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  "body": "\n## Summary\n\nThe Mackey machine is a method to find the irreducible [[unitary representations|74BPYJNZTM]] of a [[locally compact group|AHV05ZBCYW]] $G$.\nIt builds them from a closed normal subgroup $N$ and from smaller subgroups.\nIn its basic form $G = N \\rtimes H$ with $N$ abelian.\nOne lets $H$ act on the characters of $N$, picks one character in each orbit, and [[induces|6Q7AHAZ6Q4]] from its stabiliser.\nWhen the orbits are well behaved, this gives every irreducible representation exactly once up to equivalence.\nIt explains the representations of the $ax + b$ group, of the Euclidean motion groups and of the Poincaré group.\n\n## Explanations\n\n### L1 human\n\nMany groups are built from a simple abelian part, like translations, and a group that acts on it, like rotations.\nThe translations alone have one-dimensional representations, labelled by frequencies.\nThe rotations move these frequencies around, and frequencies in the same orbit give the same representation of the whole group.\nSo one picks one frequency per orbit, looks at the rotations that fix it, and combines the two pieces by induction.\nFor the motion group of the plane, the orbits are the circles of radius $r \u003e 0$ and the origin, and each circle gives one infinite-dimensional irreducible representation.\n",
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