Non-symmetric Jacobi polynomials of type BC1 as vector-valued polynomials, Part 1: Spherical functions

We study non-symmetric Jacobi polynomials of type $B{C}_1$ 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 $B{C}_1$ in terms of the symmetric Jacobi polynomials of type $B{C}_1$. 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 $B{C}_1$.

We show that the non-symmetric Jacobi polynomials of type $B{C}_1$ 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.