Mackeys Imprimitivity Theorem — Examples

The operator $U(s)$ is a unitary swap with $U(s{)}^2=I$, so it defines a unitary representation of the discrete group $G$; every orbit map from this discrete group is continuous. Covariance: $U(s)$ fixes ${\delta }_a$ and swaps ${\delta }_b,{\delta }_c$, so $U(s)P(\{a\})U(s{)}^{-1}=P(\{a\})=P(s\cdot \{a\})$, $U(s)P(\{b\})U(s{)}^{-1}=P(\{c\})=P(s\cdot \{b\})$, and $U(s)P(\{c\})U(s{)}^{-1}=P(\{b\})=P(s\cdot \{c\})$; for the identity the identity is trivial, and covariance extends to all subsets since the three singletons generate the power set and both sides are PVM-valued. Hence $(U,P)$ is a system of imprimitivity; it is defined on the standard Borel three-point space of [F3].

Invariant projections and non-ergodicity: $P(\{a\})$ and $P(\{b,c\})=P(\{b\})+P(\{c\})$ are nonzero and different from $I$, and both are invariant under $U$, since the orbits are $\{a\}$ and $\{b,c\}$; thus the system is not ergodic.

Nontransitivity: the orbits of the action are the singleton $\{a\}$ and the two-point set $\{b,c\}$, while a homogeneous space of $\mathbb{Z}\mathrm{/}2$ has one or two points by [F2]; a transitive system on a homogeneous space is concentrated on a single orbit, so the three-point base with two orbits cannot be equivariantly identified with any $G\mathrm{/}H$. Hence the system is not transitive.

Correct decomposition: $P(\{a\})$ and $P(\{b,c\})$ are complementary invariant projections, and on their ranges the system restricts to the transitive system on the single orbit $\{a\}$ (the one-point homogeneous space $G\mathrm{/}G$ with the trivial representation) and to the transitive system on $\{b,c\}$ (the two-point homogeneous space $G\mathrm{/}\{e\}$ with the regular representation), respectively. So $(U,P)$ is the direct sum of the two transitive systems, and the failure above is exactly the failure of a direct sum of transitive systems to be classified by one subgroup.

Non-classification by one subgroup: the classification data $(H,\sigma )$ determine a system whose base is the homogeneous space $G\mathrm{/}H$, of one or two points by [F2], and whose imprimitivity measure is concentrated on the orbits of that base; no such data can reproduce the three-point base with two orbits, since equivariant Borel isomorphisms preserve cardinalities and orbit counts. Therefore the nontransitive system is not classified by a single closed subgroup and a representation of it.

The explicit finite computation therefore exhibits a system of imprimitivity that is neither transitive nor ergodic and is not classified by one stabilizer subgroup; transitivity (or ergodicity) is essential to the one-subgroup form of Mackey's imprimitivity theorem.

For $n=0$, the base and group are singletons, $L^2({\mathbb{R}}^0)=\mathbb{C}$ with unit mass, $U=I$ and $P$ is the one-point PVM; induction from the trivial group gives this system, and there are no coordinate operators. Thus all claims hold in that case. Assume $n\ge 1$ for the remaining steps. Then $U$ is a strongly continuous unitary representation and $P$ is a projection-valued measure by [F1] and [F2]. Covariance is a direct computation from $(U_{t}f)(x)=f(x-t)$: $U_{t}P(E)U_t^{-1}=M_{{\mathbf{1}}_E(\cdot -t)}=P(t+E)$. The action is transitive and the stabilizer of the origin is $\{0\}$ by [F3], so the base is ${\mathbb{R}}^n\mathrm{/}\{0\}={\mathbb{R}}^n$ with the trivial subgroup.

For each $j$, $M_{x_j}$ has domain $\{f\in L^2:x_{j}f\in L^2\}$ and is an unbounded self-adjoint real multiplication operator by the multiplier result of [F5]. Its spectral projections are $M_{{\mathbf{1}}_{\{x:x_j\in A\}}}$ for Borel $A\subseteq \mathbb{R}$. They commute, and the joint multiplication PVM is $P$; the unbounded coordinate integral $\int x_{j}dP$ equals $M_{x_j}$ on its stated domain, while bounded Borel functions of the coordinates act by the bounded integrals of [F2].

$U$ is induced from the trivial representation of the trivial subgroup: by the $H=\{e\}$ clause of [F4], ${\mathrm{Ind}}_{\{0\}}^{{\mathbb{R}}^n}1$ is the left regular representation on $L^2({\mathbb{R}}^n)$, and the canonical system of that induction is the multiplication system $P$.

