On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C(V,L(^2d)), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.