We study a certain category of bimodules over a filtered algebra

quantizing the algebra of functions on a conical symplectic singularity.

The bimodules we care about are so called Harish-Chandra bimodules. This notion

first appeared in the case of universal enveloping algebras of semisimple Lie

algebras in the work of Harish-Chandra on representations of the corresponding

complex Lie groups. Since then it was generalized to filtered quantizations

of algebras of functions on affine Poisson varieties. The goal of this talk

is to explain a classification of the simple Harish-Chandra bimodules with full

support over quantizations of conical symplectic singularities (that have

no slices of type E_8). We will see that these irreducible bimodules are in

one-to-one correspondence with the irreducible representations of a suitable

finite group. The group in question arises as the quotient of the algebraic

fundamental group of the open leaf by a normal subgroup depending on

the quantization parameter in a way that will be explained in the talk.

The talk is based on arXiv:1810.07625. I will not assume any preliminary

knowledge of conical symplectic singularities, their quantizations etc.