Macdonald polynomials and algebraic integrability

We construct explicitly (nonpolynomial) eigenfunctions of the difference operators by Macdonald in the case t = q(k), k epsilon 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 = q(k) (k epsilon 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 A(n) root system where the previously known methods do not work. (C) 2002 Elsevier Science (USA).