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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 talk is based on arXiv:1810.07625. I will not assume any preliminary knowledge of conical symplectic singularities, their quantizations etc.