Spherical transform and Jacobi polynomials on root systems of type BC

Abstract

Let R be a root system of type BC in a = Rr of general positive multiplicity. We introduce certain canonical weight function on Rr 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 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. Introduction The Gaussian functions and the Hermite polynomials play an important role in Fourier transform on Euclidean spaces; the Hermite polynomials diagonalize the harmonic oscillator and the corresponding Hermite type function diagonalize the Fourier transform, which make the Plancherel theory more transparent. The generalization of the Plancherel theory to any non-compact symmetric symmetric spaces has been studied intensively and there are still no general theory generalizing corresponding the results of Hermite polynomials and their Fourier transform, and above all, no concrete orthogonal systems of functions with explicit spherical transforms are constructed. In the present paper we will establish such a theory for root system of type BC. Now associated with any root system in R there are two kinds of remarkable orthogonal polynomials, namely the Heckman-Opdam orthogonal polynomials giving the spectral decomposition of the algebra of the Weyl group invariant polynomials of Cherednik operators acting on certain L-space on a compact torus T, and the MacdonaldKoornwinder polynomials orthogonal with respect to certain weight functions on R defined as a product of Gamma functions. Part of the product is in fact the HarishChandra Plancherel measure for the spectral decomposition of the algebra of Cherednik operators on the non-compact space R. In the present paper we will introduce certain canonical weight function f−2ν(t) for root system of type BC, and we compute its spherical transform f−2ν(λ). The product of f−2ν(λ) 2 with the Harish-Chandra measure |c(λ)| gives precisely weight functions in the Macdonald-Koornwinder orthogonality Research supported by the Swedish Science Research Council (VR). 1