Intertwining Functors and Irreducibility of Standard Harish—Chandra Sheaves

  • Dragan Miličić
Chapter
Part of the Progress in Mathematics book series (PM, volume 101)

Abstract

Let g be a complex semisimple Lie algebra and σ an involution of g. Denote by t the fixed point set of this involution. Let K be a connected algebraic group and ϕ a morphism of K into the group G = Int(g) of inner automorphisms of g such that its differential is injective and identifies the Lie algebra of K with t. Let X be the flag variety of g, i.e. the variety of all Borel subalgebras in g. Then K acts algebraically on X, and it has finitely many orbits which are locally closed smooth sub varieties. The typical situation is the following: g is the complexification of the Lie algebra of a connected real semisimple Lie group G 0 with finite center, K is the complexification of a maximal compact subgroup of G 0, and σ the corresponding Cartan involution.

Keywords

Teal 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Beilinson, J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris, Ser. I 292 (1981), 15–18.MathSciNetMATHGoogle Scholar
  2. 2.
    A. Beilinson, J. Bernstein, A generalization of Casselman’s submodule theorem, Representation Theory of Reductive Groups, Birkhauser, Boston, 1983, pp. 35–52.Google Scholar
  3. 3.
    A. Beilinson, Localization of representations of reductive Lie Algebras, Proceedings of International Congress of Mathematicians, August 16–24, 1983, Warszawa, pp. 699–710.Google Scholar
  4. 4.
    A. Borel et al., Algebraic D-modules, Academic Press, Boston, 1987.MATHGoogle Scholar
  5. 5.
    H. Hecht, D. Miličić, W. Schmid, J. A. Wolf, Localization and standard modules for real semisimple Lie groups I: The duality theorem, Inventiones Math. 90 (1987), 297-332.MATHCrossRefGoogle Scholar
  6. 6.
    H. Hecht, D. Milicic, W. Schmid, J. A. Wolf, Localization and standard modules for real semisimple Lie groups II: Irreducibility, vanishing theorems and classification, in preparation.Google Scholar
  7. 8.
    T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 332–357.MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Milčić, W. Soergel, Twisted Harish-Chandra sheaves and Whittaker modules, in preparation.Google Scholar
  9. 9.
    D. Miličić, Localization and representation theory of reductive Lie groups, (manuscript), to appear.Google Scholar
  10. 10.
    I. Mirković, Classification of irreducible tempered representations of semisimple Lie groups, Ph. D. Thesis, University of Utah, 1986.Google Scholar
  11. W. Schmid, Construction and classification of irreducible Harish-Chandra modules, in this volume.Google Scholar
  12. 12.
    B. Speh, D. Vogan, Reducibility of generalized principal series representations, Acta Math. 145 (1980), 227–299.MathSciNetMATHCrossRefGoogle Scholar
  13. D. Vogan, Irreducible characters of semisimple Lie groups III: proof of the Kazh-dan-Lusztig conjectures in the integral case, Inventiones Math. 71 (1983), 381–417.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Dragan Miličić
    • 1
  1. 1.Department of MathrmaticsUniversity of UtahSalt Lake CityUSA

Personalised recommendations