Mackey Theory

Introduction

Mackey theory is the body of results, due to George Mackey, that analyses the unitary representations of a locally compact group in terms of induced representations. Its central results are the imprimitivity theorem, which characterises induced representations by the presence of a system of imprimitivity on a homogeneous space, the theorem comparing two induced representations $\operatorname{Ind}_H^G\sigma$ and $\operatorname{Ind}_{H'}^G\sigma'$ and deciding when they are equivalent, the irreducibility criterion, and the Mackey machine for a group with a normal subgroup — a procedure that reduces the classification of the irreducible unitary representations of $G$ to the same problem for smaller groups. Applied to a semidirect product it produces the little-group method; applied to a nilpotent Lie group it produces Kirillov's orbit method, in which the irreducibles are parametrised by the coadjoint orbits.

The article develops these results in the frame of the previous article: the induced representation is $\operatorname{Ind}_H^G\sigma$ in the function-space or balanced-product realisation of Induced Representations of Locally Compact Groups, the unitary dual is $\operatorname{Irr}(G)$ of Representation Theory of Locally Compact Groups, and the Haar measure and the quasi-invariant measure on a homogeneous space are those of Locally Compact Groups and Haar Measure in Part III. The boundary with Part III is the one fixed in the representation-theoretic block: the constructions and classification statements — systems of imprimitivity, the imprimitivity theorem, the intertwining number theorem, the little-group method, the orbit correspondence, the obstruction cocycle — are the subject of this article, while the harmonic analysis — the completion of the induced spaces, the direct-integral decomposition of an induced representation, the Plancherel theorem and the measure-theoretic details of the orbit correspondence — belongs to Analysis on Groups in Part III, where the measure and the limit are available. The cohomological classification of the obstruction uses the group cohomology of Group Cohomology in Part I, and the operator-algebraic frame — the imprimitivity theorem read as a Morita equivalence of $C^*$-algebras — is that and is used only as a pointer. No physics is invoked.

Systems of Imprimitivity

Definition

Let $G$ act on a locally compact space $X$ by homeomorphisms. A system of imprimitivity for a unitary representation $\pi$ of $G$ based on $X$ consists of a $\sigma$-algebra of subsets of $X$ stable under $G$ together with a projection-valued measure

$$ P : \mathcal{B}(X) \longrightarrow \operatorname{Proj}(\mathcal{H}_\pi) $$

from the measurable sets of $X$ to the orthogonal projections of $\mathcal{H}_\pi$, such that

$$ \pi(g)\, P(E)\, \pi(g)^{-1} = P(gE) \qquad \text{for all } g \in G,\ E \subseteq X \text{ measurable} . $$

Equivalently, in the transitive case $X = G/H$, the data consist of a unitary representation $\pi$ of $G$ and a projection-valued measure $P$ on the quotient $G/H$ on which $G$ acts such that $\pi(g)P(E)\pi(g)^{-1} = P(gE)$; when the measure is a point mass at the coset $H$ one recovers the imprimitivity condition

$$ P(H)\,\pi(h) = \pi(h)\,P(H) \qquad \text{for all } h \in H , $$

with $P(H)$ the projection onto the fibre over the coset $H$, which therefore commutes with the operators $\pi(h)$, $h \in H$; the general system is generated by this one projection together with the action.

Definition. A representation $\pi$ of $G$ is monomial if it is induced from a one-dimensional representation of a closed subgroup, and of finite type if it is induced from a finite-dimensional representation of a closed subgroup.

The Imprimitivity Theorem

Theorem (imprimitivity; Mackey). Let $G$ be a locally compact group acting transitively on a standard Borel space $X$ with isotropy subgroups conjugate to a fixed closed subgroup $H$, and let $\rho$ be a unitary representation of $H$. Then the induced representation $\operatorname{Ind}_H^G\rho$ carries a system of imprimitivity based on $X = G/H$ by

$$ P(E)\, f = 1_E\, f , \qquad E \subseteq G/H , $$

the projection being multiplication by the indicator function of $E$ in the function-space realisation. Conversely, every unitary representation $\pi$ of $G$ with a system of imprimitivity based on $G/H$ is unitarily equivalent to an induced representation $\operatorname{Ind}_H^G\rho$, where $\rho$ is the restriction of $\pi$ to the isotropy subgroup $H$ on the range of the projection $P(H)$.

