Spherical transform and Jacobi polynomials on root systems of type BC

2005/03/31 by Genkai Zhang, Zhang, Genkai
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Representation Theory (math.RT) #math.CA #math.RT

paper · pdf · doi:10.48550/arxiv.math/0503735

arxiv created 2005/04/26 · arxiv updated 2009/12/01

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.

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