Equivariant K-theory of Hilbert schemes via shuffle algebra

In this paper we construct the action of Ding-Iohara and shuffle algebras on the sum of localized equivariant K-groups of Hilbert schemes of points on C(2). We show that commutative elements K(i) of shuffle algebra act through vertex operators over the positive part {h(i)}(i>0) of the Heisenberg algebra in these K-groups. Hence we get an action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space C[h(1), h(2), .....].