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This paper is a continuation of math.QA/9907181 and math.QA/9908115. Weconsider traces of intertwiners between certain representations of thequantized enveloping algebra associated to a semisimple complex Lie algebra g,which are twisted by a ``generalized Belavin-Drinfeld triple'', i.e a tripleconsisting of two subdiagrams of the Dynkin diagram of g together with anisomorphism between them. The generating functions F(lambda,mu) for such tracesdepend on two weights lambda and mu. We show that F(lambda,mu) satisfy two setsof difference equations in the variable lambda: the Macdonald-Ruijsenaars (MR)equations and the quantum Knizhnik-Zamolodchikov  (qKZB) equations. These equations involve as a main ingredient the quantumdynamical R-matrices constructed in math.QA/9912009. When the generalizedBelavin-Drinfeld triple is an automorphism, we show that F(lambda,mu) satisfyanother two sets of difference equations with respect to the weight mu. Thesedual MR and dual qKZB equations involve the usual Felder's dynamical R-matrix.These results were first obtained by the first author and A. Varchenko in thespecial case of the trivial Belavin-Drinfeld triple. However, the symmetrybetween lambda and mu which exists in that case is destroyed in the twistedsetting. At the end, we brielfly treat the (simialr) case of Kac-Moody algebrasg and derive the classical limits of all the previous results.Comment: 30 pages, late

Twisted Traces of Quantum Intertwiners¶and Quantum Dynamical R-Matrices¶Corresponding to Generalized Belavin–Drinfeld Triples