Holomorphic H-spherical distribution vectors in principal series representations
Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L2(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 Ξ. In this article we construct a minimal G-invariant subdomain ΞH of Ξ with G/H as Shilov boundary. Let π be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of π, which admit a holomorphic extension to ΞH, 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 L2(G/H) mc in a space of holomorphic functions. © Springer-Verlag 2004.
Publication Source (Journal or Book title)
Gindikin, S., Krötz, B., & Ólafsson, G. (2004). Holomorphic H-spherical distribution vectors in principal series representations. Inventiones Mathematicae, 158 (3), 643-682. https://doi.org/10.1007/s00222-004-0376-1