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

We define on an ordered semi simple symmetric space =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 is a symmetric space of rank one.

Sur un espace symétrique semi simple ordonné =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 est un espace symétrique de rang un.

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {3},
     year = {1994},
     pages = {927-965},
     doi = {10.5802/aif.1421},
     zbl = {0810.43003},
     mrnumber = {96a:43012},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_3_927_0}
}
Faraut, Jacques; Hilgert, Joachim; Ólafsson, Gestur. Spherical functions on ordered symmetric spaces. Annales de l'Institut Fourier, Volume 44 (1994) no. 3, pp. 927-965. doi : 10.5802/aif.1421. http://www.numdam.org/item/AIF_1994__44_3_927_0/

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