Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group (Q973944)

Let \(G=\exp (g)\) be a completely solvable Lie group. It is well known that the unitary dual \(\widehat{G}\) of \(G\), the set of equivalence classes of irreducible unitary representations of \(G\), is homeomorphic to the space of co-adjoint orbits of \(G\). In addition, every irreducible representation of the \(C^{*}\)-algebra \(C^{*}(G)\) of \(G\) is uniquely determined by its kernel in \(C^{*}(G)\). In this paper, the authors give a description of the kernel \(\ker\pi\) in \(\;L^{1}(G)\) of a unitary irreducible representation \(\pi\) of \(G\) in terms of the corresponding co-adjoint orbit \(O_{\pi}\). They prove that \(\ker\pi\) is given by the functions whose abelian Fourier transform vanish on the Kirillov orbit \(O_{\pi}\) if and only if this orbit \(O_{\pi}\) is affine linear. This is a generalization of a result obtained earlier for nilpotent Lie groups.