Arkiv för Matematik

  • Ark. Mat.
  • Volume 20, Number 1-2 (1982), 69-85.

Spherical functions and invariant differential operators on complex Grassmann manifolds

Bob Hoogenboom

Full-text: Open access

Abstract

Proofs are given of two theorems of Berezin and Karpelevič, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.

Article information

Source
Ark. Mat. Volume 20, Number 1-2 (1982), 69-85.

Dates
Received: 27 October 1980
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.afm/1485896970

Digital Object Identifier
doi:10.1007/BF02390499

Zentralblatt MATH identifier
0496.33010

Rights
1982 © Institut Mittag Leffler

Citation

Hoogenboom, Bob. Spherical functions and invariant differential operators on complex Grassmann manifolds. Ark. Mat. 20 (1982), no. 1-2, 69--85. doi:10.1007/BF02390499. http://projecteuclid.org/euclid.afm/1485896970.


Export citation

References

  • Berezin, F. A. & F. I. Karpelevič, Zonal spherical functions and Laplace operators on some symmetric spaces, Dokl. Akad. Nauk. SSSR (N. S.) 118 (1958), 9–12. (In Russian.)
  • Erdélyi, A., W. Magnus, F. Oberhettinger & F. Tricombi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, 1953.
  • Harish-Chandra, Spherical Functions on a Semisimple Lie Group, I. Am. J. Math. 80 (1958), 241–310.
  • Helgason, S., Analysis on Lie Groups and Homogeneous Spaces, Conf. Board of the Math. Sci. Regional Conf. Ser. Math. no. 14. Am. Math. Soc., Providence, R.I. 1972.
  • Helgason, S., Functions on Symmetric Spaces, in Proceedings of Symposia in Pure Mathematics, vol. XXVI; Harmonic Analysis on Homogeneous Spaces, Am. Math. Soc. Providence, R.I. 1973.
  • Hua, L. K., Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Am. Math. Soc., Providence, R.I. 1963.
  • Koornwinder, T. H., Orthogonal Polynomials in Two Variables which are eigenfunctions of two Algebraically Independent Partial Differential Operators, I, II. Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), pp. 48–58, 59–66.
  • Koornwinder, T. H., A new proof of a Paley-Wiener Type Theorem for the Jacobi Transform. Ark. Mat. 13 (1975), 145–159.
  • Takahashi, R., Fonctions Sphériques zonales sur U(n,n+k; F), in “Séminaire d'Analyse Harmonique” (1976–77) Faculté des Sciences de Tunis, Departement de Mathematique, 1977.