On the complex geometry of invariant domains in complexified symmetric spaces

Neeb, Karl-Hermann. "On the complex geometry of invariant domains in complexified symmetric spaces." Annales de l'institut Fourier 49.1 (1999): 177-225. <http://eudml.org/doc/75332>.

@article{Neeb1999,
abstract = {Let $M=G/H$ be a real symmetric space and $\{\frak g\}=\{\frak h\} + \{\frak q\}$ the corresponding decomposition of the Lie algebra. To each open $H$-invariant domain $D_\{\frak q\}\subseteq i\{\frak q\}$ consisting of real ad-diagonalizable elements, we associate a complex manifold $\Xi (D_\{\frak q\})$ which is a curved analog of a tube domain with base $D_\{\frak q\}$, and we have a natural action of $G$ by holomorphic mappings. We show that $\Xi (D_\{\frak q\})$ is a Stein manifold if and only if $D_\{\frak q\}$ is convex, that the envelope of holomorphy is schlicht and that $G$-invariant plurisubharmonic functions correspond to convex $H$-invariant functions on $D_\{\frak q\}$. Finally we apply these results to obtain an integral decomposition for $G$-invariant Hilbert spaces of holomorphic functions on $\Xi (D_\{\frak q\})$.},
author = {Neeb, Karl-Hermann},
journal = {Annales de l'institut Fourier},
keywords = {semigroup; Lie group; compact symmetric group; compact symmetric space; plurisubharmonic functions; representation theory; causal symmetric spaces},
language = {eng},
number = {1},
pages = {177-225},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the complex geometry of invariant domains in complexified symmetric spaces},
url = {http://eudml.org/doc/75332},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Neeb, Karl-Hermann
TI - On the complex geometry of invariant domains in complexified symmetric spaces
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 1
SP - 177
EP - 225
AB - Let $M=G/H$ be a real symmetric space and ${\frak g}={\frak h} + {\frak q}$ the corresponding decomposition of the Lie algebra. To each open $H$-invariant domain $D_{\frak q}\subseteq i{\frak q}$ consisting of real ad-diagonalizable elements, we associate a complex manifold $\Xi (D_{\frak q})$ which is a curved analog of a tube domain with base $D_{\frak q}$, and we have a natural action of $G$ by holomorphic mappings. We show that $\Xi (D_{\frak q})$ is a Stein manifold if and only if $D_{\frak q}$ is convex, that the envelope of holomorphy is schlicht and that $G$-invariant plurisubharmonic functions correspond to convex $H$-invariant functions on $D_{\frak q}$. Finally we apply these results to obtain an integral decomposition for $G$-invariant Hilbert spaces of holomorphic functions on $\Xi (D_{\frak q})$.
LA - eng
KW - semigroup; Lie group; compact symmetric group; compact symmetric space; plurisubharmonic functions; representation theory; causal symmetric spaces
UR - http://eudml.org/doc/75332
ER -