  The earlier work of the first and the third named 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 schemes of points on the plane. In this
presentation, the fixed points of torus action correspond to generalized
Macdonald polynomials and the the matrix elements of the operators have
explicit combinatorial presentation.
