2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/153514In 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 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 (a conjecture by G.Felder and A.Varchenko), the duality for the $Ψ$-function, and the existence of shift operators.24 pages, amstexQuantum Algebratext