Non-symmetric Jacobi polynomials of type $BC_{1}$ as vector-valued polynomials Part 1: spherical functions

TL;DR

This paper presents a new vector-valued and matrix-valued interpretation of non-symmetric Jacobi polynomials of type BC1, linking them to spherical functions on spheres and revealing connections to the Dirac operator.

Contribution

It introduces a novel matrix-valued framework for non-symmetric Jacobi polynomials, connecting them to spherical functions and Dirac operators on spheres.

Findings

Expressed non-symmetric Jacobi polynomials in terms of symmetric ones.

Identified these polynomials with spherical functions on spheres.

Linked the Cherednik operator to the Dirac operator on spinor fields.

Abstract

We study non-symmetric Jacobi polynomials of type $BC_{1}$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials yields a new expression of the non-symmetric Jacobi polynomials of type $BC_1$ in terms of the symmetric Jacobi polynomials of type $BC_{1}$. In this interpretation, the Cherednik operator, that has the non-symmetric Jacobi polynomials as eigenfunctions, corresponds to two shift operators for the symmetric Jacobi polynomials of type $BC_{1}$. We show that the non-symmetric Jacobi polynomials of type $BC_{1}$ with so-called geometric root multiplicities, interpreted as vector-valued polynomials, can be identified with spherical functions on the sphere $S^{2m+1}=\mathrm{Spin}(2m+2)/\mathrm{Spin}(2m+1)$ associated with the fundamental spin-representation of $\mathrm{Spin}(2m+1)$. The Cherednik operator corresponds to the Dirac operator for the spinors on $S^{2m+1}$ in this interpretation.