Espace de Schwartz pour la transformation de Fourier hypergéométrique

Résumé be a reduced root system in an euclidean vector space E, W the Weyl group of R. One determines the hypergeometric Fourier transform (cf. E. Opdam, Cuspidal hypergeometric functions, preprint) of the related Schwartz space of W-invariant functions on E, introduced by Tinfou. The answer is very natural and quite similar to results of Harish-Chandra. The proof requires a theory of the constant term for W-invariant functions satisfying the hypergeometric system. This is analogous to Harish-Chandraʹs theory, once one has realized that simple difference operators play here the role of some elements of the unipotent radical of a parabolic subalgebra. Also, the fact that the parameter of cuspidal hypergeometric functions might be singular introduces new difficulties. This theory being established, one can take advantage of the techniques we used for real reductive symmetric spaces (cf. the introduction to the article by the author, Formule de Plancherel pour les espaces symétriques réductifs, Ann. Math.147 (1998), 417–452), especially the truncation and its consequences.