Proof sketch. The forward construction is immediate: the indicator $1_E$ is a bounded multiplication operator on the induced space, the family $E \mapsto 1_E$ is a projection-valued measure on $G/H$, and the identity $\pi(g)1_E\pi(g)^{-1} = 1_{gE}$ is the equivariance of left translation. For the converse, let $\mathcal{H}_0 = P(H)\mathcal{H}_\pi$; the imprimitivity condition says that $\mathcal{H}_0$ is invariant under $H$, and $\rho = \pi|_H$ restricted to $\mathcal{H}_0$ is a unitary representation of $H$. The map sending $v \in \mathcal{H}_0$ to the $H$-equivariant function $g \mapsto (\pi(g)^{-1}v)$ lands in the induced space, and it intertwines the actions; it is unitary because the projection-valued measure on $G/H$ is the spectral measure of the induced functions' supports. The measure-theoretic care — the existence of a quasi-invariant measure on $G/H$, the $\sigma$-finiteness and the identification of the range of $P$ — uses the measure theory of Part III and the quasi-invariant measure of Locally Compact Groups and Haar Measure, §Quotients and Homogeneous Spaces.

Corollary (the $C^*$-algebraic form). The imprimitivity theorem is equivalent to a Morita equivalence between the crossed-product $C^*$-algebra $C_0(X)\rtimes G$ and the group algebra $C^*(H)$, and the induction functor corresponds under this equivalence to the balanced product $\otimes_{C^*(H)}$. This is the form of the theorem used in noncommutative geometry and in the classification of $C^*$-algebras, and the $C^*$-algebraic framework is that in Part II .

Example (the transitive system of the quasiregular representation). The quasiregular representation $\lambda_{G/H}$ of Induced Representations of Locally Compact Groups, §Basic Properties of Induction, is the induced representation $\operatorname{Ind}_H^G1_H$, and the system of imprimitivity based on $G/H$ is given by multiplication by indicators. It is the model system: every transitive system is obtained from a quasiregular one by multiplying by a representation of the isotropy subgroup.

Mackey's Analysis of Induced Representations

Equivalence of Induced Representations

Two induced representations may be equivalent even when the inducing data look different; the comparison is made by Mackey's double-coset analysis, which reduces the problem to a finite or countable sum over double cosets and then to a conjugacy question for representations of a single subgroup.

Theorem (intertwining number theorem; Mackey). Let $H_1, H_2$ be closed subgroups of $G$ and let $\rho_i$ be unitary representations of $H_i$. Under the standard regularity hypotheses on the double coset space $H_2\backslash G/H_1$ — which hold, in particular, when the space is countable and the orbit maps are sufficiently regular, as for a Lie group with closed subgroups — the intertwining space is given by

$$ \dim\operatorname{Hom}_G\bigl(\operatorname{Ind}_{H_1}^G\rho_1,\ \operatorname{Ind}_{H_2}^G\rho_2\bigr) \;=\; \sum_{s \in H_2\backslash G/H_1} \dim\operatorname{Hom}_{H_2\cap sH_1s^{-1}}\bigl(\rho_1^s,\ \rho_2\bigr) , $$

with $\rho_1^s(x) = \rho_1(s^{-1}xs)$, the two representations $\rho_1^s$ and $\rho_2$ being read on the common subgroup $H_2\cap sH_1s^{-1}$. Consequently $\operatorname{Ind}_{H_1}^G\rho_1 \cong \operatorname{Ind}_{H_2}^G\rho_2$ precisely when the summands match: for a bijection $s \leftrightarrow s'$ between the double cosets carrying nonzero summands, the constituents $\operatorname{Ind}_{H_2\cap sH_1s^{-1}}^{H_2}\rho_1^s$ of the first representation and the corresponding constituents of the second are equivalent, all remaining summands of both sums being zero.

