Invariant Differential Operators and Weyl Group Invariants

Part of the Progress in Mathematics book series (PM, volume 101)


In this research announcement we describe the relationship between the algebra Z(G) of bi-invariant differential operators on a simple noncompact Lie group G and the algebra D(G/K )of invariant differential operators on the symmetric space G/K associated with G (cf. §4). The natural map µ: Z(G) → D(G/K) turns out to be surjective except in exactly four cases. These cases involve the exceptional groups and for them the relationship between Z(G) and D(G/K) is quite complicated. However for all cases of G/K, each DD(G/K) is the “ratio” of two µ(Z 1) and µ (Z 2) (with Z 1,Z 2Z(G)). See Theorem 4.1.


Symmetric Space Spherical Function Quotient Field Noncompact Type Invariant Differential Operator 
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. [Be]
    F. A. Berezin, Laplace operators on semisimple Lie groups, Amer. Math. Soc. Transl. 21 (1962), 239–339.MathSciNetGoogle Scholar
  2. [Bi]
    F.V, Bien, D-modules and spherical representations, Math. Notes, Princeton Univ. Press, Princeton, N. J., 1990.Google Scholar
  3. [DL]
    J. Delsarte and J.L. Lions, Moyennes Généraliées, Comm. Math. Helv. 33 (1959), 59–69.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [E]
    L. Ehrenpreis, The use of partial differential equations for the study of group representations, Proc. Symp. Pure Math. Vol. XXVI, Amer. Math. Soc. 1973, 317–320.Google Scholar
  5. [F-J]
    M. Flensted-Jensen, Spherical functions on a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106–146.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [HC1]
    Harish-Chandra, The characters of semisimple Lie groups. Trans. Amer. Math. Soc. 83 (1956) 98–163.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [HC2]
    Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math. 801 (1958), 241–310.MathSciNetCrossRefGoogle Scholar
  8. [Hi]
    S. Helgason, Duality and Radon transform for symmetric spaces, Amer. J. Math. 85 (1963), 667–692.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [H2]
    S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565–601.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [H3]
    S. Helgason, Radon-Fourier transforms on symmetric spaces and related group representations, Bull. Amer. Math. Soc. 71 (1965), 757–763.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [H4]
    S. Helgason, Some results on Radon transforms, Huygens principle and X-ray transforms, Contemp. Math. 63 (1987), 151–177.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [H5]
    S. Helgason, Some results on invariant differential operators on symmetric spaces, Amer. J. Math, (to appear).Google Scholar
  13. [H6]
    S. Helgason, Geometric Analysis on Symmetric Spaces, monograph in preparation.Google Scholar
  14. [L]
    C.Y. Lee, Invariant polynomials of Weyl groups and applications to the centers of universal enveloping algebras, Can. J. Math. XXVI (1974), 583–592.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations