On the dual of an exponential solvable Lie group

Let $G$ be a connected, simply connected exponential solvable Lie group with Lie algebra $\mathfrak{g}$. The Kirillov mapping $\eta :\,\,\mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G) \to \hat G$ gives a natural parametrization of $\hat G$ by co-adjoint orbits and is known to be continuous. In this paper a finite partition of $\mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G)$ is defined by means of an explicit construction which gives the partition a natural total ordering, such that the minimal element is open and dense. Given $\pi \in \hat G$, elements in the enveloping algebra of ${\mathfrak{g}_c}$ are constructed whose images under $\pi$ are scalar and give crucial information about the associated orbit. This information is then used to show that the restriction of $\eta$ to each element of the above-mentioned partition is a homeomorphism.