manuscripta mathematica

, Volume 88, Issue 1, pp 1–24 | Cite as

Representations of double coset hypergroups and induced representations

  • Peter Hermann


The principal goal of this paper is to investigate the representation theory of double coset hypergroups. IfK=G//H is a double coset hypergroup, representations ofK can canonically be obtained from those ofG. However, not every representation ofK originates from this construction in general, i.e., extends to a representation ofG. Properties of this construction are discussed, and as the main result it turns out that extending representations ofK is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation ofK always admits an extension toG. Furthermore, we realize the Gelfand pair\(SL(2,\mathfrak{K})//SL(2,R)\) (where\(\mathfrak{K}\) are a local field andR its ring of integers) as a polynomial hypergroup on ℕ0 and characterize the (proper) subset of its dual consisting of extensible representations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Benson, C., Jenkins, J., Ratcliff, G.,On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc.321, 85–116 (1990)CrossRefMathSciNetGoogle Scholar
  2. [2]
    Cartier, P.,Representations of p-adic groups, Automorphic Forms, Representations andL-Functions, Proc. Sympos. Pure Math., vol. 33, part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155.Google Scholar
  3. [3]
    Dunkl, C.F.,The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc.179, 331–348 (1979)CrossRefMathSciNetGoogle Scholar
  4. [4]
    Fell, J.M.G., Doran, R.S.,Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 2, San Diego: Academic Press 1988Google Scholar
  5. [5]
    Gel'fand, I.M., Graev, M.I., Piatetski-Shapiro, I.I.,Automorphic Functions, Philadelphia: Saunders 1965Google Scholar
  6. [6]
    Hauenschild, W., Kaniuth, E., Kumar, A.,Harmonic analysis on central hypergroups and induced representations, Pac. J. Math.110, 83–112 (1984)MathSciNetGoogle Scholar
  7. [7]
    Hermann, P.,Induced representations of hypergroups, Math. Z.211, 687–699 (1992)MathSciNetGoogle Scholar
  8. [8]
    hermann, P.,Induced representations and hypergroup homorphisms, Mh. Math.116, 245–262 (1993)CrossRefMathSciNetGoogle Scholar
  9. [9]
    Hermann, P., Voit, M.,Induced representations and duality results for commutative hypergroups, Forum Math., to appearGoogle Scholar
  10. [10]
    Jewett, R.I.,Spaces with an abstract convolution of measures, Adv. Math.18, 1–101 (1975)CrossRefMathSciNetGoogle Scholar
  11. [11]
    Lasser, R.,Orthogonal polynomials and hypergroups, Rend. Mat. Appl.2, 185–209 (1983)MathSciNetGoogle Scholar
  12. [12]
    Rieffel, M.A.,Induced Representations of C *-Algebras, Adv. Math.13, 176–257 (1974)CrossRefMathSciNetGoogle Scholar
  13. [13]
    Spector, R.,Mesures invariants sur les hypergroupes, Trans. Am. Math. Soc.239, 147–166 (1978)CrossRefMathSciNetGoogle Scholar
  14. [14]
    Voit, M.,Positive characters on commutative hypergroups and some applications, Math. Z.,198, 405–421 (1988)CrossRefMathSciNetGoogle Scholar
  15. [15]
    Voit, M.,On the Fourier transformation of positive, positive definite measures on commutative hypergroups, and dual convolution structures, man. math.72, 141–153 (1991)MathSciNetGoogle Scholar
  16. [16]
    Voit, M., Central limit theorems for random walks on ℕ0 that are associated with orthogonal polynomials, J. Multivariate Anal.34, 290–322 (1990)CrossRefMathSciNetGoogle Scholar
  17. [17]
    Voit, M.,Factorization of probability measures on symmetric hypergroups, J. Austr. Math. Soc. (Series A)50, 417–467 (1991)MathSciNetGoogle Scholar
  18. [18]
    Vrem, R.C.,Harmonic analysis on compact hypergroups, Pac. J. Math.85, 239–251 (1979)MathSciNetGoogle Scholar
  19. [19]
    Weil, A.,Basic number theory, Springer Verlag, Berlin 1973MATHGoogle Scholar
  20. [20]
    Zeuner, Hm.,Properties of the cosh hypergroup, In: Heyer, H. (ed.) Probability Measures on Groups IX. Proc. Conf., Oberwolfach 1988, (Lect. Notes Math., vol. 1379, pp. 425–434) Berlin Heidelberg New York: Springer 1989CrossRefGoogle Scholar
  21. [21]
    Zeuner, Hm.,Limit theorems for one-dimensional hypergroups, Habilitationsschrift, Tübingen 1991Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Peter Hermann
    • 1
  1. 1.Fachbereich Mathematik-InformatikUniversität-Gesamthochschule PaderbornPaderbornFederal Republic of Germany

Personalised recommendations