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,...].