2007/11/09 by Genkai Zhang, Zhang, Genkai
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Holomorphic and Operator Theory #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.0711.1480
arxiv created 2007/11/09 · openalex publication_date 2007/11/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X=H/L be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain D=G/K. The intersection S of the Shilov boundary of D with X defines a distinguished subset of the topological boundary of X and is invariant under H and can also be realized as S=H/P for certain parabolic subgroup P of H. We study the spherical representations IndPH(\lam) of H induced from P. We find formulas for the spherical functions in terms of the Macdonald 2F1 hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces D. We consider a class of H-invariant integral intertwining operators from the representations IndPH(\lam) on L2(S) to the holomorphic representations of G on D restricted to H. We construct a new class of complementary series for the groups H=SO(n, m), SU(n, m) (with n-m >2) and Sp(n, m) (with n-m>1). We realize them as a discrete component in the branching rule of the analytic continuation of the holomorphic discrete series of G=SU(n, m), SU(n, m)× SU(n, m) and SU(2n, 2m) respectively.