Inventiones mathematicae

, Volume 98, Issue 2, pp 281–292 | Cite as

Convexity and unitary representations of nilpotent Lie groups

  • N. J. Wildberger


The moment map of symplectic geometry is extended to associate to any unitary representation of a nilpotent Lie group aG-invariant subset of the dual of the Lie algebra. We prove that this subset is the closed conex hull of the Kirillov orbit of the representation.


Hull Unitary Representation Symplectic Geometry Conex 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnal, D., Cortet, J.C. [1]: Nilpotent Fourier transform and applications. Lett. Math. Phys.9, 25–34 (1985)Google Scholar
  2. 2.
    Atiyah, M.F. [1]: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc.14, 1–15 (1982)Google Scholar
  3. 3.
    Atiyah, M.F. [2]: Angular Momentum, Convex polyhedra and Algebraic Geometry. Proc. Edinburgh Math. Soc.26, 121–138 (1983)Google Scholar
  4. 4.
    Grünbaum, B. [1]: Convex Polytopes. New York: John Wiley and Sons 1967Google Scholar
  5. 5.
    Guillemin, V., Sternberg, S. [1]: Convexity Properties of the moment mapping. Invent. Math.67, 491–513 (1982)Google Scholar
  6. 6.
    Guillemin, V., Sternberg, S. [2]: Convexity Properties of the moment mapping, II. Invent. Math.77, 533–546 (1984)Google Scholar
  7. 7.
    Howe, R. [1]: Quantum mechanics and partial differential equations. J. Funct. Anal.38, 188–254 (1980)Google Scholar
  8. 8.
    Kirillov, A.A. [1]: Representations of Nilpotent Lie Groups. Russian Math. Surveys17, 53–103 (1962)Google Scholar
  9. 9.
    Kirwan, F. [1]: Convexity Properties of the moment mapping, III. Invent. Math.77, 547–552 (1984)Google Scholar
  10. 10.
    Kostant, B. [1]: Quantization and Unitary Representations. Lect. Notes Math. vol. 170, Berlin Heidelberg New York: Springer, 1970, pp. 87–208Google Scholar
  11. 11.
    Souriau, J.M. [1]: Structure des systèmes dynamiques. Paris: Dunod, 1970Google Scholar
  12. 12.
    Wildberger, N.J. [1]: Quantization and Harmonic Analysis on Nilpotent Lie groups. Thesis, Yale University 1983Google Scholar
  13. 13.
    Wildberger, N.J. [2]: The moment map of a Lie group representation (preprint)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • N. J. Wildberger
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations