Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Koszul Duality Patterns
in Representation Theory

Authors: Alexander Beilinson, Victor Ginzburg and Wolfgang Soergel
Journal: J. Amer. Math. Soc. 9 (1996), 473-527
MSC (1991): Primary 17B10; Secondary 16A03
MathSciNet review: 1322847
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain ${\mathbb Z}$-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category ${\mathcal O}$ that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category ${\mathcal O}$ again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.

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

  • [ABV92] Jeffrey Adams, Dan Barbasch, and David A. Vogan, Jr, The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser, 1992.MR 93j:22001
  • [Bac82] Jörgen Backelin, A distributiveness property of augmented algebras and some related homological results, Ph. D. Thesis, Stockholm, 1982.
  • [Bar83] Dan Barbasch, Filtrations on Verma modules, Ann. Scient. Éc. Norm. Sup. 16 (1983), 489--494.MR 85j:17013
  • [Bas68] Hyman Bass, Algebraic K-theory, Benjamin, 1968.MR 40:2736
  • [BB81] Alexander A. Beilinson and Joseph N. Bernstein, Localisation de ${{\mathfrak g}}$-modules, C. R. Acad. Sci. Paris, Sér. 1 292 (1981), 15--18. MR 82k:14015
  • [BB86] Alexander A. Beilinson and Joseph N. Bernstein, A proof of Jantzen conjectures, Preprint, 1986; finally published in I. M. Gelfand Seminar, Advances in Soviet Math., vol. 16, part 1, Amer. Math. Soc., Providence, RI, 1993, pp. 1--50.MR 95a:22022
  • [BBD82] Alexander A. Beilinson, Joseph N. Bernstein, and Pierre Deligne, Faisceaux pervers, Astérisque 100 (1982).MR 86g:32015
  • [Bea83] Arnault Beauville, Variétés kähleriennes dont la première classe de Chern est nulle, J. Diff. Geometry 18 (1983), 755--782.MR 86c:32030
  • [Ber90] Joseph N. Bernstein, Trace in categories, Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory (Alain Connes, Michel Duflo, Anthony Joseph, and Rudolph Rentschler, eds.), Birkhäuser, 1990, pp. 417--423.MR 92d:17010
  • [BF85] Jörgen Backelin and Ralf Fröberg, Koszul algebras, Veronese subrings and rings with linear resolutions, Rev. Roumaine Math. Pures Appl. 30 (1985), 85--97.MR 87c:16002
  • [BG80] Joseph N. Bernstein and Sergei I. Gelfand, Tensor products of finite and infinite representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245--285.MR 82c:17003
  • [BG86] Alexander A. Beilinson and Victor Ginsburg, Mixed categories, Ext-duality and representations (results and conjectures), Preprint, 1986.
  • [BGG73] Joseph N. Bernstein, Israel M. Gelfand, and Sergei I. Gelfand, Schubert cells and cohomology of the spaces ${G/P}$, Russian Math. Surveys 28 (1973), no. 3, 1--26.MR 55:2941
  • [BGG78] Joseph N. Bernstein, Israel M. Gelfand, and Sergei I. Gelfand, Algebraic bundles over ${{\mathbb P}^n}$ and problems of linear algebra, Functional Analysis and its App. 12 (1978), no. 3, 214--216.MR 80c:14010a
  • [BGS88] Alexander A. Beilinson, Victor Ginsburg, and Vadim V. Schechtman, Koszul duality, Journal of Geometry and Physics 5 (1988), 317--350.MR 91c:18011
  • [BMS87] Alexander Beilinson, Robert MacPherson, and Vadik Schechtmann, Notes on motivic cohomology, Duke Math. Journal 54 (1987), 679--710.MR 88f:14021
  • [Bor84] Armand Borel (ed.), Intersection cohomology, Progress in Mathematics, vol. 50, Birkhäuser, 1984.MR 88d:32024
  • [CC87] Luis G. Casian and David H. Collingwood, The Kazhdan-Lusztig conjecture for generalized Verma modules, Math. Z. 195 (1987), 581--600.MR 88i:17008
