---
tag: 972X7QM34D
kind: definition
name: Mackey machine
msc: [22D30, 22D10]
defines: 12CEWPMSG4
uses: [AHV05ZBCYW, 74BPYJNZTM, 6Q7AHAZ6Q4, 0YQPZB5Q1G]
sources:
  - url: https://en.wikipedia.org/wiki/System_of_imprimitivity
---

## Definition

Let $G = N \rtimes H$ be a second countable [[locally compact group|AHV05ZBCYW]], with $N$ closed, normal and abelian, and $H$ closed.
Let $\widehat{N}$ be the group of continuous characters $\chi \colon N \to \set{z \in \C : \abs{z} = 1}$.
The group $H$ acts on $\widehat{N}$ by $(h \cdot \chi)(n) = \chi(h^{-1} n h)$.
For $\chi \in \widehat{N}$, let $H_\chi$ be its stabiliser in $H$, and let $\sigma$ be an irreducible [[unitary representation|74BPYJNZTM]] of $H_\chi$.
The formula $(\chi \otimes \sigma)(n h) = \chi(n) \sigma(h)$ for $n \in N$ and $h \in H_\chi$ defines a unitary representation of $N \rtimes H_\chi$.
The *Mackey machine* assigns to the pair $(\chi, \sigma)$ the [[induced representation|6Q7AHAZ6Q4]]
$$\pi_{\chi, \sigma} = \operatorname{Ind}_{N \rtimes H_\chi}^{G} (\chi \otimes \sigma).$$

## Remarks

Mackey's theorem: suppose the action of $H$ on $\widehat{N}$ is regular, meaning that the [[orbit|0YQPZB5Q1G]] 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 $\pi_{\chi, \sigma}$.