Proof sketch. The double coset decomposition of $G$ relative to $H_1$ and $H_2$ fibres the restriction: $\operatorname{Ind}_{H_1}^G\rho_1$ restricted to $H_2$ decomposes as the sum, over the double cosets, of the representations induced from $\rho_1^s$ on the intersections, which is the restriction formula proved below by the orbit decomposition of $G/H_1$ under $H_2$ together with induction in stages. Taking intertwiners into $\operatorname{Ind}_{H_2}^G\rho_2$ and using Frobenius reciprocity of Induced Representations of Locally Compact Groups twice reduces the computation to the stated sum of Hom spaces on the intersections; the identity is exact in the finite case and in the type I case, and is quoted as standard in general. The statement is recorded in the form of a comparison of families of representations of $H_2$, which is the form used in the little-group method below.

Irreducibility Criterion

Theorem (irreducibility criterion; Mackey). Let $H \leq G$ be closed and let $\rho$ be an irreducible unitary representation of $H$. Under the standard regularity hypotheses on the double cosets $H\backslash G/H$, the induced representation $\operatorname{Ind}_H^G\rho$ is irreducible if and only if for every $g \in G \smallsetminus H$ the representations $\rho$ and $\rho^g$ of $H \cap gHg^{-1}$, defined by $\rho^g(x) = \rho(g^{-1}xg)$ restricted to the intersection, are disjoint; equivalently, the stabiliser of the class of $\rho$ in the conjugation action of $G$ on $\operatorname{Irr}(H)$ is exactly $H$.

Proof. The restriction formula applied with $H_1 = H_2 = H$ expresses $\operatorname{Ind}_H^G\rho$ restricted to $H$ as the direct sum over the double cosets $H\backslash G/H$ of the representations induced from $\rho^g$ on $H \cap gHg^{-1}$. The double coset $H$ itself contributes $\rho$, while the others contribute representations orthogonal to $\rho$ or not, and $\operatorname{Ind}_H^G\rho$ is irreducible exactly when its endomorphism algebra is $\mathbb{C}$. Computing $\operatorname{End}_G(\operatorname{Ind}_H^G\rho)$ by Frobenius reciprocity of Induced Representations of Locally Compact Groups gives $\operatorname{Hom}_H(\operatorname{Ind}_H^G\rho|_H,\rho)$, which by the restriction formula is $\operatorname{Hom}_H(\bigoplus_{g}\rho^g, \rho)$, a space of dimension one exactly when only the identity double coset contributes, which is exactly the disjointness condition on $\rho$ and its conjugates. The general statement is the theorem of Mackey; the counting is exact in the finite case and in the type I case, and is quoted as standard in general.

Restriction and Tensor Products

Theorem (restriction formula). Let $H_1, H_2 \leq G$ be closed subgroups and let $\rho$ be a unitary representation of $H_1$. Then, as a representation of $H_2$,

$$ \Bigl(\operatorname{Ind}_{H_1}^G\rho\Bigr)\Big|_{H_2} \;\cong\; \bigoplus_{s \in H_2\backslash G/H_1} \operatorname{Ind}_{H_2\cap sH_1s^{-1}}^{H_2}\rho^s , $$

the sum running over the double cosets and $\rho^s(h) = \rho(s^{-1}hs)$ for $h \in H_2\cap sH_1s^{-1}$.

Theorem (tensor product formula). Let $H_1, H_2 \leq G$ be closed subgroups and let $\rho_i$ be unitary representations of $H_i$. Then

$$ \operatorname{Ind}_{H_1}^G\rho_1 \otimes \operatorname{Ind}_{H_2}^G\rho_2 \;\cong\; \bigoplus_{s \in H_2\backslash G/H_1} \operatorname{Ind}_{H_2\cap sH_1s^{-1}}^{G}\bigl(\rho_1^s \otimes \rho_2\bigr) , $$

where $\rho_1^s$ and the restriction of $\rho_2$ are both read on the intersection group $H_2\cap sH_1s^{-1}$, and the constituents are indexed by the double cosets.

