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    Ezov, V. V. and Schmalz, G. 1998. Automorphisms and holomorphic mappings of standard CR-manifolds and Siegel domains. Journal of Mathematical Sciences, Vol. 92, Issue. 2, p. 3712.
    1978. Vol. 80, Issue. , p. 587.
    Alekseevskii, D. V. 1975. Lie groups and homogeneous spaces. Journal of Soviet Mathematics, Vol. 4, Issue. 5, p. 483.
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On Canonical Realizations of Bounded Symmetric Domains as Matrix-Spaces 1)

  • Mikio Ise (a1)
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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).

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Copyright
COPYRIGHT: © Editorial Board of Nagoya Mathematical Journal 1971
Footnotes
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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.

Footnotes
References
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[1] Cartan, E., Sur les domaines bornès homogènes des l’espace de n-variables complexes. Abhandlungen Math. Sem. Hambourg, 11 (1935), 116–162.
[2] Cartan, H., Ouverts fondamentaux pour le groupe modulaire, Seminaire H. Cartan, 1957–58. Expose III.
[3] Helgason, S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London, 1962.
[4] Hermann, R., Geometric aspects of potential theory in the symmetric bounded domains, II, Math. Annalen, 151 (1963), 143–149.
[5] Ise, M., Some properties of complex analytic vector bundles over compact, complex homogeneous spaces, Osaka Math. J., 12 (1960), 217–252.
[6] Ise, M., Realization of irreducible bounded symmetric domain of type (V), Proc. Jap. Acad. Sci., 45 (1969), 233–237.
[7] Ise, M., Realization of irreducible bounded symmetric domain of type (VI), ibid., 846–849.
[8] Klingen, H., Diskontinuierliche Gruppen in symmetrischen Räurnen, I, Math. Annalen, 129 (1955), 345–369.
[9] Klingen, H., Über analytischen Abbildungen verallgemeinerter Einheitskreis auf sich, Math. Ananlen, 132 (1956), 134–144.
[10] Langlands, R., The dimension of spaces of automorphic forms, Amer. J. Math., 85 (1963), 99–125.
[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.
[12] Moore, C.C., Compactifications of symmetric spaces, II, Amer. J. Math., 86 (1964), 201–218.
[13] Nagano, T., Transformation groups on compact symmetric spaces, Trans. Amer. Math. Soc, 118 (1965), 428–453.
[14] Siegel, C.L., Analytic Functions of Several Complex Variables, Princeton, 1949.
[15] Takahashi, R., Sur les représentations unitaires des groupes de Lorentz généralisés. Bull. Soc. Math, de France, 91 (1962), 289–433.
[16] Takeuchi, M., Cell decompositions and Morse equalities on certain symmetric spaces, J. Fac. Sci. Univ. of Tokyo, 12 (1965), 81–192.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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