For Schwartz $f$, the difference quotient $(U_{t{e}_j}f-f)\mathrm{/}t$ tends in $L^2$ to $-{\mathrm{\partial }}_{j}f$, by the fundamental theorem of calculus and a Schwartz majorant. To identify the full domain, apply the unitary Fourier transform of [F5]: coordinate translation becomes multiplication by $e^{-2\pi it{\xi }_j}$ in the usual Fourier normalization. The derivative limit exists precisely when ${\xi }_j\widehat{f}\in L^2$: sufficiency follows from $\mathrm{\mid }(e^{-2\pi it{\xi }_j}-1)\mathrm{/}t\mathrm{\mid }\le 2\pi \mathrm{\mid }{\xi }_j\mathrm{\mid }$ and dominated convergence, and necessity from an almost-everywhere convergent subsequence of any $L^2$ limit of the quotients, whose pointwise limit is $-2\pi i{\xi }_j\widehat{f}$. Define $P_j={\mathcal{F}}_2^{-1}{M}_{2\pi {\xi }_j}{\mathcal{F}}_2$ on $D_j=\{f:{\xi }_j{\mathcal{F}}_{2}f\in L^2\}$. The multiplier theorem makes it self-adjoint and its transported exponential is $U_{t{e}_j}=e^{-it{P}_j}$. Thus $G_j=-i{P}_j$ and $T_j=-i{G}_j=-P_j$ on the full domain $D_j$. On Schwartz functions the Fourier differentiation identity gives $P_j=-i{\mathrm{\partial }}_j$, recovering $G_j=-{\mathrm{\partial }}_j$ and $T_j=i{\mathrm{\partial }}_j$ there. No pointwise derivative of a general $L^2$ class is used.

Standard Borel and Polish: by [F6] the metric $d_{\mathrm{\infty }}$ makes ${\mathbb{R}}^n$ complete with countable dense subset ${\mathbb{Q}}^n$, so ${\mathbb{R}}^n$ is Polish and its Borel space is standard Borel, while the singleton case $n=0$ uses the zero metric as in step 1.1; the base with its standard Borel structure is the one used by the system.

Steps 1.1, 1.2, 1.3, 1.4 and 2.1 verify all the displayed claims: transitivity with trivial stabilizer, the PVM as joint spectral measure of position, the induced-representation identification, the momentum generators, and the standard-Borel base.

$(U,P)$ is a system of imprimitivity: by [F1] $U$ is a strongly continuous unitary representation on the finite-dimensional space ${\mathrm{\ell }}^2(X)$, and by [F2] $P$ is a projection-valued measure with $UP(E)U^{-1}=P(gE)$ for all $g$ and all $E\subseteq X$. The action is transitive, so the system is transitive on the finite homogeneous space $G\mathrm{/}H$ by [F3], which is a standard Borel space by [F1].

The equivalence preserves both parts of the system. For $v\in {\mathrm{\ell }}^2(X)$ set $F_v(g)=v(g{x}_0)$. Then $F_v(gh)=F_v(g)$ for $h\in H$, and ${\sum }_{gH}\mathrm{\mid }{F}_v(g){\mathrm{\mid }}^2={\sum }_{x\in X}\mathrm{\mid }v(x){\mathrm{\mid }}^2$, so this is a unitary onto the covariant model of ${\mathrm{Ind}}_H^G{1}_H$ with counting quotient measure. It sends $(U_{a}v)(x)=v(a^{-1}x)$ to $F_v(a^{-1}g)$ and sends $P(E)$ to multiplication by ${\mathbf{1}}_E(g{x}_0)$. Thus the permutation system is the canonical induced system of $1_H$, not merely an equivalent group representation.

Fibre dimension: for a finite-dimensional unitary representation carrying a transitive system, the fibres $P(\{x\})\mathcal{H}$ are mutually orthogonal (the singletons are disjoint) and sum to $\mathcal{H}$; transitivity of $U$ transports $P(\{x\})$ to $P(\{gx\})$, so all fibres have the same dimension $d$; hence $\mathrm{\mid }X\mathrm{\mid }d=\dim \mathcal{H}$ and $d=\dim (\mathcal{H})\mathrm{/}\mathrm{\mid }X\mathrm{\mid }$. The general theorem identifies the representation with the induction of a unitary representation of $H$ on one fibre, of that dimension; the induced space has the original total dimension.

Steps 1.1, 2.1 and 3.1 verify the claims: the permutation system is a transitive system of imprimitivity on the finite standard Borel space, it is equivalent to the canonical system of the trivial representation of the stabilizer, the finite computation of the induction is the permutation representation, and the fibre dimension of a general finite-dimensional transitive system is $\dim (\mathcal{H})\mathrm{/}\mathrm{\mid }X\mathrm{\mid }$.

