# Harish-Chandra bimodules for type A rational Cherednik algebras

## TL;DR

This paper investigates Harish-Chandra bimodules for type A rational Cherednik algebras, establishing embeddings into category O, describing irreducibles, and constructing a duality that elucidates their structure and tensor products.

## Contribution

It introduces a fully faithful embedding of Harish-Chandra bimodules into category O and constructs a duality applicable to Nakajima quiver varieties, advancing understanding of their structure.

## Key findings

- Embedding of bimodules into category O established
- Explicit description of irreducible bimodules provided
- Duality and tensor product structures characterized

## 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.