  Intermediate Jacobi polynomials for a root system $R$ with Weyl group $W$ are
orthogonal polynomials that are invariant under a parabolic subgroup of $W$.
The extreme cases are the symmetric and the non-symmetric Jacobi polynomials
studied by Heckman and Opdam. Intermediate Jacobi polynomials can also be
understood as vector-valued orthogonal polynomials.
  We study the rank one case in this paper. The interpretation of the
non-symmetric Jacobi polynomials as vector-valued polynomials has interesting
consequences. The first is that the non-symmetric Jacobi polynomials can be
expressed in terms of the symmetric ones. The second is that we recover a shift
operator for the symmetric Jacobi polynomials that comes from the Hecke algebra
representation. Thirdly, for geometric root multiplicities the vector-valued
polynomials can be identified with spherical functions on the sphere $S^{2n}$
associated to the fundamental spin-representation. In this way the
Cherednik-operator appears as a Dirac operator for the spinors on this space.
