Geodesic spheres and non radial eigenfunctions on Damek-Ricci spaces

Let S=NA be a Damek-Ricci space with its standard metric γS. Let C:S→B be the Cayley transform from S onto the unit ball B in s. We compute the transported metric γB=C−1∗(γS). By separating variables in geodesic polar coordinates, we then compute the non-radial M-invariant eigenfunctions of the Laplacian on S, where M is the group of automorphisms of S preserving the inner product on s. The “radial” part of these eigenfunctions is given by (associated) Jacobi functions. The “angular” part is given by certain orthogonal polynomials in two variables studied by Koornwinder.