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Pseudo-Hermitian Symmetric Spaces

  • Soji Kaneyuki
Chapter
Part of the Progress in Mathematics book series (PM, volume 185)

Abstract

Let G/H be an almost effective symmetric coset space of a connected Lie group G. We do not assume H to be compact. If G is simple, G/H is called a simple symmetric space. If the linear isotropy representation of H is irreducible (resp. reducible), then G/H is called simple irreducible (resp. reducible). If G/H admits a G-invariant complex structure J and a G-invariant pseudo-Hermitian metric (with respect to J), then a G/H is called pseudo-Hermitian. Simple symmetric spaces were classified infinitesimally by Berger [1].

Keywords

Symmetric Space Equivariant Action Hermitian Symmetric Space Siegel Domain Hermitian Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    M. Berger, Les espaces symétriques non compacts Ann. Sci. École Norm. Sup., 74 (1957), 85–177.MATHGoogle Scholar
  2. [2]
    W. Bertram, Un théorème de Liouville pour les algèbres de Jordan Bull. Soc. Math. France, 124 (1996), 299–327.MathSciNetMATHGoogle Scholar
  3. [3]
    H. Braun and M. Koecher Jordan-Algebren, Springer-Verlag, Heidelberg, 1965.Google Scholar
  4. [4]
    J. E. D’Atri and S. Gindikin, Siegel domain realization of pseudo-Hermitian symmetric manifolds Geometriae Dedicata, 46, (1993), 91–126.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    B. A. Dubrovin, A. T. Fomenko and S. P. Novikov Modern Geometry-Methods and Applications, Part I, Springer-Verlag, New York, 1984.MATHGoogle Scholar
  6. [6]
    J. Faraut and S. Gindikin, Pseudo-Hermitian symmetric spaces of tube type, in Topics in Geometry, In Memory of J. D’Atri (S. Gindikin, Ed.), Birkhäuser, Boston, 1996, pp.123–154.Google Scholar
  7. [7]
    S. Gindikin, Integral geometry, twistors and generalized conformal structures, J. Geom. and Physics, 5, (1988), 19–35.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    S. Gindikin and S. Kaneyuki, On the automorphism group of the generalized conformal structure of a symmetric R-space, Differential Geom. Appl., 8, (1998), 21–33.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    L. K. Hua, Causality and Lorentz group, Proc. R. Soc. London, A380, (1982), 487–488.MATHCrossRefGoogle Scholar
  10. [10]
    S. Kaneyuki, On orbit structure of compactifications of parahermitian symmetricspaces, Japan J. Math., 13, (1987) 333–370.MathSciNetMATHGoogle Scholar
  11. [11]
    S. Kaneyuki, On Siegel domains of finite type, J. Math. Soc. Japan, 39, (1987), 597–607.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    S. Kaneyuki, The Sylvester’s law of inertia for Jordan algebras, Proc. Japan Acad. Ser. A, 64, (1988), 311–313.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    S. Kaneyuki, On the causal structures of the Silov boundaries of symmetricbounded domains, in Prospects in Complex Geometry Lect. Notes. in Math. 1468, Springer-Verlag, New York, 1991, pp.127–159.Google Scholar
  14. [14]
    S. Kaneyuki, Pseudo-Hermitian symmetric spaces and Siegel domains overnon-degenerate cones, Hokkaido Math. J., 20, (1991) 213–239.MathSciNetMATHGoogle Scholar
  15. [15]
    S. Kaneyuki, On the subalgebras go and gevof semisimple graded Lie algebras, J. Math. Soc. Japan, 45, (1993) 1–19.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    S.Kaneyuki, The Sylvester’s law of inertia in simple graded Lie algebras, J. Math. Soc. Japan, 50, (1998), 593–614.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    S. Kaneyuki, On the automorphism groups of parahermitian symmetric spaces, Preprint.Google Scholar
  18. [18]
    S. Kaneyuki and H. Asano, Graded Lie algebras and generalized Jordan triple systems, Nagoya Math. J., 112, (1988), 81–115.MathSciNetMATHGoogle Scholar
  19. [19]
