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Harish-Chandra bimodules over rational Cherednik algebras

José Simental

Abstract

We study Harish-Chandra bimodules over the rational Cherednik algebra $H_{c}(W)$ associated to a complex reflection group $W$ with parameter $c$. Our results allow us to partially reduce the study of these bimodules to smaller algebras. We classify those pairs of parameters $(c,c')$ for which there exist fully supported Harish-Chandra bimodules, and give a description of the category of all Harish-Chandra bimodules modulo those without full support. When $W$ is a symmetric group we are able to classify all irreducible Harish-Chandra bimodules. Our proofs are based on localization techniques, the action of the Namikawa-Weyl group on the set of parameters, and the study of partial KZ functors.

Harish-Chandra bimodules over rational Cherednik algebras

Abstract

We study Harish-Chandra bimodules over the rational Cherednik algebra associated to a complex reflection group with parameter . Our results allow us to partially reduce the study of these bimodules to smaller algebras. We classify those pairs of parameters for which there exist fully supported Harish-Chandra bimodules, and give a description of the category of all Harish-Chandra bimodules modulo those without full support. When is a symmetric group we are able to classify all irreducible Harish-Chandra bimodules. Our proofs are based on localization techniques, the action of the Namikawa-Weyl group on the set of parameters, and the study of partial KZ functors.

Paper Structure

This paper contains 36 sections, 56 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rational Cherednik Algebras.
  4. Homogeneous rational Cherednik algebras.
  5. Category $\mathcal{O}_{c}$
  6. Hecke algebras.
  7. KZ functor
  8. Lattice $\mathfrak{p}_{\mathbb{Z}}$
  9. Restriction functors for category $\mathcal{O}$
  10. Harish-Chandra bimodules.
  11. Induction and restriction functors for HC bimodules.
  12. Reduction to corank 1
  13. Localization
  14. Bezrukavnikov-Etingof isomorphisms
  15. Supports and symplectic leaves
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $W$ be a complex reflection group, and let $c, c' \in \mathbb{C}[\mathcal{S}]$ be conjugation invariant functions. The following is true.

Theorems & Definitions (95)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5: losev_bernstein, Lemma 4.2
  • Lemma 2.6
  • proof
  • ...and 85 more