Holomorphic H-spherical distribution vectors in principal series representations

• 成果类型: Article
• 署名作者: Gindikin, S; Krötz, B; Olafsson, G
• 署名单位: Rutgers University System; Rutgers University New Brunswick; University of Oregon; Louisiana State University System; Louisiana State University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-004-0376-1
• 发表日期: 2004
• 页码: 643-682
• 关键词: symmetric-spaces; fourier

摘要

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L-2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown Xi. In this article we construct a minimal G-invariant subdomain Xi(H) of Xi with G/H as Shilov boundary. Let pi be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of pi, which admit a holomorphic extension to Xi(H), is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L-2(G/H)(mc) in a space of holomorphic functions.