Inventiones mathematicae

, Volume 154, Issue 3, pp 617–651

On the category 𝒪 for rational Cherednik algebras


DOI: 10.1007/s00222-003-0313-8

Cite this article as:
Ginzburg, V., Guay, N., Opdam, E. et al. Invent. math. (2003) 154: 617. doi:10.1007/s00222-003-0313-8


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.

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