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 E8. More precisely, consider the quantization 𝒜⋋ with parameter ⋋. We show that the top quotient \( \overline{\mathrm{HC}}\left(\mathcal{A}\lambda \right) \) of the category of Harish-Chandra 𝒜⋋-bimodules embeds into the category of representations of the algebraic fundamental group, Γ, of the open leaf. The image coincides with the representations of Γ/Γ⋋, where Γ⋋ is a normal subgroup of Γ that can be recovered from the quantization parameter ⋋ combinatorially. 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.