On the complex geometry of invariant domains in complexified symmetric spaces

Neeb, Karl-Hermann

Annales de l'Institut Fourier, Tome 49 (1999), p. 177-225 / Harvested from Numdam

Résumé

Soit $M=G/H$ un espace symétrique réel et ${\frak g}={\frak h} + {\frak q}$ la décomposition correspondante de l’algèbre de Lie. À tout domaine ouvert et $H$-invariant $D_{\frak q}\subseteq i{\frak q}$ formé d’éléments réels ad-diagonalisables, on associe une variété complexe $\Xi (D_{\frak q})$ qui est une généralisation non-linéaire d’un domaine tube à base $D_{\frak q}$ et nous avons une action naturelle de $G$ par des applications holomorphes. On montre que $\Xi (D_{\frak q})$ est une variété de Stein si et seulement si $D_{\frak q}$ est convexe, que l’enveloppe d’holomorphie est schlicht et que les fonctions $G$-invariantes plurisousharmoniques correspondent aux fonctions $H$-invariantes convexes sur $D_{\frak q}$. Finalement on applique ces résultats pour démontrer l’existence d’une décomposition intégrale pour les espaces de Hilbert $G$-invariants de fonctions holomorphes sur $\Xi (D_{\frak q})$.

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 $\Xi (D_{\frak q})$.

Détail

Publié le : 1999-01-01

MR 2000i:32040 | Zbl 0921.22003

DOI : https://doi.org/10.5802/aif.1671

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