Abstract
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 D ∈ D(G/K) is the “ratio” of two µ(Z 1) and µ (Z 2) (with Z 1,Z 2 ∈ Z(G)). See Theorem 4.1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. A. Berezin, Laplace operators on semisimple Lie groups, Amer. Math. Soc. Transl. 21 (1962), 239–339.
F.V, Bien, D-modules and spherical representations, Math. Notes, Princeton Univ. Press, Princeton, N. J., 1990.
J. Delsarte and J.L. Lions, Moyennes Généraliées, Comm. Math. Helv. 33 (1959), 59–69.
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.
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.
Harish-Chandra, The characters of semisimple Lie groups. Trans. Amer. Math. Soc. 83 (1956) 98–163.
Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math. 801 (1958), 241–310.
S. Helgason, Duality and Radon transform for symmetric spaces, Amer. J. Math. 85 (1963), 667–692.
S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565–601.
S. Helgason, Radon-Fourier transforms on symmetric spaces and related group representations, Bull. Amer. Math. Soc. 71 (1965), 757–763.
S. Helgason, Some results on Radon transforms, Huygens ’ principle and X-ray transforms, Contemp. Math. 63 (1987), 151–177.
S. Helgason, Some results on invariant differential operators on symmetric spaces, Amer. J. Math, (to appear).
S. Helgason, Geometric Analysis on Symmetric Spaces, monograph in preparation.
C.Y. Lee, Invariant polynomials of Weyl groups and applications to the centers of universal enveloping algebras, Can. J. Math. XXVI (1974), 583–592.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Helgason, S. (1991). Invariant Differential Operators and Weyl Group Invariants. In: Barker, W.H., Sally, P.J. (eds) Harmonic Analysis on Reductive Groups. Progress in Mathematics, vol 101. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0455-8_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0455-8_9
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6768-3
Online ISBN: 978-1-4612-0455-8
eBook Packages: Springer Book Archive