Nilpotent Lie groups: Fourier inversion and prime ideals

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups $$G= \hbox {exp}({\mathfrak {g}})$$ by showing that operator fields defined on suitable sub-manifolds of $${\mathfrak {g}}^*$$ are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of $$L^1(G)$$ as kernels of sets of irreducible representations of G.