Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$

TL;DR

This paper derives explicit formulas for intertwining operators of the quantum group $U_q(sl_2)$, connecting them to little $q$-Jacobi polynomials and $q$-deformed Rankin--Cohen operators, revealing new algebraic structures.

Contribution

It provides explicit formulas for symmetry breaking operators and holographic operators for $U_q(sl_2)$ using little $q$-Jacobi polynomials, including a $q$-deformation of classical operators.

Findings

Explicit formulas for intertwining operators in terms of $q$-polynomials

Connection between symmetry breaking operators and $q$-Hahn polynomials

Realization of Verma modules on polynomial spaces with non-commuting variables

Abstract

This paper presents explicit formulas for intertwining operators of the quantum group $U_q(sl_2)$ acting on tensor products of Verma modules. We express a first set of intertwining operators (the holographic operators) in terms of the little $q$-Jacobi polynomials, and we obtain for the dual set (the symmetry breaking operators) a $q$-deformation of the Rankin--Cohen operators. The Verma modules are realised on polynomial spaces and, interestingly, we find along the way the need to work with non-commuting variables. Explicit connections are given with the Clebsch--Gordan coefficients of $U_q(sl_2)$ expressed with the $q$-Hahn polynomials.