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On the category đť’Ş for rational Cherednik algebras

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, IL 60637, Chicago, USA

    Victor Ginzburg & Nicolas Guay

  2. Korteweg de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, 1018TV, The Netherlands

    Eric Opdam

  3. UFR de Mathématiques et Institut de Mathématiques de Jussieu (CNRS UMR 7586), Université Paris VII, 2 place Jussieu, 75251, Paris Cedex 05, France

    Raphaël Rouquier

Authors
  1. Victor Ginzburg
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  2. Nicolas Guay
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  3. Eric Opdam
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  4. Raphaël Rouquier
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Correspondence to Victor Ginzburg, Nicolas Guay, Eric Opdam or Raphaël Rouquier.

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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

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