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  "body": "\n## Definition\n\nLet $G = N \\rtimes H$ be a second countable [[locally compact group|AHV05ZBCYW]], with $N$ closed, normal and abelian, and $H$ closed.\nLet $\\widehat{N}$ be the group of continuous characters $\\chi \\colon N \\to \\set{z \\in \\C : \\abs{z} = 1}$.\nThe group $H$ acts on $\\widehat{N}$ by $(h \\cdot \\chi)(n) = \\chi(h^{-1} n h)$.\nFor $\\chi \\in \\widehat{N}$, let $H_\\chi$ be its stabiliser in $H$, and let $\\sigma$ be an irreducible [[unitary representation|74BPYJNZTM]] of $H_\\chi$.\nThe formula $(\\chi \\otimes \\sigma)(n h) = \\chi(n) \\sigma(h)$ for $n \\in N$ and $h \\in H_\\chi$ defines a unitary representation of $N \\rtimes H_\\chi$.\nThe *Mackey machine* assigns to the pair $(\\chi, \\sigma)$ the [[induced representation|6Q7AHAZ6Q4]]\n$$\\pi_{\\chi, \\sigma} = \\operatorname{Ind}_{N \\rtimes H_\\chi}^{G} (\\chi \\otimes \\sigma).$$\n\n## Remarks\n\nMackey's theorem: suppose the action of $H$ on $\\widehat{N}$ is regular, meaning that the [[orbit|0YQPZB5Q1G]] space $\\widehat{N}/H$ is countably separated.\nThen each $\\pi_{\\chi, \\sigma}$ is irreducible, and every irreducible unitary representation of $G$ is equivalent to some $\\pi_{\\chi, \\sigma}$.\nTwo of them are equivalent exactly when the characters lie in one orbit and the representations $\\sigma$ correspond under conjugation.\n\nExample: the $ax + b$ group $G = \\R \\rtimes \\R_{\u003e0}$, where $a \u003e 0$ acts on $\\R$ by $b \\mapsto ab$.\nHere $\\widehat{N} \\cong \\R$ through $\\chi_\\xi(b) = e^{i \\xi b}$, and $a$ sends $\\chi_\\xi$ to $\\chi_{\\xi / a}$.\nThere are three orbits: $\\set{0}$, $(0, \\infty)$ and $(-\\infty, 0)$.\nThe orbit $\\set{0}$ has stabiliser $\\R_{\u003e0}$ and gives the one-dimensional characters of $\\R_{\u003e0}$, trivial on $N$.\nThe other two orbits have trivial stabiliser and give two infinite-dimensional irreducible representations $\\operatorname{Ind}_N^G \\chi_{\\pm 1}$.\n\nNon-example: let $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ act on $N = \\Z^2$, and let $G = \\Z^2 \\rtimes_A \\Z$.\nThen $\\widehat{N}$ 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.\nThe regularity hypothesis fails, and some irreducible unitary representations of $G$ are not of the form $\\pi_{\\chi, \\sigma}$.\n",
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