Algebraic integrability of Macdonald operators and representations of quantum groups

In this paper we construct examples of commutative rings of difference oper ators with matrix coefficients from representation theory of quantum groups , generalizing the results of our previous paper [ES] to the q-deformed cas e. A generalized Baker-Akhiezer function Psi is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators. In particular, we obtain the following result in Macd onald theory: at integer values of the Mac donald parameter Ic, there exist difference operators commuting with Macdonald operators which are not poly nomials of Macdonald operators. This result generalizes an analogous result of Chalyh and Veselov for the case q = I, to arbitrary q. As a by-product, we prove a generalized Weyl character formula for Macdonald polynomials (= Conjecture 8.2 from [FV]), the duality for the Psi-function, and the exist ence of shift operators.