no. 3
Spherical functions on ordered symmetric spaces
Annales de l'Institut Fourier, Tome 44 (1994) no. 3, pp. 927-965.

Sur un espace symétrique semi simple ordonné ℳ=G/H nous définissons une famille de fonctions sphériques par une représentation intégrale semblable à la représentation intégrale de Harish Chandra des fonctions sphériques sur un espace riemannien symétrique de type non compact. Puis nous associons à ces fonctions sphériques une transformation de Laplace sphérique. Dans cette transformation le produit de composition de deux noyaux causaux invariants a pour image le produit ordinaire de leurs transformées. Nous établissons une formule d’inversion pour cette transformation lorsque G est un groupe complexe et H une forme réelle de G, et lorsque ℳ est un espace symétrique de rang un.

We define on an ordered semi simple symmetric space ℳ=G/H a family of spherical functions by an integral formula similar to the Harish-Chandra integral formula for spherical functions on a Riemannian symmetric space of non compact type. Associated with these spherical functions we define a spherical Laplace transform. This transform carries the composition product of invariant causal kernels onto the ordinary product. We invert this transform when G is a complex group, H a real form of G, and when ℳ is a symmetric space of rank one.

@article{AIF_1994__44_3_927_0,
     author = {Faraut, Jacques and Hilgert, Joachim and \'Olafsson, Gestur},
     title = {Spherical functions on ordered symmetric spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {927--965},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {3},
     year = {1994},
     doi = {10.5802/aif.1421},
     zbl = {0810.43003},
     mrnumber = {96a:43012},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1421/}
}
TY  - JOUR
AU  - Faraut, Jacques
AU  - Hilgert, Joachim
AU  - Ólafsson, Gestur
TI  - Spherical functions on ordered symmetric spaces
JO  - Annales de l'Institut Fourier
PY  - 1994
SP  - 927
EP  - 965
VL  - 44
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1421/
DO  - 10.5802/aif.1421
LA  - en
ID  - AIF_1994__44_3_927_0
ER  - 
%0 Journal Article
%A Faraut, Jacques
%A Hilgert, Joachim
%A Ólafsson, Gestur
%T Spherical functions on ordered symmetric spaces
%J Annales de l'Institut Fourier
%D 1994
%P 927-965
%V 44
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1421/
%R 10.5802/aif.1421
%G en
%F AIF_1994__44_3_927_0
Faraut, Jacques; Hilgert, Joachim; Ólafsson, Gestur. Spherical functions on ordered symmetric spaces. Annales de l'Institut Fourier, Tome 44 (1994) no. 3, pp. 927-965. doi : 10.5802/aif.1421. https://aif.centre-mersenne.org/articles/10.5802/aif.1421/

[Ba88] E. Van Den Ban, The principal series for a reductive symmetric space I. H-fixed distribution vectors, Ann. Sci. E.N.S., 21 (1988), 359-412. | Numdam | MR | Zbl

[CK72] C. Cronström, and W.H. Klink Generalized O (1,2) expansions of multiparticle amplitudes, Annals of Physics, 69 (1972), 218-278.

[De90] P. Delorme, Coefficients généralisés de séries principales sphériques et distributions sphériques sur Gℂ/Gℝ, Invent. Math., 105 (1991), 305-346. | MR | Zbl

[Er53a] A. Erdelyi, et. al., Higher transcendental functions I, Mc Graw-Hill, New York, 1953. | MR | Zbl

[Er53b] A. Erdelyi, et al., Higher transcendental functions II, Mc Graw-Hill, New York, 1953. | MR | Zbl

[Fa87] J. Faraut, Algèbres de Volterra et transformation de Laplace sphérique sur certains espaces symétriques ordonnés, Symp. Math., 29 (1987), 183-196. | MR | Zbl

[Fa91] J. Faraut, Espaces symétriques ordonnés et algèbres de Volterra, J. Math. Soc. Japan, 43 (1991), 133-147. | MR | Zbl

[FV86] J. Faraut, and G.A. Viano, Volterra algebra and the Bethe-Salpeter equation, J. Math. Phys., 27 (1986), 840-848. | MR | Zbl

[He78] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York/London, 1978. | MR | Zbl

[He84] S. Helgason, Groups and Geometric Analysis, Academic Press, New York/London, 1984. | Zbl

[KnZu82] A.W. Knapp, and J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math., 116 (1982), 389-455. | MR | Zbl

[La89] J.D. Lawson, Ordered manifolds, invariant cone fields, and semigroups, Forum Math., 1 (1989), 273-308. | EuDML | MR | Zbl

[Mi83] M. Mizony, Une transformation de Laplace-Jacobi, SIAM J. Math., 14 (1983), 987-1003. | MR | Zbl

[Ne91] K.H. Neeb, A convexity theorem for semisimple symmetric spaces, to appear in Pac. J. Math. | Zbl

[Óla85] G. Ólafsson, Integral formulas and induced representations associated to an affine symmetric space, Math. Gotting., 33 (1985).

[Óla87] G. Ólafsson, Fourier and Poisson transformation associated to a semisimple symmetric space, Invent. Math., 90 (1987), 605-629. | EuDML | MR | Zbl

[Óla90] G. Ólafsson, Causal symmetric spaces, Math. Gotting., 15 (1990).

[ÓH92] G. Ólafsson, and J. Hilgert, Causal symmetric spaces, book in preparation. | Zbl

[Ol81] G. I. Ol'Shanskii, Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series, Funct. Anal. and Appl., 15 (1981), 275-285. | MR | Zbl

[Ol82] G. I. Ol'Shanskii, Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal (order) structures on pseudo-Riemannian symmetric spaces, Sov. Math. Dokl., 26 (1982), 97-101. | Zbl

[Vi80] G.A. Viano, On the harmonic analysis of the elastic scattering amplitude of two spinless particles at fixed momentum transfer, Ann. Inst. H. Poincaré, A, 32 (1980), 109-123. | EuDML | Numdam

[Wo72] J. Wolf, The fine structure of Hermitean symmetric spaces, in Symmetric spaces, Boothby and Weiss ed., Marcel Dekker, New York, (1972). | Zbl

Cité par Sources :