Proof. The two formulas are the two faces of one computation. The tensor product $\operatorname{Ind}_{H_1}^G\rho_1\otimes\operatorname{Ind}_{H_2}^G\rho_2$ is the restriction to the diagonal of the representation $\operatorname{Ind}_{H_1\times H_2}^{G\times G}(\rho_1\boxtimes\rho_2)$ of $G \times G$, and the restriction formula applied to the diagonal subgroup $\Delta G \subseteq G\times G$ expresses it as a sum over the double cosets of $\Delta G$ in $G\times G$ against $H_1\times H_2$; those double cosets are parametrised by the double cosets $H_2\backslash G/H_1$, and the constituent at $s$ is induced from $\rho_1^s\otimes\rho_2$ on the intersection. The restriction formula is proved directly by the orbit decomposition of $G/H_1$ under $H_2$. Both are stated and proved as standard in the literature; the details are combinatorial in nature and are exact.

Example (finite groups). For finite $G$ the formulas are the classical ones of Representations of Groups: the restriction formula is Mackey's theorem on the restriction of an induced representation, and the tensor product formula computes the tensor product of two induced representations as a sum over double cosets. The character identities, with sums over $G$ replaced by averages, are the finite content of the two theorems.

The Mackey Machine

Groups with a Normal Subgroup

The Mackey machine classifies the irreducible unitary representations of a group $G$ with a closed normal subgroup $N$ such that the quotient $G/N$ is well behaved, in terms of the action of $G$ on $\operatorname{Irr}(N)$ and the representations of certain stabilisers. The machinery proceeds in three steps.

Theorem (Mackey's analysis). Let $N \trianglelefteq G$ be a closed normal subgroup and let $\pi$ be an irreducible unitary representation of $G$. Then:

(a) The restriction $\pi|_N$ is a direct sum (or, in general, a direct integral) of irreducibles of $N$, and the set of those occurring is a $G$-orbit in $\operatorname{Irr}(N)$ — the orbit of $\pi$ — because $\pi(g)\pi(n)\pi(g)^{-1} = \pi(gng^{-1})$ shows that the set of classes occurring is invariant under conjugation.

(b) Fix an irreducible $\rho \in \operatorname{Irr}(N)$ in that orbit and let

$$ G_\rho = \{g \in G : \rho^g \cong \rho\} $$

be its stabiliser or little group. There is a projective representation $\sigma$ of $G_\rho$ on $\mathcal{H}_\rho$ extending $\rho$, with cocycle

$$ \sigma(g)\sigma(h) = \omega(g,h)\,\sigma(gh) , \qquad \omega(g,h) \in \mathbb{T} , $$

and $\pi$ is equivalent to $\operatorname{Ind}_{G_\rho}^G(\sigma_\pi)$ for an irreducible projective representation $\sigma_\pi$ of $G_\rho$ whose cocycle is $\omega$.

(c) The irreducible representations of $G$ with orbit $\mathcal{O}$ correspond to the irreducible projective representations of $G_\rho$ with cocycle $\omega$ whose restriction to $N$ is a multiple of $\rho$, and these may be read on $G_\rho/\ker\rho$ with the cocycle induced by $\omega$. In particular, when $\omega$ is a coboundary — which holds in particular when the extension $1 \to N \to G_\rho \to G_\rho/N \to 1$ splits over the quotient — the projective representations are ordinary representations of $G_\rho$, and the classification is by the irreducibles of the little group.

