Inventiones mathematicae

, Volume 121, Issue 1, pp 531–578

Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra

  • W. Rossmann
Article

Abstract

The paper develops a Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra and analyzes the resulting monodromy representation of the Weyl group.

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References

  1. V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko: Singularities of Differentiable Maps, Volume II. Birkhäuser, Boston-Basel-Berlin, 1988Google Scholar
  2. D. Barbasch, D. Vogan: Primitive ideals and orbital integrals in complex classical groups. Math. Ann.259, 153–199 (1982)Google Scholar
  3. A. Borel, J.C. Moore: Homology theory for locally compact spaces. Michigan Math. J.7, 137–159 (1960)Google Scholar
  4. E. Brieskorn: Über die Auflösung gewisser Singularitäten von holomorphen Addildungen. Math. Ann.166, 76–102 (1966)Google Scholar
  5. E. Brieskorn: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manus. Math.2, 103–161 (1970)Google Scholar
  6. D. Collingwood, W. McGovern: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold, New York, 1993Google Scholar
  7. C. De Concini, G. Lusztig, C. Procesi: Homology of the zero-set of a nilpotent vector field on a flag manifold. J. Amer. Math. Soc.1, 15–34 (1988)Google Scholar
  8. P. Deligne, N. Katz: Sèminaire de Géometrie Algébrique du Bois-Marie 1967–1969. SGA7II. Lecture Notes in Math.340, Springer, 1970Google Scholar
  9. R. Hotta: On Joseph's construction of Weyl group representations. Tôhoku Math. J.36, 49–74 (1984)Google Scholar
  10. R. Hotta, T.A. Springer: A specialization theorem form certain Weyl group representations and an application to Green polynomials. Invent math.41, 113–127 (1977)Google Scholar
  11. M. Kashiwara, P. Shapira. Sheaves on manifolds. Springer, New York, 1990Google Scholar
  12. B. Kostant, S. Rallis: Orbits and representations associated with symmetric spaces. Am. J. Math.93, 753–809 (1971)Google Scholar
  13. T.Y. Lam: Young diagrams, Schur functions, the Gale-Ryser theorem, and a conjecture of Snapper. J. Pure Appl. Algebra10, 81–94 (1977)Google Scholar
  14. S. Lefschetz: L'Analyse Situs et la Géometrie Algébrique. Gauthier-Villars. Paris, 1924. Reprinted in Selected Papers, Chelsea, New York, 1971Google Scholar
  15. S. Lefschetz: Topology, Amer. Math. Soc. Colloquium Publications, New York, 1930. Reprinted by Chelsea, New York, 1965Google Scholar
  16. T. Matsuki: The orbits on affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan31, 331–357 (1979)Google Scholar
  17. J. Milnor: Singular Points of Complex Hypersurfaces. Annals of Math. Studies 61, Princeton U. Press, Princeton, 1968Google Scholar
  18. I. Mirković, T. Uzawa, K. Vilonen: Matsuki correspondence for sheaves. Invent. math.109, 231–245 (1992)Google Scholar
  19. F. Pham: Singularités des systèmes différentiels de Gauss-Manin. Progress in Mathematics, Birkhäuser, Boston-Basel-Stuttgart, 1979Google Scholar
  20. É. Picard, G. Simart: Théorie des Fonctions Algébriques de Deux Variables Indépendantes. Gauthier-Villars, Paris 1897Google Scholar
  21. R.W. Richardson, T.A. Springer: The Bruhat order on symmetric varieties. Preprint, University of Utrecht, 1989Google Scholar
  22. R.W. Richardson, T.A. Springer: Combinatorics and geometry ofK-orbits on the flag manifold. Preprint, Australian National University, 1992Google Scholar
  23. W. Rossmann: The structure of semisimple symmetric spaces. Can. J. Math.31, 157–180 (1979)Google Scholar
  24. W. Rossmann: Characters as contour integrals. In Lie Group Representations III, R. Herb et al., editors, Lecture Notes in Math.1077, Springer, 1984 375–388Google Scholar
  25. W. Rossmann: Nilpotent orbital integrals in a real semisimple Lie algebra and representations of Weyl groups. In Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Actes du colloque en l'honneur de Jacques Dixmier. A. Connes, et al., editors. Progess in Mathematics vol. 92, Birkhäuser 263–287 1990Google Scholar
  26. W. Rossmann: Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety I: an integral formula; II: representations of Weyl groups. J. Funct.96, 130–154, 155–192 (1991)Google Scholar
  27. J. Sekiguchi: Remarks on nilpotent orbits of a symmetric pair. J. Math. Soc. Japan39, 127–138 (1987)Google Scholar
  28. P. Slodowy: Four lectures on simple groups and singularities. Communications of the Math. Inst., Rijksuniversiteit Utrecht, v. 11, (1) 1980Google Scholar
  29. P. Slodowy: Simple Singularities and Simple Algebraic Groups. Lecture Notes in Mathematics815 (2), Springer, 1980Google Scholar
  30. N. Spaltenstein: Classes Unipotentes et Sous-groupes de Borel. Lecture Notes in Mathematics815, Springer, 1982Google Scholar
  31. T.A. Springer: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. math.36, 173–207 (1976)Google Scholar
  32. T.A. Springer: A construction of representations of Weyl groups. Invent. math.44, 279–293 (1978)Google Scholar
  33. T.A. Springer: A generalization of the orthogonality relations of Green functions. Preprint (1993)Google Scholar
  34. R. Steinberg: On the desingularization of the unipotent variety. Invent. math.36, 209–312 (1976)Google Scholar
  35. R. Thom: Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc.75, 240–284 (1969)Google Scholar
  36. N. Wallach: Real Reductive Groups I. Academic Press, Inc., New York, 1988Google Scholar
  37. J. Wolf: The action of a real semisimple Lie group on a complex flag manifold 7. Bulletin A.M.S.75, 1121–1237 (1969)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • W. Rossmann
    • 1
  1. 1.Department of MathematicsUniversity of OttawaOttawaCanada

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