Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fonctions sphériques des espaces symétriques compacts


Author: Jean-Louis Clerc
Journal: Trans. Amer. Math. Soc. 306 (1988), 421-431
MSC: Primary 43A90; Secondary 22E46
DOI: https://doi.org/10.1090/S0002-9947-1988-0927699-4
MathSciNet review: 927699
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An integral formula, similar to Harish-Chandra's formula for spherical functions on a noncompact Riemannian symmetric space $ G/K$ is given for the spherical functions of the compact dual $ U/K$. As a consequence, an asymptotic expansion, as the parameter tends to infinity, is obtained, by using the (complex) stationary phase method.

RÉSUMÉ. On démontre une formule intégrale pour les fonctions sphériques d'un espace symétrique de type compact $ U/K$, analogue de la formule d'Harish-Chandra pour le dual non-compact $ G/K$. En conséquence on obtient un équivalent asymptotique lorsque le paramètre tend vers l'infini, en utilisant la méthode de la phase stationnaire complexe.


References [Enhancements On Off] (What's this?)

  • [1] D. Barlet and J. L. Clerc, Le comportement à l'infini des fonctions de Bessel généralisées. I, Adv. in Math, (à paraître).
  • [2] J. L. Clerc, Une formule asymptotique de type Melher-Heine pour les zonales d'un espace riemannien symétrique, Studia Math. 57 (1976), 27-32. MR 0425521 (54:13476)
  • [3] -, Transformées de Fourier des mesures orbitales dans l'espace tangent d'un espace symétrique (à paraître).
  • [4] J. J. Duistermaat, J. A. C. Kolk and V. S. Varadarajan, Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semi simple Lie groups, Compositio Math. 49 (1983), 309-398. MR 707179 (85e:58150)
  • [5] Harish-Chandra, Spherical functions on a semi-simple Lie group. I, Amer. J. Math. 80 (1958), 241-310. MR 0094407 (20:925)
  • [6] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978. MR 514561 (80k:53081)
  • [7] -, Groups and geometric analysis, Academic Press, New York, 1984. MR 754767 (86c:22017)
  • [8] A. Melin and J. Sjöstrand, Fourier integral operators with complex valued phase functions, Lecture Notes in Math., vol. 459, Springer-Verlag, pp. 120-223. MR 0431289 (55:4290)
  • [9] T. O. Sherman, Fourier analysis on compact symmetric space, Bull. Amer. Math. Soc. 83 (1977), 378-380. MR 0445236 (56:3580)
  • [10] R. J. Stanton, On mean convergence of Fourier series on compact Lie groups, Trans. Amer. Math. Soc. 218 (1976), 61-87. MR 0420158 (54:8173)
  • [11] M. Sugiura, Representations of compact groups realized by spherical functions on symmetric spaces, Proc. Japan Acad. 38 (1962), 111-113. MR 0142689 (26:258)
  • [12] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939.
  • [13] E. P. Van den Ban, Asymptotic expansions and integral formulas for eigenfunctions on a semisimple Lie groups, Thesis, Utrecht, 1983.
  • [14] L. Vretare, Elementary spherical functions on symmetric spaces, Math. Scand. 39 (1976), 343-358. MR 0447979 (56:6289)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A90, 22E46

Retrieve articles in all journals with MSC: 43A90, 22E46


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0927699-4
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society