Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum
Abstract
Let $G/H$ be a semisimple symmetric space. Then the space $L^2(G/H)$ can be decomposed into a finite sum of series representations induced from parabolic subgroups of $G$. The most continuous part of the spectrum of $L^2(G/H)$ is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions living on a tube type bounded symmetric domain, containing $G/H$ as a boundary component. A boundary value map is constructed and we show that it induces an $G$-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part $L_{\rm mc}^2(G/H)$ of $L^2(G/H)$. We also relate our Hardy space with the classical Hardy space on the tube domain.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- November 2001
- DOI:
- 10.48550/arXiv.math/0111172
- arXiv:
- arXiv:math/0111172
- Bibcode:
- 2001math.....11172G
- Keywords:
-
- Mathematics - Representation Theory;
- Mathematics - Complex Variables;
- 22E30;
- 32M15;
- 43A85
- E-Print:
- Revised version: readability improved, minor errors corrected. To appear in Math. Annalen, 36 pages