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Eigenspaces of the Laplace-Beltrami operator on a hyperboloid

Published online by Cambridge University Press:  22 January 2016

Jiro Sekiguchi
Jiro Sekiguchi*
Affiliation:
Department of Mathematics, Tokyo Metropolitan University
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Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on the unit disk is represented by the Poisson integral of a hyperfunction on the unit circle, much interest has been arisen to the study of the Poisson integral representation of joint eigenfunctions of all invariant differential operators on a symmetric space X. In particular, his original idea of expanding eigenfunctions into K-finite functions has proved to be generalizable up to the case where X is a Riemannian symmetric space of rank one (cf. [4], [5], [11]). Presently, extension to arbitrary rank has been completed by quite a different formalism which views the present problem as a boundary-value problem for the differential equations. It should be recalled that along this line of approach a general theory of the systems of differential equations with regular singularities was successfully established by Kashiwara-Oshima (cf. [6], [7]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 79 , October 1980 , pp. 151 - 185
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

References

[ 1 ] Faraut, J., Distributions spheriques sur les espace pseudo-Riemanniens et les hyperboloides (preprint).Google Scholar
[ 2 ] Gel’fand, I. M. and Shilov, G. E., Generalized functions, vol. 1, 1964, Academic Press, New York and London.Google Scholar
[ 3 ] Harish-Chandra, , Spherical functions on a semisimple Lie group I, Amer. J. Math., 80 (1958), 241–310.Google Scholar
[ 4 ] Helgason, S., A duality for symmetric spaces with applications to group representations, Advances in Math., 5 (1970), 1–154.Google Scholar
[ 5 ] Helgason, S., Invariant differential equations on homogeneous manifolds, Bull. Amer. Math. Soc. vol. 83 (1977), 751–774.Google Scholar
[ 6 ] Kashiwara, M., Kowata, K., Minemura, K., Okamoto, K., Oshima, T., and Tanaka, M., Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math., 107 (1978), 1–39.Google Scholar
[ 7 ] Kashiwara, M. and Oshima, T., Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math., 106 (1977), 145–200.Google Scholar
[ 8 ] Koh, S. S., On affine symmetric spaces, Trans. Amer. Math. Soc, 119 (1965), 291–309.Google Scholar
[ 9 ] Matsuki, T., The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan, 31, No. 2 (1979), 331–357.CrossRefGoogle Scholar
[10] Matsumoto, S., Hiraoka, K. and Okamoto, K., Eigenfunctions of the laplacian on a real hyperboloid of one sheet, Hiroshima Math. J., 7 (1978), 855–864.Google Scholar
[11] Minemura, K., Eigenfunctions of the laplacian on a real hyperbolic space, J. Math. Soc. Japan, 27 (1975), 82–105.Google Scholar
[12] Oshima, T., Boundary value problem for symmetric spaces corresponding to various boundaries, RIMS Kōkyūroku, 281 (1976), 211–226 (in Japanese).Google Scholar
[13] Oshima, T., On a realization of Riemannian symmetric spaces, J. Math. Soc. Japan, 30 (1978), 117–132.Google Scholar
[14] Oshima, T. and Sekiguchi, J., Eigenspace of invariant differential operators on an affine symmetric space, preprint.Google Scholar
[15] Rossmann, W., Analysis on real hyperbolic spaces, J. Functional Analysis, 30 (1978), 448–477.CrossRefGoogle Scholar
[16] Wallach, N. R., Harmonic analysis on homogeneous spaces, Marcel Dekker, Inc., New York.Google Scholar