We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups G= \exp(\g) by showing that operator fields defined on suitable sub-manifolds of 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.
Lin, Y-F., Ludwig, J., & Molitor-Braun, C. (2019). Nilpotent Lie groups: Fourier inversion and prime ideals. Journal of Fourier Analysis and Applications , 25(2), 345-376. https://doi.org/10.1007/s00041-017-9586-y