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

TL;DR

This paper constructs and analyzes intertwining operators on tensor products of $U'_q(\\widehat{sl}(2))$ modules, relating them to the vector-valued big $q$-Jacobi transform and generalizing unitarity properties.

Contribution

It introduces new intertwining operators for $U'_q(\widehat{sl}(2))$ representations and connects their matrix elements to the big $q$-Jacobi transform, extending previous unitarity results.

Findings

Operators satisfy intertwining conditions on tensor products.

Explicit evaluation of infinite sums related to these operators.

Bilinear summation formulas generalize unitarity properties.

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.