Inventiones mathematicae

, Volume 154, Issue 3, pp 617–651

On the category 𝒪 for rational Cherednik algebras

Article

Abstract

We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arkhipov, S.: Semi-infinite cohomology of associative algebras and bar duality. Int. Math. Res. Not. 17, 833–863 (1997) CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex exceptional groups. J. Algebra 80, 350–382 (1983) MathSciNetMATHGoogle Scholar
  3. 3.
    Berest, Y., Etingof, P., Ginzburg, V.: Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. 118, 279–337 (2003) Google Scholar
  4. 4.
    Bjork, J.-E.: Rings of differential operators. North-Holland Math. Libr. 21. Amsterdam, New York: North-Holland Publishing Co. 1979 Google Scholar
  5. 5.
    Borel, A., et al.: Algebraic D-modules. Academic Press 1987 Google Scholar
  6. 6.
    Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998) MathSciNetGoogle Scholar
  7. 7.
    Broué, M., Michel, J.: Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées. In: Finite reductive groups, 73–139. Birkhäuser 1997 Google Scholar
  8. 8.
    Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988) MathSciNetMATHGoogle Scholar
  9. 9.
    Cline, E., Parshall, B., Scott, L.: Stratifying endomorphism algebras. Mem. Am. Math. Soc. 124 (1996) Google Scholar
  10. 10.
    Deligne, P.: Equations différentielles à points singuliers réguliers. Lect. Notes Math. 163. Springer 1970 Google Scholar
  11. 11.
    Dipper, R., James, G.: Representations of Hecke Algebras of General Linear Groups. Proc. Lond. Math. Soc., III. Ser. 52, 20–52 (1986) Google Scholar
  12. 12.
    Dunkl, C., Opdam, E.: Dunkl operators for complex reflection groups. Proc. Lond. Math. Soc., III. Ser. 86, 70–108 (2003) Google Scholar
  13. 13.
    Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space and deformed Harish-Chandra homomorphism. Invent. Math. 147, 243–348 (2002) CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Garsia, A.M., McLarnan, T.J.: Relations between Young’s natural and the Kazhdan-Lusztig representations of Sn. Adv. Math. 69, 32–92 (1988) MathSciNetMATHGoogle Scholar
  15. 15.
    Geck, M., Rouquier, R.: Filtrations on projective modules for Iwahori-Hecke algebras. In: Modular representation theory of finite groups, 211–221. de Gruyter 2001 Google Scholar
  16. 16.
    Guay, N.: Projective modules in the category 𝒪 for the Cherednik algebra. To appear in Commun. Pure Appl. Algebra (2003) Google Scholar
  17. 17.
    Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979) MathSciNetMATHGoogle Scholar
  18. 18.
    Lusztig, G.: Characters of reductive groups over a finite field. Ann. Math. Stud. 107. Princeton: Princeton Univ. Press 1984 Google Scholar
  19. 19.
    Lusztig, G.: Cells in affine Weyl groups. In: Algebraic groups and related topics. Adv. Studies in Pure Math., Vol. 6, pp. 255–287. Kinokuniya, North-Holland 1985 Google Scholar
  20. 20.
    Lusztig, G.: Cells in affine Weyl groups, III. J. Fac. Sci., Univ. Tokyo, Sect. IA, Math. 34, 223–243 (1987) Google Scholar
  21. 21.
    Müller, J.: letter, 25 October 2002 Google Scholar
  22. 22.
    Naruse, H.: On an isomorphism between Specht module and left cell of 𝔖n. Tokyo J. Math. 12, 247–267 (1989) MathSciNetMATHGoogle Scholar
  23. 23.
    Ringel, C.M.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z. 208, 209–223 (1991) MathSciNetMATHGoogle Scholar
  24. 24.
    Soergel, W.: Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Am. Math. Soc. 3, 421–445 (1990) MathSciNetMATHGoogle Scholar
  25. 25.
    Soergel, W.: Charakterformeln für Kipp-Moduln über Kac-Moody Algebren. Represent. Theory 1, 115–132 (1997)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Korteweg de Vries Institute for MathematicsUniversity of AmsterdamThe Netherlands
  3. 3.UFR de Mathématiques et Institut de Mathématiques de Jussieu (CNRS UMR 7586)Université Paris VIIParis Cedex 05France

Personalised recommendations