Abstract
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.
Highlights
Connected, nilpotent Lie group G, the description of its spectrum and the Fourier inversion theorem are due to Kirillov [3], who showed that the dual space G of G is in one-to-one correspondence with the space g∗/G of co-adjoint orbits of G
We study a general version of the Fourier inversion theorem for nilpotent Lie groups
The paper is organised in the following way: in Sect. 2 we recall the definition of induced representations and of kernel functions, we explain the notion of variable nilpotent Lie groups and their Lie algebras, of index sets for co-adjoint orbits and of adapted kernel functions on a G-invariant sub-manifold of g∗
Summary
Connected, nilpotent Lie group G, the description of its spectrum and the Fourier inversion theorem are due to Kirillov [3], who showed that the dual space G of G is in one-to-one correspondence with the space g∗/G of co-adjoint orbits of G. Once we have the Retract Theorem, we can apply it to study the G-prime ideals of the Banach algebra L1(G). Using the methods in [7], it follows that every G-prime ideal in L1(G) is the kernel of such a G-orbit This result can be used, for instance, in the study of bounded irreducible representations (π, X ) of a Lie group G on a Banach space X. Exponential Lie group G, the G-prime ideals are kernels of G-orbits In this way the bounded irreducible Banach space representations of an exponential Lie group could be determined. 2 we recall the definition of induced representations and of kernel functions, we explain the notion of variable nilpotent Lie groups and their Lie algebras, of index sets for co-adjoint orbits and of adapted kernel functions on a G-invariant sub-manifold of g∗. As an application of the Retract Theorem, in the last section (Sect. 5) we show that every G-prime ideal in L1(G) is the kernel of a G-orbit
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