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.