Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

We continue our study of the AGT correspondence between instanton counting on C/Zp and Conformal field theories with the symmetry algebra A(r,p). In the cases r = 1, p = 2 and r = 2, p = 2 this algebra specialized to: A(1,2) = H sI(2)1 and A(2,2) = H sI(2)2 NSR. As the main tool we use a new construction of the algebra A(r, 2) as the limit of the toroidal aI(1) algebra for q, t tend to -1. We claim that the basis of the representation of the algebra A(r/2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q, t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the N=1 Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q, t tend to -1. © 2013 SISSA, Trieste, Italy.