Macdonald Polynomials and Algebraic Integrability

Abstract

We construct explicitly (nonpolynomial) eigenfunctions of the difference operators by Macdonald in the case t=qk, k∈Z. This leads to a new, more elementary proof of several Macdonald conjectures, proved first by Cherednik. We also establish the algebraic integrability of Macdonald operators at t=qk (k∈Z), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including the BCn case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of the An root system where the previously known methods do not work.