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Spherical functions on ordered symmetric spaces. (English) Zbl 0810.43003

Summary: We define on an ordered semisimple symmetric space \({\mathcal M}=G/H\) 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 \({\mathcal M}\) is a symmetric space of rank one.

MSC:

43A90 Harmonic analysis and spherical functions
44A10 Laplace transform
53C35 Differential geometry of symmetric spaces
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