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.

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     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},
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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. http://www.numdam.org/articles/10.5802/aif.1421/

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