Let G be a p-adic reductive group and let U0 be the unipotent radical of a minimal parabolic subgroup of G. We introduce a Fourier transform defined on the space of smooth Whittaker functions on G which are compactly supported modulo U0. We determine its image. The proof follows the proof of Heiermann for the functions on the group.
During the proof, we establish an inversion formula. This formula allows us to prove that an irreducible smooth representation of G, which has a Whittaker model in the space of smooth Whittaker functions on G which are compactly supported modulo U0, is cuspidal.
This work gave us the opportunity to prepare a framework for the study of harmonic analysis on p-adic reductive symmetric spaces: B-matrices and constant term; a study of wave packets.
