Characters, Bi-Modules and Representations in Lie Group Harmonic Analysis

  • N. J. Wildberger
Conference paper
Part of the Trends in Mathematics book series (TM)


This paper is a personal look at some issues in the representation theory of Lie groups having to do with the role of commutative hypergroups, bi-modules, and the construction of representations. We begin by considering Frobenius’ original approach to the character theory of a finite group and extending it to the Lie group setting, and then introduce bi-modules as objects intermediate between characters and representations in the theory. A simplified way of understanding the formalism of geometric quantization, at least for compact Lie groups, is presented, which leads to a canonical bi-module of functions on an integral coadjoint orbit. Some meta-mathematical issues relating to the construction of representations are considered.


Line Bundle Irreducible Representation Irreducible Character Coadjoint Orbit Circle Bundle 
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]
    D. Arnal and J. Ludwig, La convexité de l’application moment d’un groupe de Lie, J. Funct. Anal. 105 (1992), 256–300.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Z. Arad and H. Blau, On table algebras and applications to products of characters, J. of Alg. 138 (1991), 186–194.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    F. Bayen, C. Flato, C. Fronsdal, A. Lichnerowicz, and D. Stern-heimer, Deformation Theory and Quantization I. Deformations of symplectic structures, Annals of Physics 111 (1978), 61–110.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, (1995).zbMATHCrossRefGoogle Scholar
  5. [5]
    M. Cahen, S. Gutt and J. Rawnsley, Quantization of Kahler manifolds. I. Geometric interpretation of Berezin’s quantization, , J. Geom. Phys. 7 (1990), no. 1, 45–62.MathSciNetzbMATHGoogle Scholar
  6. [6]
    A. H. Dooley and N. J. Wildberger, Harmonic analysis and the global explonential map for compact Lie groups, (Russian) Funk-tsional. Anal, i Prilozhen. 27 (1993), no. 1, 25–32. tranlsation in Functional Anal. Appl. 27 (1993), no. 1, 21–27MathSciNetzbMATHGoogle Scholar
  7. [7]
    C. F. Dunkl, The measure algebra of a locally compact hypergroup, , Trans. Amer. Math. Soc. 179 (1973), 331–348.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    G. Frobenius, Über Gruppencharaktere, Gesammelte Abhandlungen, Vol. Ill, Springer-Ver lag, Berlin, 1–37.Google Scholar
  9. [9]
    V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, 1984, Cambridge.zbMATHGoogle Scholar
  10. [10]
    N. E. Hurt, Geometric Quantization in Action, Mathematics and its Applications Vol. 8, Reidel, 1983, Dordrecht.zbMATHCrossRefGoogle Scholar
  11. [11]
    R. I. Jewett, Spaces with an Abstract convolution of measures. Adv. Math. 18 (1975), 1–101.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    A. A. Kirillov, Elements of the Theory of Representations, Grundlehren der math. Wissenschaften 220, Springer-Verlag, Berlia, 1976.zbMATHCrossRefGoogle Scholar
  13. [13]
    G. Mackey, Harmonic analysis as exploitation of Symmetry, Bull. Amer. Math. Soc. 3 number 1 (July, 1980), 543–698.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    J. Sniatycki, Geometric Quantization and Quantum Mechanics, 1980, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  15. [15]
    R. Spector, Mesures invariantes sur les hypergroupes, Trans. Amer. Math. Soc. 239 (1978), 147–165.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    V. S. Sunder and N. J. Wildberger, On discrete hypergroups and their actions on sets, preprint (1996).Google Scholar
  17. [17]
    R. Thompson, Author vs Referee: A case history for middle level mathematicians, Amer. Math. Monthly 90 No. 10 (1983), 661–668.MathSciNetCrossRefGoogle Scholar
  18. [18]
    M. Vergne, A Plancherel formula without group representations, , in Operator Algebras and Group Representations Vol. II, Neptune, (1980), 217–226.Google Scholar
  19. [19]
    N. J. Wildberger, Hypergroups and Harmonic Analysis, Centre Math. Anal. (ANU) 29 (1992), 238–253.MathSciNetGoogle Scholar
  20. [20]
    N. J. Wildberger, Finite commutative hypergroups and applications from group theory to conformai field theory, Contemp. Math. 183 (1995), 413–434.MathSciNetCrossRefGoogle Scholar
  21. [21]
    N. J. Wildberger, On the Fourier transform of a compact semisim-ple Lie group, J. Austral. Math. Soc. (Series A) 56 (1994), 64–116.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    N. J. Wildberger, Convexity and representations of nilpotent Lie groups, Invent. Math. 98 (1989), 281–292.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    N. J. Wildberger, The moment map of a Lie group representation, Trans. Amer. Math. Soc. 330 (1992), 257–268.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    N. J. Wildberger, Hypergroups, Symmetrie spaces, and wrapping maps, in Probability Measures on Groups and related structures, Proc. Oberwolfach 1994, World Scientific, Singapore, 1995.Google Scholar
  25. [25]
    N. M. J. Woodhouse, Geometric Quantization, Clarendon Press, 1992, Oxford.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • N. J. Wildberger
    • 1
  1. 1.School of MathematicsUniversity of New South WalesSydneyAustralia

Personalised recommendations