  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 to the $q$-deformed
case. 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
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 $\Psi$-function,
and the existence of shift operators.
