Equivariant K-theory of Hilbert schemes via shuffle algebra

Abstract

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 ${\mathbb{C}}^2$. We show that commutative elements $K_i$ of shuffle algebra act through vertex operators over the positive part $\{{\mathfrak{h}}_i{\}}_{i>0}$ 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 $\mathbb{C}[{\mathfrak{h}}_1,{\mathfrak{h}}_2,\dots ]$.