no. 1
On the complex geometry of invariant domains in complexified symmetric spaces
Annales de l'Institut Fourier, Tome 49 (1999) no. 1, pp. 177-225.

Soit M=G/H un espace symétrique réel et 𝔤=𝔥+𝔮 la décomposition correspondante de l’algèbre de Lie. À tout domaine ouvert et H-invariant D 𝔮 ⊆i𝔮 formé d’éléments réels ad-diagonalisables, on associe une variété complexe Ξ(D 𝔮 ) qui est une généralisation non-linéaire d’un domaine tube à base D 𝔮 et nous avons une action naturelle de G par des applications holomorphes. On montre que Ξ(D 𝔮 ) est une variété de Stein si et seulement si D 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 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 Ξ(D 𝔮 ).

Let M=G/H be a real symmetric space and 𝔤=𝔥+𝔮 the corresponding decomposition of the Lie algebra. To each open H-invariant domain D 𝔮 ⊆i𝔮 consisting of real ad-diagonalizable elements, we associate a complex manifold Ξ(D 𝔮 ) which is a curved analog of a tube domain with base D 𝔮 , and we have a natural action of G by holomorphic mappings. We show that Ξ(D 𝔮 ) is a Stein manifold if and only if D 𝔮 is convex, that the envelope of holomorphy is schlicht and that G-invariant plurisubharmonic functions correspond to convex H-invariant functions on D 𝔮 . Finally we apply these results to obtain an integral decomposition for G-invariant Hilbert spaces of holomorphic functions on Ξ(D 𝔮 ).

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     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},
     zbl = {0921.22003},
     mrnumber = {2000i:32040},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1671/}
}
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Neeb, Karl-Hermann. On the complex geometry of invariant domains in complexified symmetric spaces. Annales de l'Institut Fourier, Tome 49 (1999) no. 1, pp. 177-225. doi : 10.5802/aif.1671. https://aif.centre-mersenne.org/articles/10.5802/aif.1671/

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