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  "body": "\n## Statement\n\nLet $G$ be a second countable [[locally compact group|AHV05ZBCYW]], and let $H$ be a closed subgroup of $G$.\nLet $\\pi$ be a [[unitary representation|74BPYJNZTM]] of $G$ on a [[Hilbert space|HM2HHE3J51]] $\\mathcal{H}$.\nA *system of imprimitivity* for $\\pi$ based on $G/H$ is a [[projection-valued measure|ATGT25E5FF]] $P$ on the Borel sets of $G/H$, acting on $\\mathcal{H}$.\nIt must satisfy\n$$\\pi(g) P(E) \\pi(g)^{-1} = P(gE)$$\nfor all $g \\in G$ and all Borel sets $E \\subseteq G/H$.\nEvery [[induced representation|6Q7AHAZ6Q4]] $\\operatorname{Ind}_H^G \\sigma$ has a canonical system of imprimitivity, where $P(E)$ multiplies a function by the indicator of $E$.\nLet $(\\pi, P)$ be any system of imprimitivity based on $G/H$.\nThen there is a unitary representation $\\sigma$ of $H$ such that $(\\pi, P)$ is unitarily equivalent to the canonical system of $\\operatorname{Ind}_H^G \\sigma$.\nThe representation $\\sigma$ is unique up to unitary equivalence.\n\n## Remarks\n\nIn operator-algebra terms, a system of imprimitivity is a covariant representation of the action of $G$ on $C_0(G/H)$.\nSuch pairs correspond to representations of the crossed product $C_0(G/H) \\rtimes G$.\nThis crossed product is Morita equivalent to the group $C^*$-algebra of $H$, and this equivalence gives one modern proof.\n\nExample: take $G = \\R$ and $H = \\set{0}$, so $G/H = \\R$.\nA system of imprimitivity is a unitary group $\\pi(t)$ together with a self-adjoint operator $Q$, through the spectral measure $P$ of $Q$.\nThe covariance condition is the Weyl form of the canonical commutation relation.\nThe theorem says that every such pair is a multiple of translations and multiplication by $x$ on $L^2(\\R)$.\nThis is the [[Stone–von Neumann theorem|Z877XDBMQ3]].\n\nThe theorem also gives a criterion for being induced: $\\pi$ is induced from $H$ exactly when it admits a system of imprimitivity based on $G/H$.\n\n## Explanations\n\n### L1 human\n\nSome representations come with a way of cutting the space into pieces labelled by points of $G/H$, and $G$ permutes the pieces.\nThe theorem says that this extra structure is exactly what induced representations have.\nSo to recognise an induced representation, one only has to find such a compatible way of cutting the space.\nIn quantum mechanics, the pieces are the positions of a particle, and the group moves the positions.\n\n### L2 human\n\nThe imprimitivity theorem is the engine of the [[Mackey machine|12CEWPMSG4]].\nIf $N$ is a closed abelian normal subgroup, the spectral measure of $\\pi|_N$ lives on the dual $\\widehat{N}$, and $G$ permutes it.\nWhen $\\pi$ 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 $G_\\chi$.\nWigner's classification of the particles through representations of the Poincaré group is an instance.\n",
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