191 research outputs found
Spherical transform and Jacobi polynomials on root systems of type BC
Let be a root system of type BC in of general
positive multiplicity. We introduce certain canonical weight function on
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 . 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
Branching coefficients of holomorphic representations and Segal-Bargmann transform
Let be a complex bounded symmetric domain of tube type in a
Jordan algebra , and let be its real
form in a Jordan algebra . The analytic continuation of
the holomorphic discrete series on forms a family of interesting
representations of . We consider the restriction on of the scalar
holomorphic representations of , as a representation of . The unitary
part of the restriction map gives then a generalization of the Segal-Bargmann
transform. The group is a spherical subgroup of and we find a canonical
basis of -invariant polynomials in components of the Schmid decomposition
and we express them in terms of the Jack symmetric polynomials. We prove that
the Segal-Bargmann transform of those -invariant polynomials are, under the
spherical transform on , multi-variable Wilson type polynomials and we give
a simple alternative proof of their orthogonality relation. We find the
expansion of the spherical functions on , when extended to a neighborhood in
, in terms of the -spherical holomorphic polynomials on , the coefficients being the Wilson polynomials
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