Nilpotent Lie Groups: Fourier Inversion and Prime Ideals

Abstract

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.