Mackey machine

Remarks

Mackey's theorem: suppose the action of $H$ on $\widehat{N}$ is regular, meaning that the orbit space $\widehat{N}/H$ is countably separated. Then each $\pi_{\chi, \sigma}$ is irreducible, and every irreducible unitary representation of $G$ is equivalent to some $\pi_{\chi, \sigma}$. Two of them are equivalent exactly when the characters lie in one orbit and the representations $\sigma$ correspond under conjugation.

Example: the $ax + b$ group $G = \R \rtimes \R_{>0}$, where $a > 0$ acts on $\R$ by $b \mapsto ab$. Here $\widehat{N} \cong \R$ through $\chi_\xi(b) = e^{i \xi b}$, and $a$ sends $\chi_\xi$ to $\chi_{\xi / a}$. There are three orbits: $\set{0}$, $(0, \infty)$ and $(-\infty, 0)$. The orbit $\set{0}$ has stabiliser $\R_{>0}$ and gives the one-dimensional characters of $\R_{>0}$, trivial on $N$. The other two orbits have trivial stabiliser and give two infinite-dimensional irreducible representations $\operatorname{Ind}_N^G \chi_{\pm 1}$.

Non-example: let $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ act on $N = \Z^2$, and let $G = \Z^2 \rtimes_A \Z$. Then $\widehat{N}$ is the torus $\R^2 / \Z^2$, and the dual action of $A$ on it is ergodic, so the orbit space is not countably separated. The regularity hypothesis fails, and some irreducible unitary representations of $G$ are not of the form $\pi_{\chi, \sigma}$.