Abstract. We generalize some aspects of the classical Fourier inversion theorem to the class of connected, simply connected, nilpotent Lie groups. In this setting, the generalized Fourier transform is the operator valued map $f \mapstochar \rightarrow (\pi _l(f))_{l \in {\eufm g}^*/Ad^*G}$. These operators are characterized by operator kernels. We construct a retract to the generalized Fourier transform which maps into the Schwartz space ${\cal S}(G)$, by limiting ourselves to a suitable set of families of operator kernels. This is done via variable Lie structures.

AMS Subject Classification (1991): 22E30, 22E27, 43A20

Keyword(s): nilpotent Lie group, Fourier inversion theorem, operator kernel, minimal ideals

Received November 21, 2005, and in revised form October 12, 2006. (Registered under 6451/2009.)