In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker–Akhiezer function Ψ 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 Macdonald theory: at integer values of the Macdonald parameter k, there exist difference operators commuting with Macdonald operators which are not polynomials of Macdonald operators. This result generalizes an analogous result of Chalyh and Veselov for the case q = 1, 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 Ψ-function, and the existence of shift operators.