Associated to a finite cyclic subgroup G of SL_2 (C), there is a family of noncommutative algebras O^{\tau} = O^{\tau}(C^2//G) representing a universal deformation of the coordinate ring of the classical Kleinian singularity C^2//G .
Earlier, in his thesis, F. Eshmatov constructed an isomorphism between the moduli space of rank one
projective modules (noncommutative line bundles) over O^{\tau} and a certain class of Nakajima quiver varieties M^{\tau} associated to G via the McKay correspondence. He showed that the varieties M^{\tau} carry a natural action of the automorphism group Aut[O^{\tau}] of the algebra O^{\tau}, and the above isomorphism is equivariant under this
action. In this talk, we will prove that the action of Aut[O^{\tau}] on M^{\tau} is actually transitive, and will use
this result to give a geometric classification of algebras Morita equivalent to O^{\tau}. We will also compute the Picard group of auto-equivalences of the abelian category of O^{\tau}-modules.
(This is joint work with X. Chen, F. Eshmatov and V. Futorny)