Spherical transform and Jacobi polynomials on root systems of type BC

TL;DR

This paper introduces a canonical weight function on root systems of type BC, computes its spherical transform, and relates associated Jacobi-type polynomials to Macdonald-Koornwinder polynomials, generalizing previous results.

Contribution

It generalizes the spherical transform and Bernstein-Sato formula for root systems of type BC, connecting Jacobi polynomials with Macdonald-Koornwinder polynomials.

Findings

Computed the spherical transform of the canonical weight function.

Proved a Bernstein-Sato type formula for the weight function.

Established the relation between Jacobi-type polynomials and Macdonald-Koornwinder polynomials.

Abstract

Let $R$ be a root system of type BC in $\mathfrak a=\mathbb R^r$ of general positive multiplicity. We introduce certain canonical weight function on $\mathbb R^r$ which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove certain Bernstein-Sato type formula. This generalizes earlier work of Unterberger-Upmeier, van Dijk-Pevsner, Neretin and the author. Associated to the weight functions 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 non-compact space $\mathfrak a$. We consider their spherical transform and prove that they are the Macdonald-Koornwinder polynomials multiplied by the spherical transform of the canonical weight function. For rank one case this was proved earlier by Koornwinder.