Intertwiners of $U'_q\bigl(\widehat{sl}(2)\bigr)$-representations and the vector-valued big $q$-Jacobi transform

Abstract

Linear operators $R$ are introduced on tensor products of evaluation modules of $U'_q\bigl(\widehat{sl}(2)\bigr)$ obtained from the complementary and strange series representations. The operators $R$ satisfy the intertwining condition on finite linear combinations of the canonical basis elements of the tensor products. Infinite sums associated with the action of $R$ on six pairs of tensor products are evaluated. For two pairs, the sums are related to the vector-valued big $q$-Jacobi transform of the matrix elements defining the operator $R$. In one case, the sums specify the action of $R$ on the irreducible representations present in the decomposition of the underlying indivisible sum of $U_q\bigl(sl(2)\bigr)$-tensor products. In both cases, bilinear summation formulae for the matrix elements of $R$ provide a generalization of the unitarity property.