Skip to main content
SpringerLink
Log in

Multiplicity formulas for finite dimensional and generalized principal series representations

  • Published:
Proceedings of the Indian Academy of Sciences - Mathematical Sciences Aims and scope Submit manuscript

Abstract

The article presents two results. (1) Let a be a reductive Lie algebra over ℂ and let b be a reductive subalgebra of a. The first result gives the formula for multiplicity with which a finite dimensional irreducible representation of b appears in a given finite dimensional irreducible representation of a in a general situation. This generalizes a known theorem due to Kostant in a special case. (2) LetG be a connected real semisimple Lie group andK a maximal compact subgroup ofG. The second result is a formula for multiplicity with which an irreducible representation ofK occurs in a generalized representation ofG arising not necessarily from fundamental Cartan subgroup ofG. This generalizes a result due to Enright and Wallach in a fundamental case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bakre M S,K-module structure of the non-fundamental principal series and related questions, Ph.D. thesis (1980) (University of Washington)

  2. Dixmier J,Enveloping Algebras (1977) (North-Holland)

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

    Article  Google Scholar 

  4. Enright T J and Wallach N R, The fundamental series of representations of a real semisimple Lie algebra,Acta Math. 140 (1978) 1–32

    Article  MATH  MathSciNet  Google Scholar 

  5. Enright T J, On the fundamental series of a real semisimple Lie algebra: Their ineducibility, resolutions and multiplicity formula,Ann. Math. 110 (1979) 1–82

    Article  MathSciNet  Google Scholar 

  6. Enright T J, Blattner-type multiplicity formulas for the fundamental series of real semisimple Lie algebras, unpublished

  7. Enright T J and Wallach N R, The fundamental series of semisimple Lie algebras and semisimple Lie groups.

  8. Harish-Chandra, Discrete series for semisimple Lie groups,Ada. Math. 113 (1965) 241–318

    MATH  MathSciNet  Google Scholar 

  9. Harish-Chandra, Discrete series for a semisimple Lie groups IIActa Math. 116 (1966) 1–111

    MATH  MathSciNet  Google Scholar 

  10. Humphreys J E, Introduction to Lie algebras and representation theory, G.T.M.9 (1972) (New York: Springer-Verlag)

    Google Scholar 

  11. Knapp W,Representation Theory of Semisimple Groups: An Overview Based on Examples (1986) (Princeton)

  12. Lepowsky J, Representations of semisimple Lie groups and an enveloping algebra decomposition, Ph.D. thesis (1970) (Manachusetts Institute of Technology)

  13. Schmid W, On the Characters of Discrete Series,Invent. Math 30 (1975) 47–144

    Article  MATH  MathSciNet  Google Scholar 

  14. Warner G, Harmonic analysis on semisimple Lie groups I, II (1972) (New York: Springer Verlag)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bombay, 400 098, Bombay, India

    M. S. Bakre

Authors
  1. M. S. Bakre
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bakre, M.S. Multiplicity formulas for finite dimensional and generalized principal series representations. Proc. Indian Acad. Sci. (Math. Sci.) 106, 379–401 (1996). https://doi.org/10.1007/BF02837695

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02837695

Keywords

Search

Navigation