Skip to main content

Highest weight modules for semisimple Lie algebras

Part of the Lecture Notes in Mathematics book series (LNM,volume 831)

Keywords

  • Weight Module
  • Discrete Series
  • Composition Factor
  • Verma Module
  • Bruhat Order

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. I. N. Bernštein, I. M. Gelfand, S. I. Gelfand, The structure of representations generated by vectors of highest weight, Functional Anal. Appl. 5 (1971), 1–9.

    CrossRef  MathSciNet  Google Scholar 

  2. _____, Differential operators on the base affine space and a study of g-modules, pp. 21–64, Lie Groups and their Representations, Halsted, New York, 1975.

    MATH  Google Scholar 

  3. _____, A category of g-modules, Functional Anal. Appl. 10 (1976), 87–92.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. P. Delorme, Extensions dans la categorie 0 de Bernštein-Gelfand-Gelfand, Applications. (preprint).

    Google Scholar 

  5. V. V. Deodhar, Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), 187–198.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. _____, On a construction of representations and a problem of Enright (preprint).

    Google Scholar 

  7. J. Dixmier, Algèbres Enveloppantes, Gauthier-Villars, Paris, 1974; English translation, North-Holland, Amsterdam, 1977.

    Google Scholar 

  8. T. J. Enright, On the algebraic construction and classification of Harish-Chandra modules, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 1063–1065.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. _____, On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae, Ann. of Math. 110 (1979), 1–82.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. _____, The representations of complex semisimple Lie groups (preprint).

    Google Scholar 

  11. T. J. Enright, V. S. Varadarajan, On an infinitesimal characterization of the discrete series, Ann. of Math. 102 (1975), 1–15.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. H. Garland, J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), 37–76.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. H. Hecht, W. Schmid, A proof of Blattner's conjecture, Invent. Math. 31 (1975), 129–154.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J. E. Humphreys, Finite and infinite dimensional modules for semisimple Lie algebras, pp. 1–64, Queen's Papers in Pure & Appl. Math. No. 48, Kingston, Ont., 1978.

    Google Scholar 

  15. J. C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), 53–65.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. _____, Moduln mit einem höchsten Gewicht, Habilitationsschrift, U. Bonn, 1977, to appear in Lect. Notes in Math.750, Springer, 1979.

    Google Scholar 

  17. A. Joseph, Dixmier's problem for Verma and principal series submodules, J. London Math. Soc. 20(1979), 193–204.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. J. Lepowsky, A generalization of the Bernštein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), 496–511.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. A. Rocha-Caridi, Splitting criteria for g-modules induced from a parabolic and the Bernštein-Gelfand-Gelfand resolution of a finite dimensional, irreducible g-module, Trans. Amer. Math. Soc., to appear.

    Google Scholar 

  21. N. N. Šapovalov, On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Funkcional. Analiz. i Priložen, 6, no. 4 (1972), 65–70 = Functional Anal. Appl. 6 (1972), 307–312.

    MathSciNet  Google Scholar 

  22. V. S. Varadarajan, Infinitesimal theory of representations of semisimple Lie groups, lectures given at NATO Advanced Study Institute, Liège, 1977.

    Google Scholar 

  23. D. A. Vogan, Irreducible characters of semisimple Lie groups II Duke Math. J. 46(1979), 805–859.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. N. R. Wallach, On the Enright-Varadarajan modules: A construction of the discrete series, Ann. Sci. École Norm Sup. (4) 9 (1976), 81–102.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1980 Springer-Verlag

About this chapter

Cite this chapter

Humphreys, J.E. (1980). Highest weight modules for semisimple Lie algebras. In: Dlab, V., Gabriel, P. (eds) Representation Theory I. Lecture Notes in Mathematics, vol 831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089779

Download citation

  • DOI: https://doi.org/10.1007/BFb0089779

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10263-2

  • Online ISBN: 978-3-540-38385-7

  • eBook Packages: Springer Book Archive