Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T05:03:13.601Z Has data issue: false hasContentIssue false
  • English
  • Français

On Canonical Realizations of Bounded Symmetric Domains as Matrix-Spaces1)

Published online by Cambridge University Press:  22 January 2016

Mikio Ise
Mikio Ise*
Affiliation:
University of Tokyo
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is the purpose of the present paper to give a natural method of realizing bounded symmetric domains as matrix-spaces. Our method yields, as special cases, the well-known bounded models of irreducible bounded symmetric domains of classical type (I)-(IV), as were already described in the original paper of E. Cartan [1] (see §3; we follow in this paper the classification table in [14], not in [1]). A direct application of this method will be to determine the canonical bounded models of the irreducible bounded symmetric domains of exceptional type; it will be published in another paper (see [6], [7] for the summary of the results).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 42 , March 1971 , pp. 115 - 133
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

Footnotes

1)

Part of the present work was done in 1964, when the author was staying at the Institute for Advanced Study, Princeton, under the sponsorship of the National Science Foundation.

References

[1] Cartan, E., Sur les domaines bornès homogènes des l’espace de n-variables complexes. Abhandlungen Math. Sem. Hambourg, 11 (1935), 116–162.Google Scholar
[2] Cartan, H., Ouverts fondamentaux pour le groupe modulaire, Seminaire H. Cartan, 1957–58. Expose III.Google Scholar
[3] Helgason, S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London, 1962.Google Scholar
[4] Hermann, R., Geometric aspects of potential theory in the symmetric bounded domains, II, Math. Annalen, 151 (1963), 143–149.Google Scholar
[5] Ise, M., Some properties of complex analytic vector bundles over compact, complex homogeneous spaces, Osaka Math. J., 12 (1960), 217–252.Google Scholar
[6] Ise, M., Realization of irreducible bounded symmetric domain of type (V), Proc. Jap. Acad. Sci., 45 (1969), 233–237.Google Scholar
[7] Ise, M., Realization of irreducible bounded symmetric domain of type (VI), ibid., 846–849.Google Scholar
[8] Klingen, H., Diskontinuierliche Gruppen in symmetrischen Räurnen, I, Math. Annalen, 129 (1955), 345–369.Google Scholar
[9] Klingen, H., Über analytischen Abbildungen verallgemeinerter Einheitskreis auf sich, Math. Ananlen, 132 (1956), 134–144.Google Scholar
[10] Langlands, R., The dimension of spaces of automorphic forms, Amer. J. Math., 85 (1963), 99–125.Google Scholar
[11] Matsushima, Y. and Murakami, S., On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math., 78 (1963), 363–416.CrossRefGoogle Scholar
[12] Moore, C.C., Compactifications of symmetric spaces, II, Amer. J. Math., 86 (1964), 201–218.Google Scholar
[13] Nagano, T., Transformation groups on compact symmetric spaces, Trans. Amer. Math. Soc, 118 (1965), 428–453.Google Scholar
[14] Siegel, C.L., Analytic Functions of Several Complex Variables, Princeton, 1949.Google Scholar
[15] Takahashi, R., Sur les représentations unitaires des groupes de Lorentz généralisés. Bull. Soc. Math, de France, 91 (1962), 289–433.Google Scholar
[16] Takeuchi, M., Cell decompositions and Morse equalities on certain symmetric spaces, J. Fac. Sci. Univ. of Tokyo, 12 (1965), 81–192.Google Scholar