Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the category 𝒪 for rational Cherednik algebras


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.

This is a preview of subscription content, log in to check access.


  1. 1.

    Arkhipov, S.: Semi-infinite cohomology of associative algebras and bar duality. Int. Math. Res. Not. 17, 833–863 (1997)

  2. 2.

    Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex exceptional groups. J. Algebra 80, 350–382 (1983)

  3. 3.

    Berest, Y., Etingof, P., Ginzburg, V.: Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. 118, 279–337 (2003)

  4. 4.

    Bjork, J.-E.: Rings of differential operators. North-Holland Math. Libr. 21. Amsterdam, New York: North-Holland Publishing Co. 1979

  5. 5.

    Borel, A., et al.: Algebraic D-modules. Academic Press 1987

  6. 6.

    Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998)

  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

  8. 8.

    Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)

  9. 9.

    Cline, E., Parshall, B., Scott, L.: Stratifying endomorphism algebras. Mem. Am. Math. Soc. 124 (1996)

  10. 10.

    Deligne, P.: Equations différentielles à points singuliers réguliers. Lect. Notes Math. 163. Springer 1970

  11. 11.

    Dipper, R., James, G.: Representations of Hecke Algebras of General Linear Groups. Proc. Lond. Math. Soc., III. Ser. 52, 20–52 (1986)

  12. 12.

    Dunkl, C., Opdam, E.: Dunkl operators for complex reflection groups. Proc. Lond. Math. Soc., III. Ser. 86, 70–108 (2003)

  13. 13.

    Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space and deformed Harish-Chandra homomorphism. Invent. Math. 147, 243–348 (2002)

  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)

  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

  16. 16.

    Guay, N.: Projective modules in the category 𝒪 for the Cherednik algebra. To appear in Commun. Pure Appl. Algebra (2003)

  17. 17.

    Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)

  18. 18.

    Lusztig, G.: Characters of reductive groups over a finite field. Ann. Math. Stud. 107. Princeton: Princeton Univ. Press 1984

  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

  20. 20.

    Lusztig, G.: Cells in affine Weyl groups, III. J. Fac. Sci., Univ. Tokyo, Sect. IA, Math. 34, 223–243 (1987)

  21. 21.

    Müller, J.: letter, 25 October 2002

  22. 22.

    Naruse, H.: On an isomorphism between Specht module and left cell of 𝔖n. Tokyo J. Math. 12, 247–267 (1989)

  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)

  24. 24.

    Soergel, W.: Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Am. Math. Soc. 3, 421–445 (1990)

  25. 25.

    Soergel, W.: Charakterformeln für Kipp-Moduln über Kac-Moody Algebren. Represent. Theory 1, 115–132 (1997)

Download references

Author information

Correspondence to Victor Ginzburg or Nicolas Guay or Eric Opdam or Raphaël Rouquier.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ginzburg, V., Guay, N., Opdam, E. et al. On the category 𝒪 for rational Cherednik algebras. Invent. math. 154, 617–651 (2003). https://doi.org/10.1007/s00222-003-0313-8

Download citation


  • Reflection
  • Weyl Group
  • Module Category
  • Dimensional Algebra
  • Reflection Group