2001/10/19 by Genkai Zhang, Zhang, Genkai
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.math/0110212
arxiv created 2001/10/19 · openalex publication_date 2001/10/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbb D=G/K be a complex bounded symmetric domain of tube type in a Jordan algebra V\mathbb C, and let D=H/L =\mathbb D∩ V be its real form in a Jordan algebra V⊂ V\mathbb C. The analytic continuation of the holomorphic discrete series on \mathbb D forms a family of interesting representations of G. We consider the restriction on D of the scalar holomorphic representations of G, as a representation of H. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group L is a spherical subgroup of K and we find a canonical basis of L-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 L-invariant polynomials are, under the spherical transform on D, 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 D, when extended to a neighborhood in \mathbb D, in terms of the L-spherical holomorphic polynomials on \mathbb D, the coefficients being the Wilson polynomials.