Let be a real symmetric space and the corresponding decomposition of the Lie algebra. To each open -invariant domain consisting of real ad-diagonalizable elements, we associate a complex manifold which is a curved analog of a tube domain with base , and we have a natural action of by holomorphic mappings. We show that is a Stein manifold if and only if is convex, that the envelope of holomorphy is schlicht and that -invariant plurisubharmonic functions correspond to convex -invariant functions on . Finally we apply these results to obtain an integral decomposition for -invariant Hilbert spaces of holomorphic functions on .
Soit un espace symétrique réel et la décomposition correspondante de l’algèbre de Lie. À tout domaine ouvert et -invariant formé d’éléments réels ad-diagonalisables, on associe une variété complexe qui est une généralisation non-linéaire d’un domaine tube à base et nous avons une action naturelle de par des applications holomorphes. On montre que est une variété de Stein si et seulement si est convexe, que l’enveloppe d’holomorphie est schlicht et que les fonctions -invariantes plurisousharmoniques correspondent aux fonctions -invariantes convexes sur . Finalement on applique ces résultats pour démontrer l’existence d’une décomposition intégrale pour les espaces de Hilbert -invariants de fonctions holomorphes sur .
@article{AIF_1999__49_1_177_0, author = {Neeb, Karl-Hermann}, title = {On the complex geometry of invariant domains in complexified symmetric spaces}, journal = {Annales de l'Institut Fourier}, pages = {177--225}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {1}, year = {1999}, doi = {10.5802/aif.1671}, mrnumber = {2000i:32040}, zbl = {0921.22003}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1671/} }
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 SP - 177 EP - 225 VL - 49 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1671/ DO - 10.5802/aif.1671 LA - en ID - AIF_1999__49_1_177_0 ER -
%0 Journal Article %A Neeb, Karl-Hermann %T On the complex geometry of invariant domains in complexified symmetric spaces %J Annales de l'Institut Fourier %D 1999 %P 177-225 %V 49 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1671/ %R 10.5802/aif.1671 %G en %F AIF_1999__49_1_177_0
Neeb, Karl-Hermann. On the complex geometry of invariant domains in complexified symmetric spaces. Annales de l'Institut Fourier, Volume 49 (1999) no. 1, pp. 177-225. doi : 10.5802/aif.1671. http://www.numdam.org/articles/10.5802/aif.1671/
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