On the complex geometry of invariant domains in complexified symmetric spaces
Annales de l'Institut Fourier, Volume 49 (1999) no. 1, p. 177-225
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 ).
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 ).
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {1},
     year = {1999},
     pages = {177-225},
     doi = {10.5802/aif.1671},
     mrnumber = {2000i:32040},
     zbl = {0921.22003},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1999__49_1_177_0}
}
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. https://aif.centre-mersenne.org/item/AIF_1999__49_1_177_0/

[AL92] H. Azad, and J.-J. Loeb, Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math., N.S., 3(4) (1992), 365-375. | MR 94a:32014 | Zbl 0777.32008

[DN93] N. Dörr, and K.-H. Neeb, On wedges in Lie triple systems and ordered symmetric spaces, Geometriae Ded., 46 (1993), 1-34. | MR 94m:17004 | Zbl 0781.53040

[Hel84] S. Helgason, Groups and geometric analysis, Acad. Press, London, 1984.

[HÓ96] J. Hilgert, G. Ólafsson, Causal Symmetric Spaces, Geometry and Harmonic Analysis, Acad. Press, 1996. | Zbl 0931.53004

[Hö73] L. Hörmander, An introduction to complex analysis in several variables, North-Holland, 1973. | Zbl 0271.32001

[Kr97] B. Krötz, The Plancherel Theorem for Biinvariant Hilbert Spaces, Publ. R.I.M.S., to appear. | Zbl 01439660

[KN96] B. Krötz and K.-H. Neeb, On hyperbolic cones and mixed symmetric spaces, Journal of Lie Theory, 6:1 (1996), 69-146. | MR 97k:17007 | Zbl 0860.22004

[KNÓ97] B. Krötz K.-H. Neeb, and G. Ólafsson, Spherical Representations and Mixed Symmetric Spaces, Journal of Representation Theory, 1 (1997), 424-461. | MR 99a:22031 | Zbl 0887.22022

[Las78] M. Lasalle, Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact, Ann. Sci. Ec. Norm. Sup., 4e série, 11 (1978), 167-210. | Numdam | Zbl 0452.43011

[Lo69] O. Loos, Symmetric Spaces I: General Theory, Benjamin, New York, Amsterdam, 1969. | MR 39 #365a | Zbl 0175.48601

[MaMo60] Y. Matsushima, and A. Morimoto, Sur certains espaces fibrés holomorphes sur une variété de Stein, Bull. Soc. Math. France, 88 (1960), 137-155. | Numdam | MR 23 #A1061 | Zbl 0094.28104

[Ne96a] K.-H. Neeb, Invariant Convex Sets and Functions in Lie Algebras, Semigroup Forum, 53 (1996), 230-261. | MR 97j:17033 | Zbl 0873.17009

[Ne96b] K.-H. Neeb, On some classes of multiplicity free representations, Manuscripta Math., 92 (1997), 389-407. | MR 99e:22010 | Zbl 0882.43002

[Ne97] K.-H. Neeb, Representation theory and convexity, submitted. | Zbl 0964.22004

[Ne98] K.-H. Neeb, On the complex and convex geometry of Ol'shanskiĠ semigroups, Annales de l'Institut Fourier, 48-1 (1998), 149-203. | Numdam | MR 99e:22013 | Zbl 0901.22003

[Ne99] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, de Gruyter, 1999, to appear. | Zbl 0936.22001