Traces of intertwiners for quantum groups and difference equations, II
  In this paper we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights lambda, mu, three scalar parameters q, omega, k, and spectral parameters z_1,...,z_N, which may be regarded as q-analogs of conformal blocks of the Wess-Zumino-Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined in math.QA/0110081. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations -- the Macdonald-Ruijsenaars equation, the q-Knizhnik-Zamolodchikov-Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation lambda <--> mu, k <--> omega. Thus, our results here generalize our previous results, dealing with the case q = 1 and the finite dimensional case. 
Pavel Etingof, Olivier Schiffmann, and Alexander Varchenko