Harish-Chandra bimodules over quantized symplectic singularities

Abstract: In this paper we classify the irreducible Harish-Chandra bimodules with full support over filtered quantizations of conical symplectic singularities under the condition that none of the slices to codimension 2 symplectic leaves has type $E_8$. More precisely, we show that the top quotient $\overline{\operatorname{HC}}(\mathcal{A}\lambda)$ of the category of Harish-Chandra bimodules over the quantization $\mathcal{A}\lambda$ with parameter $\lambda$ embeds into the category of representations of the algebraic fundamental group, $\Gamma$, of the open leaf. The image coincides with the representations of $\Gamma/\Gamma_\lambda$, where $\Gamma_\lambda$ is a normal subgroup of $\Gamma$ that can be recovered from the quantization parameter $\lambda$. As an application of our results, we describe the Lusztig quotient group in terms of the geometry of the normalization of the orbit closure in almost all cases.