A note on tensor products of q-algebra representations and orthogonal polynomials

We work out examples of tensor products of distinct generalized slq(2) algebras with a factor from the positive discrete series of representations of one algebra and a factor from the negative discrete series of the other. We show that the equation for the common eigenfunctions of the Casimir operator and the Cartan subalgebra generator is just the three-term recurrence relation corresponding to orthogonality for special cases of the Askey-Wilson polynomials, and this connection yields an almost immediate resolution of the tensor product representation into a direct integral of irreducible representation. An identity for the matrix elements of the “group representation operators” with respect to the tensor product and the reduced bases follows easily. Cases where the measures for the orthogonal polynomials are not unique correspond to cases where the tensor products and their resolutions are also nonunique.