  • [CPS91] Edward Cline, Brian Parshall, and Leonard Scott, Abstract Kazhdan-Lusztig theories, Tôhoku Math. J. (2) 45 (1993), 511--534. MR 94k:20079
  • [Del80] Pierre Deligne, La conjecture de Weil II, IHES Publ. Math. 52 (1980), 137--252.MR 83c:14017
  • [ES87] Tom J. Enright and Brad Shelton, Categories of highest weight modules: Applications to classical hermitian symmetric pairs, Memoirs of the AMS 367 (1987).MR 88f:22052
  • [Har66] Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, vol. 20, Springer, 1966.MR 36:5145
  • [Irv88] Ron S. Irving, The socle filtration of a Verma module, Ann. Scient. Éc. Norm. Sup. 21 (1988), 47--65.MR 89h:17015
  • [Irv90] Ron S. Irving, A filtered category ${\CO_s}$ and applications, Memoirs of the AMS 419 (1990).MR 90f:17011
  • [Jan87] Jens C. Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, 1987.MR 89c:20001
  • [Kel92] Bernhard Keller, Deriving DG-categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), 63--102. MR 95e:18010
  • [KL80a] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones Math. 53 (1980), 191--213.MR 81j:20066
  • [KL80b] David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Proc. Symp. Pure Math. 36, AMS, 1980, pp. 185--203.MR 84g:14054
  • [KS90] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren, vol. 292, Springer, 1990.MR 92a:58132
  • [Löf86] Clas Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda ext-algebra, Lecture Notes in Mathematics, vol. 1183, Springer, 1986, pp. 291--338.MR 88f:16030
  • [MV87] R. Mirollo and Kari Vilonen, Bernstein-Gelfand-Gelfand reciprocity on perverse sheaves, Ann. Scient. Ec. Norm. Sup. $4^e$ série 20 (1987), 311--324.MR 89e:32017
  • [Pri70] Stewart B. Priddy, Koszul resolutions, Transactions AMS 152 (1970), 39--60.MR 42:346
  • [Sai90] Morihiko Saito, Mixed Hodge modules, Publications RIMS, Kyoto University 26 (1990), 221--333.MR 91m:14014
  • [Soe86] Wolfgang Soergel, Équivalences de certaines catégories de ${{\mathfrak g}}$-modules, C. R. Acad. Sci. Paris, Sér. 1 303 (1986), no. 15, 725--728.MR 88c:17011
  • [Soe89a] Wolfgang Soergel, ${{\mathfrak n}}$-cohomology of simple highest weight modules on walls and purity, Inventiones Math. 98 (1989), 565--580.MR 90m:22037
  • [Soe89b] Wolfgang Soergel, Parabolisch-singuläre Dualität für Kategorie ${{\mathcal O}}$, Preprint MPI/89-68, 1989.
  • [Soe90] Wolfgang Soergel, Kategorie ${{\mathcal O}},$ perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, Journal of the AMS 3 (1990), 421--445.MR 91e:17007
  • [Soe92] Wolfgang Soergel, Langlands' philosophy and Koszul duality, Preprint, 1992.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 17B10, 16A03

Retrieve articles in all journals with MSC (1991): 17B10, 16A03

Additional Information

Alexander Beilinson
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Victor Ginzburg
Affiliation: Department of Mathematics, The University of Chicago, Chicago, Illinois 60637

Wolfgang Soergel
Affiliation: Max-Planck-Institut für Mathematik Gottfried-Claren-Straße 26 D-53 Bonn 3 Germany
Address at time of publication: Mathematisches Institut, Universität Freiburg, Albertstraße 23b, D-79104 Freiburg, Germany

Received by editor(s): November 13, 1991
Received by editor(s) in revised form: February 16, 1995
Additional Notes: The first author was partially supported by an NSF grant
The second author thanks Harvard University and MIT, where part of this work was written
The third author thanks the MPI and DFG for financial support
Article copyright: © Copyright 1996 American Mathematical Society