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Arkiv för Matematik

, Volume 20, Issue 1–2, pp 69–85 | Cite as

Spherical functions and invariant differential operators on complex Grassmann manifolds

  • Bob Hoogenboom
Article

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.

Key Words & Phrases

Complex Grassmann manifold spherical function radial part of an invariant differential operator hypergeometric function Jacobi function 

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References

  1. 1.
    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.)MATHMathSciNetGoogle Scholar
  2. 2.
    Erdélyi, A., W. Magnus, F. Oberhettinger &F. Tricombi,Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, 1953.Google Scholar
  3. 3.
    Harish-Chandra, Spherical Functions on a Semisimple Lie Group, I.Am. J. Math. 80 (1958), 241–310.CrossRefMathSciNetGoogle Scholar
  4. 4.
    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.Google Scholar
  5. 5.
    Helgason, S., Functions on Symmetric Spaces, inProceedings of Symposia in Pure Mathematics, vol. XXVI; Harmonic Analysis on Homogeneous Spaces, Am. Math. Soc. Providence, R.I. 1973.Google Scholar
  6. 6.
    Hua, L. K.,Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Am. Math. Soc., Providence, R.I. 1963.Google Scholar
  7. 7.
    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.MathSciNetGoogle Scholar
  8. 8.
    Koornwinder, T. H.,A new proof of a Paley-Wiener Type Theorem for the Jacobi Transform. Ark. Mat.13 (1975), 145–159.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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.Google Scholar

Copyright information

© Institut Mittag Leffler 1982

Authors and Affiliations

  • Bob Hoogenboom
    • 1
  1. 1.Mathematisch CentrumAmsterdamThe Netherlands

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