Nilpotent Lie Groups: Fourier Inversion and Prime Ideals

Abstract

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups $$G= \\hbox {exp}({\\mathfrak {g}})$$G=exp(g) by showing that operator fields defined on suitable sub-manifolds of $${\\mathfrak {g}}^*$$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 $$L^1(G)$$L1(G) as kernels of sets of irreducible representations of G.