  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 $G$.
