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Holomorphic H-spherical distribution vectors in principal series representations

Simon Gindikin·Bernhard Kroetz·Gestur Olafsson
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Abstract

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 $Ξ$. In this article we construct a minimal G-invariant subdomain $Ξ_H$ 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 $Ξ_H$, 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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Simon Gindikin, Bernhard Kroetz, Gestur Olafsson. 2004-04-12. Holomorphic H-spherical distribution vectors in principal series representations. https://doi.org/10.1007/s00222-004-0376-1

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