| Summary: | We study Harish-Chandra bimodules for rational Cherednik algebras $H{c}(W)$ associated to a complex reflection group $W$ and parameter $c$. Our results allow to partially reduce the study of these to smaller algebras. We use this to classify those pairs of parameters $(c,c')$ for which there exist Harish-Chandra bimodules with full support, and we give a description of the category of all Harish-Chandra bimodules modulo those bimodules with proper support. When $W$ is the
symmetric group, we produce an embedding from the category of Harish-Chandra $H{c}$-bimodules to the category $\cO_{c}$, prove that its image is closed under subquotients, and find the irreducibles in its image. Finally, when $W$ is of cyclotomic type we produce a duality in the category of Harish-Chandra bimodules. We do this in the more general setting of quantized quiver varieties. Our methods are based on localization techniques, the study of partial KZ functors, the action of
Namikawa-Weyl groups and restriction functors.
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