Schwartz space for a hypergeometric Fourier transform

Let R be a reduced root system in an euclidean vector space E, W the Weyl g roup of R. One determines the hypergeometric Fourier transform (cf. E. Opda m, Cuspidal hypergeometric functions, preprint) of the related Schwartz spa ce 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 hype rgeometric system. This is analogous to Harish-Chandra's theory, once one h as realized that simple difference operators play here the role of some ele ments of the unipotent radical of a parabolic subalgebra. Also, the fact th at the parameter of cuspidal hypergeometric Functions might be singular int roduces new difficulties. This theory being established, one can take advan tage 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 symetriques reductifs, Ann. Math. 147 (1998), 417-452), especially the truncation and its consequences. (C) 1999 Academic Press.