Proof sketch. (a) The restriction decomposes over the spectral measure of the commutative algebra generated by $N$; conjugating by $g$ changes $\rho$ to $\rho^g$, so the set of classes occurring is an orbit. (b) The stabiliser $G_\rho$ acts on $\mathcal{H}_\rho$ by intertwining operators that are unitary up to scalars; the choice of a measurable section of the action produces the cocycle $\omega$, and the extension of $\rho$ to a projective representation is a measurable choice, made unique up to a character. The induction of that projective representation back to $G$ is the representation $\pi$, and the equivalence is the imprimitivity theorem applied to the system of imprimitivity carried by the orbit $\mathcal{O}$, identified with $G/G_\rho$ via the orbit map. (c) The projective representations of $G_\rho$ with cocycle $\omega$ are modules over the twisted group algebra, and the irreducible ones are classified by the little group modulo the kernel of $\rho$; the split case reduces to ordinary representation theory. The full proof, including the measurability of the choices and the direct-integral refinement, is standard and is quoted from the literature; the decomposition into a direct integral in the non-type-I case is the Part III refinement.

Semidirect Products and the Little-Group Method

Corollary (semidirect products). Let $G = N \rtimes H$ with $N$ abelian and normal. Then the irreducible unitary representations of $G$ are parametrised by the pairs $(\mathcal{O}, \sigma)$ where $\mathcal{O}$ is a $G$-orbit in the Pontryagin dual $N^\vee$ and $\sigma$ is an irreducible unitary representation of the stabiliser $H_\chi$ of a point $\chi \in \mathcal{O}$, the representation being $\operatorname{Ind}_{N\rtimes H_\chi}^G(\chi \otimes \sigma)$. Here $H_\chi = \{h \in H : h\cdot\chi = \chi\}$ is the little group of $\chi$, and the projective obstruction is absent because $H_\chi$ meets $N$ trivially in the semidirect product.

Proof. Apply the Mackey machine with $N$ abelian, so $\operatorname{Irr}(N) = N^\vee$ by Pontryagin Duality; the stabiliser of $\chi \in N^\vee$ is $G_\chi = N \rtimes H_\chi$, and the extension $\sigma$ is $\chi \otimes \sigma$ with $\sigma$ an ordinary representation of $H_\chi$ because $N$ is abelian and the action of $H_\chi$ on $N$ is by group automorphisms preserving $\chi$, so the cocycle $\omega$ can be computed to be trivial on $N$ and on a complement. The induction is from $G_\chi$ to $G$.

Example (the Euclidean group and its motion group analogue). Let $G = SO(n) \ltimes \mathbb{R}^n$, the group of orientation-preserving isometries of $\mathbb{R}^n$. The dual of the normal subgroup $\mathbb{R}^n$ is $\mathbb{R}^n$, on which $SO(n)$ acts by the defining representation. The orbits are the spheres of radius $r > 0$ together with the origin. For $r > 0$ the stabiliser of a point is $SO(n-1)$, and the irreducible unitary representations of $G$ with orbit the sphere of radius $r$ are $\operatorname{Ind}_{\mathbb{R}^n\rtimes SO(n-1)}^{G}(\chi_\xi \otimes \sigma)$ for $\sigma \in \operatorname{Irr}(SO(n-1))$; the ones with orbit the origin are the representations of $SO(n)$ pulled back. This is the complete classification of the irreducible unitary representations of the Euclidean group, and it exhibits the general pattern: a continuous family parametrised by the radius and a discrete parameter from the little group.

Example (the affine group of the line). Let $G = \mathbb{R} \rtimes \mathbb{R}_{>0}$ with the multiplication of Locally Compact Groups and Haar Measure, §The Modular Function and Unimodularity. The normal subgroup $\mathbb{R}$ has dual $\mathbb{R}$, and $\mathbb{R}_{>0}$ acts by scaling with two orbits: $\{0\}$ and $\mathbb{R}\smallsetminus\{0\}$. The orbit $\{0\}$ gives the characters of the quotient $\mathbb{R}_{>0} \cong \mathbb{R}$, a continuous family of one-dimensional representations; the two orbits $\mathbb{R}_{>0}$ and $\mathbb{R}_{<0}$ are free, so the little group is trivial and the corresponding representations are $\operatorname{Ind}_{\mathbb{R}}^G\chi_{\pm}$, the two infinite-dimensional irreducible unitary representations of the affine group, contragredient to one another and each a member of its continuous series. This is the standard computation of the unitary dual of the $ax+b$ group and the simplest non-unimodular example of the machine.

