On Unitarizable Harish-Chandra Bimodules for Deformations of Type-A Kleinian Singularities

The notion of a Harish-Chandra bimodule, that is, finitely generated U(g)-bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [9]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, that is, the algebra itself, for a deformation of Kleinian singularity of type A is unitarizable. We obtain a partial classification of unitarizable regular bimodules.