In this paper we construct the action of Ding-Iohara and shuffle algebras on the sum of localized equivariant -groups of Hilbert schemes of points on . We show that commutative elements of shuffle algebra act through vertex operators over the positive part of the Heisenberg algebra in these -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 .
"Equivariant -theory of Hilbert schemes via shuffle algebra." Kyoto J. Math. 51 (4) 831 - 854, Winter 2011. https://doi.org/10.1215/21562261-1424875