Nilpotent Groups and the Orbit Method

Theorem (Kirillov's orbit method, statement). Let $G$ be a connected simply connected nilpotent Lie group with Lie algebra $\mathrm{G}$. Then:

(a) the coadjoint action of $G$ on the dual vector space $\mathrm{G}^*$ has orbits that are symplectic manifolds under the Kirillov–Kostant form

$$ \omega_\xi(X, Y) = \langle \xi, [X, Y]\rangle , \qquad \xi \in \mathrm{G}^* ; $$

(b) every coadjoint orbit $\mathcal{O}$ carries a $G$-invariant measure and a canonical irreducible unitary representation $\pi_\mathcal{O}$ of $G$ realised on the space of square-integrable sections of a line bundle over $\mathcal{O}$ (the Kirillov correspondence);

(c) the map $\mathcal{O} \mapsto \pi_\mathcal{O}$ is a bijection between the coadjoint orbits and $\operatorname{Irr}(G)$;

(d) the representation $\pi_\mathcal{O}$ is monomial: it is induced from a one-dimensional unitary character of a closed subgroup, and it is obtained by the Mackey machine from the abelian normal subgroups of the lower central series.

Proof sketch. The proof is by induction on the dimension, along the lower central series. A nonzero nilpotent Lie algebra has a centre of positive dimension, so $G$ has a central subgroup $Z \neq \{e\}$; fix a unitary character $\chi$ of $Z$ and let $G_\chi$ be its stabiliser in $G$. The Mackey machine applied to the central subgroup $Z$ — the case in which the kernel is central, so that the action of $G$ on $\operatorname{Irr}(Z)$ is trivial and no orbit analysis beyond the stabiliser is needed — reduces the irreducibles of $G$ with central character $\chi$ to the irreducible projective representations of $G_\chi/Z$ with the Mackey cocycle $\omega(h,k) = \chi([\tilde h,\tilde k])$ determined by $\chi$ and the commutators, lifted to $G_\chi$; the representation is then induced from $G_\chi$ to $G$. When the centre acts by scalars, the alternating form $\langle\xi,[X,Y]\rangle$ attached to the corresponding $\xi \in \mathrm{G}^*$ is nondegenerate modulo its radical, a maximal isotropic subalgebra $\mathrm{M}$ — a polarisation, which exists for a nilpotent algebra — is chosen so that the character $e^{i\langle\xi,\cdot\rangle}$ is trivial on $[\mathrm{M},\mathrm{M}]$, and $\pi_\mathcal{O} = \operatorname{Ind}_M^G\chi_\xi$ is irreducible and independent of the polarisation up to equivalence; the orbit $\mathcal{O}$ is identified with $G/M$ and the Kirillov–Kostant form is the symplectic form of that homogeneous space. The full argument, including the verification that the inductive stage matches the coadjoint-orbit picture, is the theorem of Kirillov and is quoted as standard; the measure-theoretic part — the invariant measure on the orbit and the completion of the induced space — is the Part III refinement.

Example (the Heisenberg group). For the Heisenberg group with Lie algebra spanned by $X, Y, Z$ with $[X,Y] = Z$ central, the dual $\mathrm{G}^*$ has coordinates $(x, y, z)$; the coadjoint orbits are the planes $z = \lambda \neq 0$, each a single orbit (because the coadjoint action moves the pair $(x,y)$ freely while fixing $z$), together with the single points of the plane $z = 0$, on which the coadjoint action is trivial. The orbit $z = \lambda$ corresponds to the Stone–von Neumann representation $\pi_\lambda$ — realised classically on the square-integrable functions of the line, an object of Part III — and each point of the plane $z = 0$ to a character of the abelianisation. This is the model case of the orbit method and the computation of Representation Theory of Locally Compact Groups, §The General Case and the Unitary Dual.

