Spherical transform and Jacobi polynomials on root systems of type BC

Let R be a root system of type BC in a = ℝ^r with general positive multiplicities. We introduce a certain canonical weight function on a which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove a Bernstein-Sato-type formula. This generalizes earlier work of Unterberger and Upmeier, van Dijk-Pevzner, Neretin, and the author. Associated to the weight function there are Heckman-Opdam orthogonal polynomials of Jacobi type on the compact torus; after a change of variables they form an orthogonal system on the noncompact space a. We prove that their spherical transforms are the multivariable Wilson polynomials multiplied by the spherical transform of the canonical weight function. For the rank one case, this was proved earlier by Koornwinder.