Advertisement

Finite Rank Homogeneous Holomorphic Bundles in Flag Spaces

  • Tim Bratten
Chapter
Part of the Progress in Mathematics book series (PM, volume 158)

Abstract

For more than forty years the study of homogeneous holomorphic vector bundles has resulted in an important source of irreducible unitary representations for a real reductive Lie group. In the mid 1950s, Harish-Chandra realized a family of irreducible unitary representations for some semisimple groups, using the global sections of homogeneous bundles defined over Hermitian symmetric spaces [6]. At about the same time Borel and Weil constructed the irreducible representations for a connected compact Lie group as global sections of line bundles defined over complex projective homogeneous spaces [3]. More than ten years later, W. Schmid in his thesis solved a conjecture by Langlands and generalized the Borel-Weil-Bott theorem to realize discrete series representations for noncompact semisimple groups [16]. This extension is nontrivial for one thing because it requires an understanding of the representations obtained on some infinite-dimensional sheaf cohomology groups.

Keywords

Parabolic Subgroup Maximal Compact Subgroup Levi Factor Discrete Series Representation Real Reductive Group 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Beilinson and J. Bernstein, Localization de g modules, C.R. Acad. Sci. Paris, 292 (1981), pp. 15–18.MathSciNetMATHGoogle Scholar
  2. [2]
    A. Borel, ET AL., Algebraic D-Modules, no. 2 in Perspectives in Mathematics, Academic Press, Inc., 1987.Google Scholar
  3. [3]
    R. Bott, Homogeneous vector bundles, Ann. of Math., 66 (1957), pp. 203–248.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    T. bratten, Realizing representations on generalized flag manifolds, Preprint, (1995). to appear in Compositio Math.Google Scholar
  5. [5]
    J. chang, Special K-types, tempered characters and the Beilinson-Bemstein realization, Duke Math. J., 56 (1988), pp. 345–383.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Harish-Chandra, Representations of semisimple Lie groups VI, Amer. J Math., 78 (1956), pp. 564–628.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    ———, Harmonic analysis on real reductive groups I, J. Func. Anal., 19 (1975), pp. 104–204.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    H. hecht and J. taylor, Analytic localization of group representations, Advances in Math., 79 (1990), pp. 139–212.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    ———, A comparison theorem for n homology, Composito Math., 86 (1993), pp. 189–207.MathSciNetMATHGoogle Scholar
  10. [10]
    H. Hecht, D. Mil, W. Schmid and J. A. Wolf, Localization and standard modules for semisimle Lie groups I: the duality theorem, Invent. Math., 90 (1987), pp. 297–332.MATHGoogle Scholar
  11. [11]
    M. kashiwara and W. schmid, Quasi-equivariant V-modules, equivariant derived category and representations of reductive Lie groups, Research anouncement, Research Institute for Mathematical Sciences, Kyoto University (1994).Google Scholar
  12. [12]
    A. Knapp, Representation Theory of Semisimple Groups, Princeton University Press, 1986.MATHGoogle Scholar
  13. [13]
    T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan, 31 (1979), pp. 331–357.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    ———, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J., 12 (1982), pp. 307–320.MathSciNetMATHGoogle Scholar
  15. [15]
    W. schmid, Boundary value problems for group invariant differential equations, Proc. Cartan Symposium, Astérique, (1985).Google Scholar
  16. [16]
    ———, Homogeneous complex manifolds and representations of semisimple Lie groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, no. 31 in Math. Surveys and Monographs, Amer. Math. Soc., 1989.Google Scholar
  17. [17]
    W. schmid and J. A. wolf, Geometric quantization and derived functor modules for semisimple Lie groups, J. Func. Anal., 90 (1990), pp. 48–112.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    J. Serre, Un théoréme de dualité, Comment. Math. Helv., 29 (1955), pp. 9–26.MATHCrossRefGoogle Scholar
  19. [19]
    D. Vogan, Unitarizability of certain series of representations, Ann. of Math., 120 (1984), pp. 141–187.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    ———, Unitary Representations of Reductive Lie Groups, no. 118 in Annals of Math. Studies, Princeton Univ. Press, 1987.Google Scholar
  21. [21]
    J. A. Wolf, The action of a real semi-simple group on a complex flag manifold, I: orbit structure and holomorphic arc components, Bull. Amer. Math. Soc., 75 (1969), pp. 1121–1237.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    H. Wong, Dolbeault cohomologies associated with finite rank representations, Ph.D thesis, Harvard University, (1991).Google Scholar

Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Tim Bratten
    • 1
  1. 1.Facultad de Ciencias ExactasUniversidad del Centro de la Provincia de Buenos Aires Campus UniversitarioTandilArgentina

Personalised recommendations