    S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math., 8, (1985), 81–98.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    S. Kobayashi and T. Nagano, On filtered Lie algebras and geometric structures I, J. Math. Mech., 13(1964), 875–908.MathSciNetMATHGoogle Scholar
  21. [21]
    M. Koecher Jordan algebras and their applications, Lect. Notes, Univ. Minnesota, Minneapolis, 1962.MATHGoogle Scholar
  22. [22]
    M. Koecher An elementary approach to bounded symmetric domains, Lect. Notes, Rice Univ. 1969.MATHGoogle Scholar
  23. [23]
    S. Koh, On affine symmetric spaces, Trans. Amer. Math. Soc., 119, (1965), 291–309.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    A. Koranyi and J. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math., 81, (1965), 265–288.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    R. P. Langlands, The dimension of the space of automorphic forms, Amer. J. Math., 85, (1963), 99–125.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    O. Loos, Jordan triple systems, R-spaces and bounded symmetric domains, Bull. Amer. Math. Soc., 77, (1971), 558–561.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    O. Loos Bounded symmetric domains and Jordan pairs, Math. Lect. Univ. Calif., Irvine, 1977.Google Scholar
  28. [28]
    H. Matsumoto, Quelques remarques sur les groupes de Lie algébriques réels J. Math. Soc. Japan, 16, (1964), 419–446.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    T. Nagano, Transformation groups on compact symmetric spaces, Trans. Amer. Math. Soc., 118, (1965), 428–453.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math., 57, (1980), 1–81.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    T. Oshima and J. Sekiguchi, The restricted root system of a semisimple symmetric pair: Group Representations and Systems of Differential Equations Adv. Studies in Pure Math. 4, Kinokuniya, Tokyo and North-Holland, Amsterdam, 1984, 433–497.Google Scholar
  32. [32]
    G. Roos, Jordan triple systems, This volume, 425–534.Google Scholar
  33. [33]
    I. Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math., 71, (1960), 77–110.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    I. Satake Algebraic structures of symmetric domains IwanamiShoten, Tokyo and Princeton Univ. Press, Princeton, NJ, 1980.MATHGoogle Scholar
  35. [35]
    M. Takeuchi, Cell decompositions and Morse equalities on certain symmetric spaces, J. Fac. Sci. Univ. Tokyo, 12, (1965), 81192.Google Scholar
  36. [36]
    M. Takeuchi, On orbits in a compact hermitian symmetric space, Amer. J. Math., 90, (1968), 657–680.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    M. Takeuchi, On conjugate loci and cut loci of compact symmetric spaces II, Tsukuba J. Math., 3, (1979), 1–29.MATHGoogle Scholar
  38. (38]
    M. Takeuchi, Basic transiformations of symmetric R-spaces, Osaka. J. Math., 25, (1988), 259–297.MathSciNetMATHGoogle Scholar
  39. [39]
    N. Tanaka, On equivalence problems associated with a certain class of homogeneous spaces, J. Math. Soc. Japan, 17, (1965), 103–139.MathSciNetMATHCrossRefGoogle Scholar
  40. [40]
    N. Tanaka, On affine symmetric spaces and the automorphism groups of product manifolds, Hokkaido Math. J., 14, (1985), 277–351.MathSciNetMATHGoogle Scholar
  41. [41]
    T. Tsuji, Siegel domains over self-dual cones and their automorphisms, Nagoya Math. J., 55, (1974), 33–80.MathSciNetMATHGoogle Scholar
  42. [42]
    J. Wolf, The action of a real semisimple group on a complex flag manifold, I; Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc., 75, (1969), 1121–1237.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    E. C. Zeeman, Causality implies the Lorentz group, J. Math. Physics, 5, (1964), 490–493.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Soji Kaneyuki
    • 1
  1. 1.Department of MathematicsSophia UniversityTokyoJapan

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