Spherical Functions for Small K-Types

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 290)


For a connected semisimple real Lie group G of non-compact type, Wallach introduced a class of K-types called small. We classify all small K-types for all simple Lie groups and prove except just one case that each elementary spherical function for each small K-type \((\pi ,V)\) can be expressed as a product of hyperbolic cosines and a Heckman–Opdam hypergeometric function. As an application, the inversion formula for the spherical transform on \(G\times _K V\) is obtained from Opdam’s theory on hypergeometric Fourier transforms.


Small K-types Spherical functions Hypergeometric functions 

1991 Mathematics Subject Classification

22E45 33C67 43A90 



The authors wish to thank Professor Hiroyuki Ochiai for helpful comments on an earlier version of this paper.


  1. 1.
    Berger, M.: Les espaces symétriques noncompacts. Ann. Sci École Norm. Sup. 74, 85–177 (1957)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Camporesi, R.: The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces. J. Lie Theory 7, 29–60 (1997)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Camporesi, R., Pedon, E.: Harmonic analysis for spinors on real hyperbolic spaces. Colloq. Math. 87, 245–287 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cherednik, I.: A unification of Knizhnik-Zamolodchikov equations and Dunkl operators via affine Hecke algebras. Inven. Math. 106, 411–431 (1991)CrossRefGoogle Scholar
  5. 5.
    Cherednik, I.: Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations. Adv. Math. 106, 65–95 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dadok, J.: On the \(C^\infty \) Chevalley’s theorem. Adv. Math. 44, 121–131 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deitmar, A.: Invariant operators on higher \(K\)-types. J. Reine Angew. Math. 412, 97–107 (1990)MathSciNetzbMATHGoogle Scholar
  8. 8.
    van Dijk, G., Pasquale, A.: Harmonic analysis on vector bundles over \({\rm {Sp}}(1, n)/{\rm {Sp}}(1)\times {\rm {Sp}}(n)\). Enseign. Math. 45, 219–252 (1999)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dunkl, C.F.: Integral kernels with reflection group invariance. Can. J. Math. 43, 1213–1227 (1991)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dunkl, C.F., de Jeu, M.F.E., Opdam, E.M.: Singular polynomials for finite reflection groups. Trans. Am. Math. Soc. 346, 237–256 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Flensted-Jensen, M.: Spherical functions on a simply connected Lie group. II. The Paley-Wiener theorem for the rank one case. Math. Ann. 228, 65–92 (1977)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gangolli, R., Varadarajan, V.S.: Harmonic Analysis of Spherical Functions on Real Reductive Groups. Springer, Berlin (1988)CrossRefGoogle Scholar
  14. 14.
    Godement, R.: A theory of spherical functions. I. Trans. Am. Math. Soc. 73, 496–556 (1952)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants, Graduate Texts in Mathematics 255. Springer, Berlin (2009)CrossRefGoogle Scholar
  16. 16.
    Harish-Chandra: Spherical functions on a semisimple Lie group. I. Amer. J. Math. 80, 241–310 (1958)Google Scholar
  17. 17.
    Harish-Chandra: Differential equations and semisimple Lie groups, manuscript, 1960. In: Collected Papers, Vol. III, pp. 57–120. Springer, New York (1984)CrossRefGoogle Scholar
  18. 18.
    Heckman, G.J.: An elementary approach to the hypergeometric shift operators of Opdam. Inven. Math. 103, 341–350 (1991)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Heckman, G.J.: Hypergeometric and spherical functions. In: Harmonic Analysis and Special Functions on Symmetric Spaces. Perspectives in Mathematics. Academic, Boston (1994)Google Scholar
  20. 20.
    Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Comp. Math. 64, 329–352 (1987)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Helgason, S.: Differential geometry, lie groups, and symmetric spaces. Am. Math. Soc. c1978 ( 2001)Google Scholar
  22. 22.
    Helgason, S.: Groups and geometric analysis. Am. Math. Soc. c1984 (2000)Google Scholar
  23. 23.
    Knapp, A.W.: Representation Theory of Semisimple Groups. Princeton University Press, Princeton (1986)Google Scholar
  24. 24.
    Knapp, A.W.: Lie Groups Beyond an Introduction, 2nd. Progress in Mathematics, vol. 140. Birkhäuser Boston, Inc., Boston (2002)CrossRefGoogle Scholar
  25. 25.
    Koornwinder, T.: Jacobi functions and analysis on noncompact semisimple Lie groups. In: Special Functions: Group Theoretical Aspects and Applications, pp. 1–85. Mathematics and Its Applications. Reidel, Dordrecht (1984)Google Scholar
  26. 26.
    Kostant, B.: On the existence and irreducibility of certain series of representations. In: Gelfand, I.M. (ed.) Lie Groups and Their Representations, pp. 231–329. Summer School Conference, Budapest, 1971. Halsted press, New York (1975)Google Scholar
  27. 27.
    Lee, S.W.: Determinants of intertwining operators between genuine principal series representations of nonlinear real split groups. Ph.D. dissertation, University of California, San Diego (2012)Google Scholar
  28. 28.
    Lee, S.W.: Representations with small \(K\) types. arXiv:1209.5653 [v3] (2013)
  29. 29.
    Lusztig, G.: Cuspidal local systems and graded Hecke algebras, I. Publ. Math. de IHES 67, 145–202 (1988)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Opdam, E.M.: Root systems and hypergeometric functions IV. Comp. Math. 67, 191–209 (1988)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Opdam, E.M.: An analogue of the Gauss summation formula for hypergeometric functions related to root systems. Math. Z. 212, 311–336 (1993)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Opdam, E.M.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175(1), 75–121 (1995)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Opdam, E.M.: Lecture notes on Dunkl operators for real and complex reflection groups. MSJ Memoir, vol. 8. Mathematical Society of Japan, Tokyo (2000)Google Scholar
  34. 34.
    Oshima, T.: A class of completely integrable quantum systems associated with classical root systems. Indag. Math. (N.S.) 16, 655–677 (2005)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Rosenberg, J.: A quick proof of Harish-Chandra’s Plancherel theorem for spherical functions on a semisimple Lie group. Proc. Am. Math. Soc. 63(1), 143–149 (1977)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Shimeno, N.: The Plancherel formula for spherical functions with a one-dimensional \(K\)-type on simply connected simple Lie group of Hermitian type. J. Funct. Anal. 121, 330–388 (1994)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Shimeno, N.: Harmonic analysis on homogeneous vector bundles on hyperbolic spaces, Springer, 1988. Tokyo J. Math. 18, 383–400 (1995)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Takahashi, R.: Fonctions sphériques dans les group \({\rm {Sp}}(n,1)\), In: Faraut, J. (ed.) Théorie dupoentielet analyse harmonique. Lecture Notes in Math. 404, 218–228 (1974)Google Scholar
  39. 39.
    Tirao, J.A.: Spherical functions. Rev. Un. Mat. Argent. 28, 75–98 (1977)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Wallach, N.: Real Reductive Groups II. Pure and Applied Mathematics. Academic, Cambridge (1992)Google Scholar
  41. 41.
    Warner, G.: Harmonic Analysis on Semi-Simple Lie Groups II. Springer, Berlin (1972)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of EngineeringTakushoku UniversityHachiojiJapan
  2. 2.School of Science and TechnologyKwansei Gakuin UniversitySandaJapan

Personalised recommendations