AGT correspondence: Ding-Iohara algebra at roots of unity and Lepowsky-Wilson construction

It was recently conjectured that the AGT correspondence between the U(r)-instanton counting on R-4/Z(p) and the two-dimensional field theories with the conformal symmetry algebra A(r, p) can be considered as a root of unity limit of its K-theoretic analogue. From this point of view, the algebra A(r, p) and a special basis in its representation are limits of the Ding-Iohara algebra and the Macdonald polynomials respectively. In this paper we confirm this conjecture for the special case r = 1. We uncover the implicit A(1, p) symmetry in this limit. We also found that the vertex operators in the special basis have factorized AFLT form.