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

This paper is a continuation of [ES1] and [EV2]. In [EV2], A.Varchenko and the first author considered weighted traces of products of intertwining operators for quantum groups Uq(g), where g is a simple Lie algebra. They showed that the generating function FV1,...VN (λ, μ) of such traces (where λ, μ are complex weights for g) satisfies four commuting systems of difference equations – the Macdonald-Ruijsenaars (MR) system, the quantum Knizhnik-Zamolodchikov-Bernard (qKZB) system, the dual MR system, and the dual qKZB system. The first two systems are systems of difference equations with respect to λ, which involve Felder’s trigonometric dynamical R-matrix depending of λ. The second two systems are systems of difference equations respect to μ, which are obtained from the first two by the transformation λ → μ, Vi → V ∗ N−i+1. Such a symmetry is explained by the fact that the function FV1,...VN (λ, μ) is invariant under this transformation. If the quantum group Uq(g) is replaced with the Lie algebra g, these results are replaced with their classical analogs ([EV2]). Namely, the MR and qKZB equations are replaced by the classical MR and KZB equations, which are differential equations involving Felder’s classical trigonometric dynamical r-matrix. The dual MR and KZB equations retain roughly the same form, but involve the rational quantum dynamical R-matrix rather than the trigonometric one. Thus, the symmetry between λ and μ is destroyed. In [ES1], we generalized the classical MR and KZB equations to the case when the trace is twisted using a ”generalized Belavin-Drinfeld triple”, i.e. a pair of subdiagrams Γ1,Γ2 of the Dynkin diagram of g together with an isomorphism T : Γ1 → Γ2 between them. It turned out that such twisted traces also satisfy differential equations which involve a dynamical r-matrix, namely the one attached to the triple (Γ1,Γ2, T ) by the second author in [S]. After [ES1] was finished, we wanted to generalize its results to the quantum case. It was clear to us that to express the result we would need an explicit quantization of classical dynamical r-matrices from [S]. Therefore, we hoped that attempts to quantize the results of [ES1] using the approach of [EV2] could help us obtain such a quantization (which was unknown even for the usual Belavin-Drinfeld classical r-matrices). This program did, in fact, succeed, and the quantization of dynamical r-matrices from [S] was recently obtained in [ESS]. In this paper, using the results of [ESS] and methods of [EV2], we generalize the difference equations from [EV2] to the twisted case; this also provides a