Mackey machine

Summary

The Mackey machine is a method to find the irreducible unitary representations of a locally compact group $G$. It builds them from a closed normal subgroup $N$ and from smaller subgroups. In its basic form $G = N \rtimes H$ with $N$ abelian. One lets $H$ act on the characters of $N$, picks one character in each orbit, and induces from its stabiliser. When the orbits are well behaved, this gives every irreducible representation exactly once up to equivalence. It explains the representations of the $ax + b$ group, of the Euclidean motion groups and of the Poincaré group.