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

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.