Harish-Chandra bimodules over quantized symplectic singularities

TL;DR

This paper classifies irreducible Harish-Chandra bimodules over quantized symplectic singularities, revealing their structure via fundamental group representations and connecting to Lusztig quotients.

Contribution

It provides a classification of bimodules over quantizations of symplectic singularities, linking their structure to fundamental groups and orbit geometry, under specific conditions.

Findings

Embedding of bimodule categories into fundamental group representations

Description of Lusztig quotient groups via orbit normalization

Classification results under conditions excluding type E8 slices

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.