Projective Representations and the Mackey Obstruction

Cocycles and Extensions

Let $G_\rho$ be the little group of an irreducible $\rho$ of a normal subgroup $N$, and let $\omega : G_\rho \times G_\rho \to \mathbb{T}$ be the cocycle of the projective extension. Then $\omega$ is a $2$-cocycle:

$$ \omega(g,h)\,\omega(gh,k) = \omega(g,hk)\,\omega(h,k) \qquad \text{for all } g, h, k \in G_\rho , $$

which is exactly the cocycle identity of Group Cohomology. Multiplying $\sigma(g)$ by a measurable scalar $c(g)$ replaces $\omega$ by the coboundary $\omega \cdot \delta c$; hence $\omega$ determines a class

$$ [\omega] \in H^2(G_\rho, \mathbb{T}) $$

in the second cohomology of $G_\rho$ with coefficients in the circle, and this class is the Mackey obstruction of the orbit. The obstruction measures the failure of the extension

$$ 1 \to N \to G_\rho \to G_\rho/N \to 1 $$

to split over the stabiliser in the relevant sense, and it is invariant under conjugation of $\rho$.

Theorem (obstruction criterion). The Mackey obstruction $[\omega]$ vanishes if and only if the representation $\rho$ of $N$ extends to an ordinary representation of $G_\rho$; in that case the irreducible representations of $G$ with orbit $\mathcal{O}$ are the representations $\operatorname{Ind}_{G_\rho}^G(\tilde\rho \otimes \tau)$ with $\tau \in \operatorname{Irr}(G_\rho/N)$. If the obstruction is nontrivial, the classification requires the twisted group algebra or, equivalently, the corresponding central extension of $G_\rho/N$ by $\mathbb{T}$.

Proof. The vanishing of a $2$-cocycle class means that the cocycle is a coboundary $\delta c$; rescaling $\sigma$ by $c^{-1}$ makes the projective representation an ordinary one, which is precisely an extension of $\rho$ to $G_\rho$. When the obstruction vanishes, the Mackey machine's classification reduces to the ordinary irreducibles of the quotient modulo the kernel, which are the $\tau \in \operatorname{Irr}(G_\rho/N)$ twisted by the extension; when it does not, the representations are the irreducible modules over the twisted group algebra, computed by the projective representation theory of $G_\rho/N$.

Example (the obstruction for the Euclidean group). For $G = SO(n)\ltimes\mathbb{R}^n$ and $\chi \in \mathbb{R}^n\smallsetminus\{0\}$ the little group is $SO(n-1)$, which meets $\mathbb{R}^n$ trivially, so the extension splits and the obstruction vanishes; the classification is by $\operatorname{Irr}(SO(n-1))$ as stated above. The obstruction is needed for extensions in which the quotient does not lift, the classical example being a group whose normal abelian subgroup has a character whose stabiliser does not split over it; the standard finite example is the quaternion group $Q_8$ with its central subgroup of order $2$, where the obstruction is the nontrivial element of $H^2(\mathbb{Z}/2\times\mathbb{Z}/2, \mathbb{T}) \cong \mathbb{Z}/2$.

The Boundary with Analysis

The completion of the induced spaces, the direct-integral decomposition over a coadjoint orbit or over a double coset space, and the identification of the little-group induction with a space of $L^2$ sections of a line bundle are Part III; the article states the constructions on the dense algebraic subspaces.
The orbit method for solvable groups and the Kirillov character formula, which expresses the character of $\pi_\mathcal{O}$ as the Fourier transform of the orbit measure, require the distribution theory of Part III.
The operator-algebraic form of the imprimitivity theorem — the Morita equivalence of $C_0(G/H)\rtimes G$ with $C^*(H)$ — is developed with the operator algebras of Topology on Linear Algebras.
What is not deferred: the imprimitivity theorem, the intertwining number theorem, the irreducibility criterion, the restriction and tensor-product formulas, the Mackey machine, the little-group method for semidirect products, the orbit correspondence for nilpotent Lie groups and the obstruction class are developed here.

