwitz, R.; MacPherson, R. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131 (1998), no. 1, 25-83. MR 1489894 (99c:55009)
  • [K90] Kac, V.G. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990 MR 1104219 (92k:17038)
  • [KK86] Kostant, B.; Kumar, S. The nil Hecke ring and cohomology of $ G/P$ for a Kac-Moody group $ G$, Adv. in Math. 62 (1986), no. 3, 187-237. MR 866159 (88b:17025b)
  • [LO08] Laszlo, Y.; Olsson, M. The six operations for sheaves on Artin stacks. I. Finite coefficients. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 109-168; II. Adic coefficients. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 169-210. MR 2434692 (2009f:14003a)
  • [M89] Mathieu, O. Construction d'un groupe de Kac-Moody et applications. Compositio Math. 69 (1989), no. 1, 37-60. MR 986812 (90f:17012)
  • [M01] May, J. P. The additivity of traces in triangulated categories. Adv. Math. 163 (2001), no. 1, 34-73. MR 1867203 (2002k:18019)
  • [MS97] Milicic, D.; Soergel, W. The composition series of modules induced from Whittaker modules. Comment. Math. Helv. 72 (1997), no. 4, 503-520. MR 1600134 (99e:17010)
  • [O07] Olsson, M. Sheaves on Artin stacks. J. Reine Angew. Math. 603 (2007), 55-112. MR 2312554 (2008b:14002)
  • [S11] Schnürer, O. Equivariant sheaves on flag varieties. Mathematische Zeitschrift, 267 (2011), no. 1-2, 27-80. MR 2772241 (2012f:14035)
  • [So90] Soergel, W. Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421-445. MR 1029692 (91e:17007)
  • [S84] Springer, T.A. A purity result for fixed point varieties in flag manifolds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 271-282. MR 763421 (86c:14034)
  • [V63] Verdier, J-L. Catégories dérivées, Etat 0, SGA 4 $ \frac {1}{2}$. Cohomologie étale, 262-311 Lecture Notes in Mathematics, 569, Springer-Verlag, 1977. MR 0463174 (57:3132)
  • [V83] Verdier, J.-L. Spécialisation de faisceaux et monodromie modérée. In Analysis and topology on singular spaces II, III (Luminy, 1981), 332-364, Astérisque, 101-102, Soc. Math. France, Paris, 1983. MR 737938 (86f:32010)
  • [Y09] Yun, Z. Weights of mixed tilting sheaves and geometric Ringel duality. Selecta Math. (N.S.) 14 (2009), no. 2, 299-320. MR 2480718 (2010d:14025)
  • [Y10] Yun, Z. Goresky-MacPherson calculus for the affine flag varieties, Canad. J. Math. 62 (2010), no. 2, 473-480. MR 2643053 (2011d:14089)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 20G44, 14M15, 14F05

Retrieve articles in all journals with MSC (2010): 20G44, 14M15, 14F05

Additional Information

Roman Bezrukavnikov
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Zhiwei Yun
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, California 94305

Received by editor(s): January 15, 2011
Received by editor(s) in revised form: July 7, 2011, August 13, 2011, and April 11, 2012
Published electronically: January 2, 2013
Additional Notes: The first author was partly supported by the NSF grant DMS-0854764.
The second author was supported by the NSF grant DMS-0635607 and Zurich Financial Services as a member at the Institute for Advanced Study, and by the NSF grant DMS-0969470.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society