Mackey imprimitivity theorem

Remarks

In operator-algebra terms, a system of imprimitivity is a covariant representation of the action of $G$ on $C_0(G/H)$. Such pairs correspond to representations of the crossed product $C_0(G/H) \rtimes G$. This crossed product is Morita equivalent to the group $C^*$-algebra of $H$, and this equivalence gives one modern proof.

Example: take $G = \R$ and $H = \set{0}$, so $G/H = \R$. A system of imprimitivity is a unitary group $\pi(t)$ together with a self-adjoint operator $Q$, through the spectral measure $P$ of $Q$. The covariance condition is the Weyl form of the canonical commutation relation. The theorem says that every such pair is a multiple of translations and multiplication by $x$ on $L^2(\R)$. This is the Stone–von Neumann theorem.

The theorem also gives a criterion for being induced: $\pi$ is induced from $H$ exactly when it admits a system of imprimitivity based on $G/H$.

Going deeper

For a student

The imprimitivity theorem is the engine of the Mackey machine. If $N$ is a closed abelian normal subgroup, the spectral measure of $\pi|_N$ lives on the dual $\widehat{N}$, and $G$ permutes it. When $\pi$ is irreducible and the action is regular, this measure sits on one orbit $G/G_\chi$, and the theorem shows that $\pi$ is induced from the stabiliser $G_\chi$. Wigner's classification of the particles through representations of the Poincaré group is an instance.