Holomorphic H-spherical distribution vectors in principal series representations

Details Holomorphic H-spherical distribution vectors in principal series representations Holomorphic H-spherical distribution vectors in principal series representations

2003-04-14

arxiv.org/abs/math/0304175v2

Invent. math. 158(3) (2004), 643-684.

Mathematics

Representation Theory

Due to a very careful and professional referee's work we could strongly improve on both exposition and detail. To appear in In

Scientific paper

10.1007/s00222-004-0376-1

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.

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