The ({mathbb {A}}_{q,t}) algebra and parabolic flag Hilbert schemes

The earlier work of the first and the third name authors introduced the algebra $${\mathbb {A}}_{q,t}$$ and its polynomial representation. In this paper we construct an action of this algebra on the equivariant K-theory of certain smooth strata in the flag Hilbert scheme of points on the plane. In this presentation, the fixed points of the torus action correspond to generalized Macdonald polynomials, and the matrix elements of the operators have an explicit presentation.