The dual action is $(k\cdot {\chi }_{\lambda })(b)=e^{i\lambda b\mathrm{/}k}$, so the parameter is $\lambda \mathrm{/}k$. For $K={\mathbb{R}}^{\times }$ the orbits are $\{0\}$ and ${\mathbb{R}}^{\times }$, with stabilizers $K$ and $\{1\}$; for $K={\mathbb{R}}_{>0}$ they are $\{0\}$, $(0,\mathrm{\infty })$, and $(-\mathrm{\infty },0)$, again with trivial nonzero stabilizers. Each partition is finite and Borel, hence regular, so [F3] makes the corresponding little-group inductions exhaustive.

At the zero character the inducing subgroup is $G$ and $N$ acts trivially. The irreducible quotient representations are one-dimensional by [F5]. Logarithm and the sign decomposition ${\mathbb{R}}^{\times }\cong {\mathbb{R}}_{>0}\times \{\pm 1\}$ give $\mathrm{\mid }a{\mathrm{\mid }}^{it}\mathrm{sign}(a{)}^{\delta }$ for the full group and $a^{it}$ for the positive group. Distinct parameters give distinct characters: vary $\log a$ and then the sign.

For $\lambda \mathrm{\ne }0$ use quotient coordinates $k\in K$ with section $s(k)=(0,k)$ and Haar measure $dk\mathrm{/}\mathrm{\mid }k\mathrm{\mid }$ (restricted to $k>0$ for the positive group). The induced action on $L^2(K,dk\mathrm{/}\mathrm{\mid }k\mathrm{\mid })$ is $({\pi }_{\lambda }(b,a)f)(k)=e^{i\lambda b\mathrm{/}k}f(k\mathrm{/}a)\mathrm{.}$ Indeed $s(k{)}^{-1}(b,a)s(k\mathrm{/}a)=(b\mathrm{/}k,1)$, and the quotient measure is invariant under $k\mapsto ak$, so its density factor is one. Finite scalar Borel measures on the real line are regular by [F4]. The spectral PVM of $N$ is multiplication by ${\mathbf{1}}_E(\lambda \mathrm{/}k)$: it is a regular PVM under the homeomorphism $k\mapsto \lambda \mathrm{/}k$ onto the nonzero orbit, and its character integral is the displayed translation action, so [F4]'s spectral uniqueness identifies it.

Let $A$ commute with ${\pi }_{\lambda }$. It commutes with all integrated $N$-operators, hence with their $C_0$ algebra by Fourier-transform density, and then with the spectral PVM. For the last inference, each unitary in the commutant conjugates $P$ to a regular PVM with the same integrated $N$-representation, hence preserves $P$ by spectral uniqueness. For a general commutant operator, its self-adjoint real and imaginary parts commute with $N$, and $e^{itS}$ for either part $S$ is a commuting unitary, by the norm-convergent power series. Differentiating that series at $t=0$ shows that $S$ commutes with $P$, and hence so does $A$. Thus $A$ commutes with all diagonal multiplications on $K$ and is $M_u$ for some bounded scalar $u$, by the diagonal commutant theorem. Commutation with ${\pi }_{\lambda }(0,a)$ makes $u(k\mathrm{/}a)=u(k)$ a.e. for every $a\in K$. Transitive ergodicity, applied to rational superlevel sets of the real and imaginary parts, makes $u$ constant a.e. Thus the commutant is scalar; an invariant closed subspace would have a commuting orthogonal projection, so ${\pi }_{\lambda }$ is irreducible. Disjoint intervals in $\log \mathrm{\mid }k\mathrm{\mid }$ give infinitely many nonzero orthogonal indicator sections, proving infinite dimension.

For $r\in K$ put $(R_{r}f)(k)=f(rk)$. Haar invariance makes $R_r$ unitary, and the explicit formula gives $R_r{\pi }_{\lambda }(b,a)R_r^{-1}={\pi }_{\lambda \mathrm{/}r}(b,a)$. Thus parameters in one orbit give equivalent representations. For the full group $r=-1$ equates $\lambda =1$ and $\lambda =-1$; for the positive group no $r$ changes sign, and the two classes are inequivalent because their spectral PVMs have disjoint supports, by [F4]'s uniqueness. They are also inequivalent to the quotient characters, whose $N$-spectrum is $\{0\}$.

Exhaustiveness from step 1.1, the character classification of step 2.1, irreducibility and infinite dimension from step 3.1, and the equivalences of step 4.1 give $\widehat{G}=(\mathbb{R}\times \{0,1\})\bigsqcup \{\pi \}$ and $\widehat{G_0}=\mathbb{R}\bigsqcup \{{\pi }_1,{\pi }_{-1}\}$.