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Pseudo-Hermitian Symmetric Spaces

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Analysis and Geometry on Complex Homogeneous Domains

Part of the book series: Progress in Mathematics ((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].

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References

  1. M. Berger, Les espaces symétriques non compacts Ann. Sci. École Norm. Sup., 74 (1957), 85–177.

    MATH  Google Scholar 

  2. W. Bertram, Un théorème de Liouville pour les algèbres de Jordan Bull. Soc. Math. France, 124 (1996), 299–327.

    MathSciNet  MATH  Google Scholar 

  3. H. Braun and M. Koecher Jordan-Algebren, Springer-Verlag, Heidelberg, 1965.

    Google Scholar 

  4. J. E. D’Atri and S. Gindikin, Siegel domain realization of pseudo-Hermitian symmetric manifolds Geometriae Dedicata, 46, (1993), 91–126.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. A. Dubrovin, A. T. Fomenko and S. P. Novikov Modern Geometry-Methods and Applications, Part I, Springer-Verlag, New York, 1984.

    MATH  Google Scholar 

  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. S. Gindikin, Integral geometry, twistors and generalized conformal structures, J. Geom. and Physics, 5, (1988), 19–35.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. K. Hua, Causality and Lorentz group, Proc. R. Soc. London, A380, (1982), 487–488.

    Article  MATH  Google Scholar 

  10. S. Kaneyuki, On orbit structure of compactifications of parahermitian symmetricspaces, Japan J. Math., 13, (1987) 333–370.

    MathSciNet  MATH  Google Scholar 

  11. S. Kaneyuki, On Siegel domains of finite type, J. Math. Soc. Japan, 39, (1987), 597–607.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Kaneyuki, The Sylvester’s law of inertia for Jordan algebras, Proc. Japan Acad. Ser. A, 64, (1988), 311–313.

    Article  MathSciNet  MATH  Google Scholar 

  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. S. Kaneyuki, Pseudo-Hermitian symmetric spaces and Siegel domains overnon-degenerate cones, Hokkaido Math. J., 20, (1991) 213–239.

    MathSciNet  MATH  Google Scholar 

  15. S. Kaneyuki, On the subalgebras go and gevof semisimple graded Lie algebras, J. Math. Soc. Japan, 45, (1993) 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  16. S.Kaneyuki, The Sylvester’s law of inertia in simple graded Lie algebras, J. Math. Soc. Japan, 50, (1998), 593–614.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Kaneyuki, On the automorphism groups of parahermitian symmetric spaces, Preprint.

    Google Scholar 

  18. S. Kaneyuki and H. Asano, Graded Lie algebras and generalized Jordan triple systems, Nagoya Math. J., 112, (1988), 81–115.

    MathSciNet  MATH  Google Scholar 

  19. S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math., 8, (1985), 81–98.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Kobayashi and T. Nagano, On filtered Lie algebras and geometric structures I, J. Math. Mech., 13(1964), 875–908.

    MathSciNet  MATH  Google Scholar 

  21. M. Koecher Jordan algebras and their applications, Lect. Notes, Univ. Minnesota, Minneapolis, 1962.

    MATH  Google Scholar 

  22. M. Koecher An elementary approach to bounded symmetric domains, Lect. Notes, Rice Univ. 1969.

    MATH  Google Scholar 

  23. S. Koh, On affine symmetric spaces, Trans. Amer. Math. Soc., 119, (1965), 291–309.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Koranyi and J. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math., 81, (1965), 265–288.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. P. Langlands, The dimension of the space of automorphic forms, Amer. J. Math., 85, (1963), 99–125.

    Article  MathSciNet  MATH  Google Scholar 

  26. O. Loos, Jordan triple systems, R-spaces and bounded symmetric domains, Bull. Amer. Math. Soc., 77, (1971), 558–561.

    Article  MathSciNet  MATH  Google Scholar 

  27. O. Loos Bounded symmetric domains and Jordan pairs, Math. Lect. Univ. Calif., Irvine, 1977.

    Google Scholar 

  28. H. Matsumoto, Quelques remarques sur les groupes de Lie algébriques réels J. Math. Soc. Japan, 16, (1964), 419–446.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Nagano, Transformation groups on compact symmetric spaces, Trans. Amer. Math. Soc., 118, (1965), 428–453.

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math., 57, (1980), 1–81.

    Article  MathSciNet  MATH  Google Scholar 

  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. G. Roos, Jordan triple systems, This volume, 425–534.

    Google Scholar 

  33. I. Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math., 71, (1960), 77–110.

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Satake Algebraic structures of symmetric domains IwanamiShoten, Tokyo and Princeton Univ. Press, Princeton, NJ, 1980.

    MATH  Google Scholar 

  35. M. Takeuchi, Cell decompositions and Morse equalities on certain symmetric spaces, J. Fac. Sci. Univ. Tokyo, 12, (1965), 81192.

    Google Scholar 

  36. M. Takeuchi, On orbits in a compact hermitian symmetric space, Amer. J. Math., 90, (1968), 657–680.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Takeuchi, On conjugate loci and cut loci of compact symmetric spaces II, Tsukuba J. Math., 3, (1979), 1–29.

    MATH  Google Scholar 

  38. M. Takeuchi, Basic transiformations of symmetric R-spaces, Osaka. J. Math., 25, (1988), 259–297.

    MathSciNet  MATH  Google Scholar 

  39. N. Tanaka, On equivalence problems associated with a certain class of homogeneous spaces, J. Math. Soc. Japan, 17, (1965), 103–139.

    Article  MathSciNet  MATH  Google Scholar 

  40. N. Tanaka, On affine symmetric spaces and the automorphism groups of product manifolds, Hokkaido Math. J., 14, (1985), 277–351.

    MathSciNet  MATH  Google Scholar 

  41. T. Tsuji, Siegel domains over self-dual cones and their automorphisms, Nagoya Math. J., 55, (1974), 33–80.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  43. E. C. Zeeman, Causality implies the Lorentz group, J. Math. Physics, 5, (1964), 490–493.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sophia University, Chiyoda-ku, Tokyo, 102-8554, Japan

    Soji Kaneyuki

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  1. Soji Kaneyuki
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Kaneyuki, S. (2000). Pseudo-Hermitian Symmetric Spaces. In: Analysis and Geometry on Complex Homogeneous Domains. Progress in Mathematics, vol 185. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1366-6_11

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  • DOI: https://doi.org/10.1007/978-1-4612-1366-6_11

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7115-4

  • Online ISBN: 978-1-4612-1366-6

  • eBook Packages: Springer Book Archive

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