Kirillov theory for $C^*(G,\Omega)$

Let $G$ be a simply connected nilpotent Lie group with Lie algebra $\frak g$; let $\frak g^*$ be the dual of $\frak g$. Let $\Omega$ be a locally compact second countable Hausdorff space with a continuous $G$ action, and let $C^*(G,\Omega)$ be the corresponding transformation group $C^*$ algebra. We construct a continuous surjective map $\phi$ from a quotient space, $\frak g^*\times\Omega/\sim$, which is a homeomorphism from $\frak g^*\times\Omega/\sim$ to Prim$(C^*(G,\Omega))$. We also describe a character theory for $C^*(G,\Omega)$ which generalizes Kirillov character theory for $G$.