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manuscripta mathematica

, Volume 113, Issue 1, pp 107–124 | Cite as

M-spherical K-modules of a rank one semisimple Lie group

  • Leandro CaglieroEmail author
  • Juan Tirao
Article

Abstract.

Let G =K A N be an Iwasawa decomposition of a connected, noncompact real semisimple Lie group with finite center and let M be the centralizer of A in K . B. Kostant proved that for every irreducible M-spherical K-module V there exists a unique d (the Kostant degree of V) such that V can be realized as a submodule of the space of all -harmonic homogeneous polynomials of degree d on . Here is a Cartan decomposition of the complexification of the Lie algebra of G .In this paper we give an algorithm to obtain a highest weight vector from any M-invariant vector in an irreducible M-spherical K-module. This algorithm allows us to compute a sharp bound for the Kostant degree d(v) of any M-invariant vector v in a locally finite M-spherical K-module V. The method computes d(v) effectively for any V if G is locally isomorphic to SO(n,1) and for if G is locally isomorphic to SU(n,1).

Keywords

High Weight Weight Vector Homogeneous Polynomial High Weight Vector Iwasawa Decomposition 
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.

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References

  1. 1.
    Brega, O., Cagliero, L., Tirao, J.: The classifying ring of a classical rank one semisimple Lie group. Submitted for publicationGoogle Scholar
  2. 2.
    Brega, O., Tirao, J.: A transversality property of a derivation of the universal enveloping algebra , for SO(n,1) and SU(n,1). Manuscr. math. 74, 195–215 (1992)Google Scholar
  3. 3.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York 1978Google Scholar
  4. 4.
    Humphreys, J.: Introduction to Lie Algebras and Representation Theory. Springer-Verlag, New York 1972Google Scholar
  5. 5.
    Johnson, K.: Composition series and intertwining operators for the spherical principal series II. Trans. Am. Math. Soc. 215, 269–283 (1976)zbMATHGoogle Scholar
  6. 6.
    Johnson, K., Wallach, N.: Composition series and intertwining operators for the spherical principal series I. Trans. Am. Math. Soc. 229, 137–173 (1977)zbMATHGoogle Scholar
  7. 7.
    Kostant, B.: On the existence and irreducibility of certain series of representations. Lie groups and their representations, 1971 Summer School in Math., Wiley, New York, 1975, pp. 231–330Google Scholar
  8. 8.
    Kostant, B., Rallis, S.: On representations associated with symmetric spaces. Am. J. Math. 93, 753–809 (1971)zbMATHGoogle Scholar
  9. 9.
    Lepowsky, J.: Algebraic results on representations of semisimple Lie groups. Trans. Am. Math. Soc. 176, 1–44 (1973)zbMATHGoogle Scholar
  10. 10.
    Tirao, J.: A restriction theorem for semisimple Lie groups of rank one. Trans. Am. Math. Soc. 279, 651–660 (1983)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Warner, G.: Harmonic Analysis on Semisimple Lie Groups I. Springer, Berlin, 1972Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.CIEM - FaMAFUniversidad Nacional de CórdobaCórdobaArgentina

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