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)



Kategorie $ \mathcal{O}$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe

Author: Wolfgang Soergel
Journal: J. Amer. Math. Soc. 3 (1990), 421-445
MSC: Primary 17B35
MathSciNet review: 1029692
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a description of ``the algebra of category $ \mathcal{O}$'' which is explicit enough to prove that the structure of the direct summands of $ \mathcal{O}$ depends only on the integral Weyl group and the singularity of the central character, as well as to establish a weak version of the duality conjectures of Beilinson and Ginsburg [BGi]. As a byproduct we describe the intersection cohomology of Schubert varieties as modules over global cohomology ring. These are certain indecomposable graded self-dual modules over the coinvariant algebra of the Weyl group, via the Borel picture for the global cohomology ring of a flag manifold. They play a central role in this article and should have an interesting future.

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

  • [BGi] A. A. Beilinson and V. A. Ginsburg, Mixed categories, Ext-duality and representations (results and conjectures), preprint, 1986.
  • [BGS] A. A. Beilinson, V. A. Ginsburg, and V. V. Schechtman, Koszul duality, preprint, 1989. MR 1048505 (91c:18011)
  • [Be] J. N. Bernstein, The trace of projective functors, erscheint in Colloque Dixmier (Mai 1989).
  • [BGe] J. N. Bernstein and S. I. Gelfand, Tensor products of finite and infinite dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245-285. MR 581584 (82c:17003)
  • [BGG1] J. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Schubert cells and cohomology of the spaces $ G/P$, Russian Math. Surveys 28, (1973) No. 3, 1-26. MR 0429933 (55:2941)
  • [BGG2] -, Category of $ \mathfrak{g}$-modules, Functional Anal. Appl. 10 (1976), 87-92.
  • [Ga] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448. MR 0232821 (38:1144)
  • [GJ] O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), 261-302. MR 644519 (83e:17009)
  • [Ir] R. Irving, Projective modules in the category $ \mathcal{O}$, preprint, 1984.
  • [Ja] J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Springer-Verlag, Berlin and New York, 1983. MR 721170 (86c:17011)
  • [Jo] A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (1980), 191-213. MR 561970 (82f:17008)
  • [KL1] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 560412 (81j:20066)
  • [KL2] -, Schubert varieties and Poincaré duality, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 185-203. MR 573434 (84g:14054)
  • [LV] G. Lusztig and D. Vogan, Singularities of closures of $ K$-orbits on flag manifolds, Invent. Math. 71 (1983), 365-379. MR 689649 (84h:14060)
  • [So1] W. Soergel, Universelle versus relative Einhüllende: Eine geometrische Untersuchung von Quotienten von universellen Einhüllenden halbeinfacher Lie-Algebren (Dissertation, Hamburg), Math. Ann. 284 (1989), 177-198. MR 1000106 (90k:17034)
  • [So2] -, Parabolisch-singuläre Dualität für Kategorie $ \mathcal{O}$, MPI-preprint, 1989.
  • [Sp] T. A. Springer, Quelques applications de la cohomologie d'intersection (Séminaire Bourbaki 1981/82, exposé 589), Astérisque 92-93 (1982), 249-273. MR 689533 (85i:32016b)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 17B35

Retrieve articles in all journals with MSC: 17B35

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society