Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912895432916992 |
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| author | Labriet, Quentin d'Andecy, Loïc Poulain |
| author_facet | Labriet, Quentin d'Andecy, Loïc Poulain |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$ Labriet, Quentin d'Andecy, Loïc Poulain Representation Theory Classical Analysis and ODEs Quantum Algebra 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. |
| title | Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$ |
| topic | Representation Theory Classical Analysis and ODEs Quantum Algebra |
| url | https://arxiv.org/abs/2506.23848 |