Summary

A system of imprimitivity for a unitary representation $\pi$ of $G$ based on a transitive $G$-space $G/H$ is a projection-valued measure $P$ on $G/H$ with $\pi(g)P(E)\pi(g)^{-1} = P(gE)$. Mackey's imprimitivity theorem states that the induced representations $\operatorname{Ind}_H^G\rho$ are exactly the representations carrying such a system, the inducing representation being the restriction of $\pi$ to $H$ on the range of $P(H)$. The theorem has a $C^*$-algebraic form as a Morita equivalence between $C_0(G/H)\rtimes G$ and $C^*(H)$.

Mackey's intertwining number theorem computes the intertwiners between two induced representations as a sum over the double cosets of Hom spaces on the intersections, and his irreducibility criterion states that $\operatorname{Ind}_H^G\rho$ is irreducible exactly when $\rho$ is irreducible and is disjoint from all its conjugates by elements outside $H$. The restriction formula expresses $(\operatorname{Ind}_{H_1}^G\rho)|_{H_2}$ as a sum over the double cosets $H_2\backslash G/H_1$ of inductions from the intersections, and the tensor product formula is the same computation for the diagonal of $G\times G$. The Mackey machine analyses an irreducible $\pi$ of a group with a closed normal subgroup $N$ by the $G$-orbit of $\pi|_N$ in $\operatorname{Irr}(N)$, the little group of a point of the orbit, and a projective extension whose cocycle has a class in $H^2(G_\rho,\mathbb{T})$, the Mackey obstruction, vanishing exactly when the little-group representation extends.

For a semidirect product $G = N\rtimes H$ with $N$ abelian the irreducibles are $\operatorname{Ind}_{N\rtimes H_\chi}^G(\chi\otimes\sigma)$ for orbits in $N^\vee$ and $\sigma$ an irreducible of the little group $H_\chi$; for a connected simply connected nilpotent Lie group, Kirillov's orbit method identifies $\operatorname{Irr}(G)$ with the coadjoint orbits by the Kirillov correspondence, and the representations are monomial. The analytic refinements belong to Analysis on Groups in Part III.

Summary of Notation

Symbol	Meaning
$G$, $N$, $H$	Locally compact group; closed normal subgroup; closed subgroup
$G/H$, $H_2\backslash G/H_1$	Homogeneous space; double coset space indexing the restriction and tensor formulas
$P$, $P(E)$	Projection-valued measure of a system of imprimitivity
$\pi(g)P(E)\pi(g)^{-1} = P(gE)$	Imprimitivity condition
$\operatorname{Ind}_H^G\rho$	Induced representation, of Induced Representations of Locally Compact Groups
$\rho^g(x) = \rho(g^{-1}xg)$	Conjugate of a representation of $H \cap gHg^{-1}$
$\operatorname{Irr}(G)$	Unitary dual of $G$
$G_\rho = \{g : \rho^g \cong \rho\}$	Little group (stabiliser) of $\rho \in \operatorname{Irr}(N)$
$H_\chi$	Little group of a character $\chi \in N^\vee$ for $G = N\rtimes H$
$\omega(g,h)$	Cocycle of a projective representation
$[\omega] \in H^2(G_\rho,\mathbb{T})$	Mackey obstruction, in group cohomology with circle coefficients
$N^\vee$	Pontryagin dual of an abelian normal subgroup
$\mathcal{O}$	Coadjoint orbit in $\mathrm{G}^*$, or orbit in $\operatorname{Irr}(N)$
$\omega_\xi(X,Y) = \langle\xi,[X,Y]\rangle$	Kirillov–Kostant form on a coadjoint orbit
$\pi_\mathcal{O}$	Kirillov representation attached to a coadjoint orbit
$\pi_\lambda$	Stone–von Neumann representation of the Heisenberg group
$\operatorname{Ind}_{G_\rho}^G(\sigma)$	Mackey's description of $\pi$ from the little group
$\mathbb{T}$	The circle group as coefficients, identified with $S^1$