Affine analogue of Jack's polynomials for sl2 (hat)

Abstract

Affine analogue of Jack's polynomials introduced by Etingof and Kirillov Jr. is studied for the case of sl₂ (hat). Using the Wakimoto representation, we give an integral formula of elliptic Selberg type for the affine Jack's polynomials. From this integral formula, the action of SL₂(Z) on the space of the affine Jack's polynomials is computed. For simple cases, we write down the affine Jack's polynomials in terms of some modular and elliptic functions.