Fusion and exchange matrices for quantized sl(2) and associated q-special functions

The aim of this paper is to evaluate in terms of q-special functions the objects (intertwining map, fusion matrix, exchange matrix) related to the quantum dynamical Yang-Baxter equation (QDYBE) for infinite dimensional representations (Verma modules) of the quantized universal enveloping algebra Uq(g) in the case g = sl(2,C). This study is done in the framework of the exchange construction, initiated by Etingof and Varchenko, in order to find solutions to the QDYBE for Uq(g) with g a complex semisimple Lie algebra (see [13, §2], [12, §2], [11, §3]). As a result, the familiar interpretation of q-Hahn and q-Racah polynomials as q-ClebschGordan and q-Racah coefficients, respectively, for finite dimensional irreducible representations of the quantum group SUq(2) is extended to a much larger range of parameters, while moreover these interpretations are obtained in an unusual and interesting way, following the definitions of intertwining map and exchange matrix. Furthermore, we reprove, by using these explicit expressions, some properties related to the objects under study in the special case g = sl(2), which were earlier proved in [13] and [12] in a more abstract way in the case of more general g. The paper is organized as follows. In Section 2 a brief review of q-special functions and of the quantized universal algebra Uq(sl(2)) and its representations is given. Section 3 deals with the intertwining operator Φq,λ : Mq,λ −→ Mq,μ ⊗ V . Here Mq,λ, Mq,μ are Verma modules for Uq(sl(2)) with highest weight vectors xλ, xμ (λ, μ ∈ C), V is a Uq(sl(2))-module, not necessarily of finite dimension and soon assumed to be a Verma module, Φq,λ is Uq(sl(2))-intertwining, and Φq,λ(xλ) is supposed to have “highest” term xμ ⊗ v. On various places in the paper we do explicit computations first in the generic infinite dimensional case and next make transition (by continuity) to the finite dimensional case in order to connect the results with known explicit expressions in the finite dimenisonal case. These limit transitions need careful justification. The matrix elements of the intertwining operator with respect to the standard bases of Verma modules generalize the q-Clebsch-Gordan coefficients for the finite dimensional case