Bounded symmetric domains and derived geometric structures

Kaup, Wilhelm

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 13 (2002), p. 243-257 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

Every homogeneous circular convex domain $D \subset \mathbb{C}^{n}$ (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group $G = Aut(D)$ of all biholomorphic automorphisms of $D$ and its isotropy subgroup $K \subset GL(n,\mathbb{C})$ at the origin (a maximal compact subgroup of $G$). The group $G$ acts in a natural way on the compact dual $X$ of $D$ (a certain compactification of $\mathbb{C}^{n}$ that generalizes the Riemann sphere in case $D$ is the unit disk in $\mathbb{C}$). Various authors have studied the orbit structure of the $G$-space $X$, here we are interested in the Cauchy-Riemann structure of the $G$-orbits in $X$ (which in general are only real-analytic submanifolds of $X$). Also, we discuss certain $K$-orbits in the Grassmannian of all linear subspaces of $\mathbb{C}^{n}$ that are closely related to the geometry of the bounded symmetric domain $D$.

Détail

Publié le : 2002-12-01

MR 1984104 | Zbl 1098.32008

Bibliographie

[1] Aupetit, B., Sur les transformations qui conservent le spectre. 13th International Conference on Banach Algebras (Blaubeuren 1997), de Gruyter, Berlin1998, 55-78. | MR 1656598 | Zbl 0932.46038

[2] Dieudonné, J., Sur une généralisation du groupe orthogonal à quatre variables. Arch. Math., 1, 1949, 282-287. | MR 29360 | Zbl 0032.10601

[3] Douady, A., Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné. Ann. Inst. Fourier, 16, 1966, 1-95. | MR 203082 | Zbl 0146.31103

[4] Faulkner, J.R., On the Geometry of Inner Ideals. J. of Algebra, 26, 1973, 1-9. | MR 367002 | Zbl 0285.17004

[5] Friedman, Y. - Russo, B., The Gelfand-Naimark theorem for JB*-triples. Duke Math. J., 53, 1986, 139-148. | MR 835800 | Zbl 0637.46049

[6] Frobenius, G., Über die Darstellung der endlichen Gruppen durch lineare Substitutionen. I. Sitzungsberichte Königl. Preuss. Akad. Wiss., 1897, 994-1015. | JFM 28.0130.01

[7] Harris, L.A., Bounded symmetric homogeneous domains in infinite dimensional spaces. Lecture Notes in Mathematics, vol. 364, Springer-Verlag, Berlin-Heidelberg-New York1973. | MR 407330 | Zbl 0293.46049

[8] Harris, L.A. - Kaup, W., Linear algebraic groups in infinite dimensions. Ill. J. Math., 21, 1977, 666-674. | MR 460551 | Zbl 0385.22011

[9] Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York-San Francisco-London1978. | MR 514561 | Zbl 0451.53038

[10] Kaup, W., A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z., 183, 1983, 503-529. | MR 710768 | Zbl 0519.32024

[11] Kaup, W., Über die Klassifikation der symmetrischen Hermiteschen Mannigfaltigkeiten unendlicher Dimension I, II. Math. Ann., 257, 1981, 463-483; 262, 1983, 503-529. | MR 639580 | Zbl 0482.32010

[12] Kaup, W., On Grassmannians associated with JB*-triples. Math. Z., 236, 2001, 567-584. | MR 1821305 | Zbl 0988.46048

[13] Kaup, W. - Zaitsev, D., On Symmetric Cauchy-Riemann Manifolds. Adv. in Math., 149, 2000, 145-181. | MR 1742704 | Zbl 0954.32016

[14] Kaup, W. - Zaitsev, D., On the CR-structure of compact group orbits associated with bounded symmetric domains. In preparation. | Zbl 1027.32032

[15] Koecher, M., An elementary approach to bounded symmetric domains. Rice Univ., Houston1969. | MR 261032 | Zbl 0217.10901

[16] Korányi, A. - Wolf, J.A., Realization of hermitian symmetric spaces as generalized half planes. Ann. of Math., 81, 1965, 265-288. | MR 174787 | Zbl 0137.27402

[17] Loos, O., Symmetric Spaces I/II. W. A. Benjamin, Inc., New York-Amsterdam1969. | Zbl 0175.48601

[18] Loos, O., Jordan Pairs. Lecture Notes in Mathematics, vo. 460, Springer-VerlagBerlin-Heidelberg-New York1975. | MR 444721 | Zbl 0301.17003

[19] Loos, O., Bounded symmetric domains and Jordan pairs. Mathematical Lectures, University of California at Irvine, Irvine1977.

[20] Springer, T.A., On the geometric algebra of the octave planes. Indag. Math., 24, 1962, 451-468. | MR 142045 | Zbl 0113.35903

[21] Takeuchi, M., On orbits in a compact hermitian symmetric space. Am. J. Math., 90, 1968, 657-680. | MR 245827 | Zbl 0181.24304

[22] Wolf, J.A., The action of a real semisimple group on a complex flag manifold. I: Orbit structure and holomorphic arc components. Bull. Am. Math., 75, 1969, 1121-1247. | MR 251246 | Zbl 0183.50901

[23] Wolf, J.A., Fine Structure of Hermitian Symmetric Spaces. In: W.M. Boothby - G.L. Weiss (eds.), Symmetric Spaces. Pure and Applied Mathematics, 8, Marcel Dekker Inc., New York 1972, 271- 357. | MR 404716 | Zbl 0257.32014

[24] Upmeier, H., Symmetric Banach Manifolds and Jordan C*-Algebras. North-Holland1985. | MR 776786 | Zbl 0561.46032