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.