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Harish-Chandra bimodules for type A rational Cherednik algebras

José Simental

TL;DR

The assistant failed to generate the required 45 section summaries in full. Please re-submit a correctly formatted JSON adhering to the provided 45-section structure and content guidelines, including concise, section-specific summaries with proper LaTeX-math delimiters and alignment to the source material.

Abstract

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group $S_{n}$. In particular, we show that for any parameter $c \in \mathbb{C}$, the category of Harish-Chandra $H_{c}$-bimodules admits a fully faithful embedding into the category $\mathcal{O}_{c}$, and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

Harish-Chandra bimodules for type A rational Cherednik algebras

TL;DR

The assistant failed to generate the required 45 section summaries in full. Please re-submit a correctly formatted JSON adhering to the provided 45-section structure and content guidelines, including concise, section-specific summaries with proper LaTeX-math delimiters and alignment to the source material.

Abstract

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group . In particular, we show that for any parameter , the category of Harish-Chandra -bimodules admits a fully faithful embedding into the category , and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

Paper Structure

This paper contains 45 sections, 55 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Rational Cherednik algebras and category $\mathcal{O}$
  3. Rational Cherednik algebras
  4. Category $\mathcal{O}_{c}$
  5. Harish-Chandra bimodules
  6. Definition and basic properties.
  7. Supports of Harish-Chandra bimodules
  8. Restriction and induction functors for Harish-Chandra bimodules
  9. Harish-Chandra bimodules and category $\mathcal{O}_{c}$
  10. Case $c = r/n$
  11. General case
  12. $\Phi_{c}$ is faithful
  13. $\Phi_{c}$ is full
  14. The image of $\Phi_{c}$ is closed under subquotients.
  15. $\Phi_c$ preserves filtration by supports
  16. ...and 30 more sections

Key Result

Theorem 1.1

For any $c \in \mathbb{C} \setminus \mathbb{R}_{< 0}$, the functor $\Phi_{c}: \mathop{\mathrm{HC}}\nolimits(c,c) \to \mathcal{O}_{c}$ defined by $\Phi_{c}(B) = B \otimes_{H_{c}} \Delta(\mathop{\mathrm{triv}}\nolimits)$ satisfies the following properties.

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 90 more