Matrix Spherical Functions and Orthogonal Polynomials: an Instructive Example

In the scalar case, it is well known that the zonal spherical functions of any compact Riemannian symmetric space of rank one can be expressed in terms of the Jacobi polynomials. The main purpose of this paper is to revisit the matrix valued spherical functions associated to the complex projective plane to exhibit the interplay among these functions, the matrix hypergeometric functions and the matrix orthogonal polynomials. We also obtain very explicit expressions for the entries of the spherical functions in the case of 2×2 matrices and exhibit a natural sequence of matrix orthogonal polynomials, beyond the group parameters.