  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.
