Stone–von Neumann uniqueness for finite-dimensional regular Weyl systems

Fix a symplectic basis from [F1], identify $V$ with ${\mathbb{R}}^n\times {\mathbb{R}}^n$, and define$\pi (v,t):=e^{i\lambda t}W(v)((v,t)\in H(V,\omega ))\mathrm{.}$The Weyl relation and the group product in [F2] give$\pi (v,t)\pi (w,s)=e^{i\lambda (t+s)-i\lambda \omega (v,w)\mathrm{/}2}W(v+w)=\pi (v+w,t+s-\omega (v,w)\mathrm{/}2),$so $\pi$ is a unitary group representation with central character $\pi (0,t)=e^{i\lambda t}I$. It is strongly continuous: if $v={\sum }_{j=1}^n(q_j{e}_j+p_j{f}_j)$, repeatedly apply the Weyl relation in this fixed coordinate order to express $W(v)$ as a finite product of $W(q_j{e}_j)$ and $W(p_j{f}_j)$ times a scalar phase. That phase is an exponential of a finite sum of bilinear coordinate terms, hence is continuous; each line factor is strongly continuous by regularity. Finite products of strongly continuous unitaries are strongly continuous by inserting and subtracting one factor at a time. Its invariant closed subspaces are exactly those of $W$, since the central factor is scalar, so $\pi$ is irreducible.

The representation space is separable. Choose $\xi \mathrm{\ne }0$. The closed span of $\pi (G)\xi$ is a nonzero invariant subspace, hence equals $\mathcal{H}$ by irreducibility. The group $G=H(V,\omega )$ is homeomorphic to ${\mathbb{R}}^{2n+1}$, so the rational-coordinate points give a countable dense subset $D$. Strong continuity makes $\{\pi (g)\xi :g\in D\}$ dense in the orbit of $\xi$; its finite rational-complex linear span is countable and dense in $\mathcal{H}$. Therefore [F3] applies.

Write $G=N⋊K$ as in [F2], and let $\tau (q)=\pi (q,0,0)$ denote the representation of $K$. Conjugation in the semidirect product gives $\tau (q)\pi (n)\tau (q{)}^{-1}=\pi ({\alpha }_q(n))$ for every $n\in N$. By [F3] there is a unique regular PVM $P$ on $\widehat{N}$ with$\pi (n)={\int }_{\widehat{N}}\chi (n)dP(\chi ),\tau (q)P(E)\tau (q{)}^{-1}=P(q\cdot E)\mathrm{.}$

The central subgroup is $z_t=(0,0,t)$, and $\pi (z_t)=e^{i\lambda t}I$. For each rational $t$, let $E_t=\{\chi \in \widehat{N}:\chi (z_t)=e^{i\lambda t}\}$, a Borel set. The SNAG formula in [F3] gives ${\mathrm{\Phi }}_P(\chi \mapsto \chi (z_t)-e^{i\lambda t})=0$. By the norm-square identity in [F9], each scalar measure $P_{\xi }$ assigns integral zero to $\mathrm{\mid }\chi (z_t)-e^{i\lambda t}{\mathrm{\mid }}^2$, so it is supported on $E_t$ and $P(E_t)=I$. Since $\mathbb{Q}$ is countable and $P$ is countably additive, $P({\bigcap }_{t\in \mathbb{Q}}{E}_t)=I$. Every character of $N\cong {\mathbb{R}}^{n+1}$ is continuous in its real parameter, so equality on rational $t$ implies that this intersection is precisely ${\widehat{N}}_{\lambda }$ from [F2]. Hence $P({\widehat{N}}_{\lambda })=I$.

The orbit map $q\mapsto q\cdot {\chi }_{0,\lambda }={\chi }_{\lambda q,\lambda }$ is a bijection by [F2]. It is a homeomorphism for the compact-open topology: $\mathrm{\mid }{e}^{iu}-e^{iv}\mathrm{\mid }\le \mathrm{\mid }u-v\mathrm{\mid }$ gives uniform convergence on each compact subset of $N$ when the frequency parameters converge. Conversely, for $\delta >0$ and any frequency coordinate, uniform distance less than $1$ from a fixed character on the compact coordinate segment $\mathrm{\mid }t\mathrm{\mid }\le \pi \mathrm{/}\delta$ forces the corresponding frequency difference $b$ to satisfy $\mathrm{\mid }b\mathrm{\mid }<\delta$: otherwise $t=\pi \mathrm{/}b$ lies in that segment and the distance is $2$. The finite-product dual identification in [F2] therefore has continuous inverse, and scaling by $\lambda \mathrm{\ne }0$ identifies $G\mathrm{/}N\cong K$ homeomorphically with ${\widehat{N}}_{\lambda }$. Since $G\mathrm{/}N\cong K\cong {\mathbb{R}}^n$, [F10] gives its standard Borel structure and Borel $G$-action. Push $P$ through this Borel isomorphism to obtain a PVM on $G\mathrm{/}N$. Its covariance is the second identity in step 2.2, and its total projection is $I$ by step 3.1. Thus $(\pi ,P)$ is a transitive system of imprimitivity on $G\mathrm{/}N$ in the sense of [F4]. Mackey's theorem yields a strongly continuous unitary representation $\sigma :N\to U(K_0)$ on a separable Hilbert space and a unitary intertwiner from $\pi$ to ${\mathrm{Ind}}_N^G\sigma$.

Let $s(q)=(q,0,0)$ be the section of $G\mathrm{/}N$. The normalized measure $\mu$ in the induced model is Radon and finite on compact sets by [F4]. Under $G\mathrm{/}N\cong {\mathbb{R}}^n$, the compact boxes $[-m,m{]}^n$ (or the singleton when $n=0$) form a countable cover, so $\mu$ is $\sigma$-finite. The base is standard Borel by [F10], and the constant fiber $K_0$ is separable, so the hypotheses of [F7] hold. For each $n\in N$, Mackey's PVM identification in [F4] sends $P(E)$ to multiplication by ${\mathbf{1}}_E$; the bounded Borel integral result [F9] therefore sends the SNAG integral of $\chi \mapsto \chi (n)$ to multiplication by $q\mapsto {\chi }_{\lambda q,\lambda }(n)$. In the induced model the action of $n$ on the fiber at $q$ is $\sigma (s(q{)}^{-1}ns(q))=\sigma ({\alpha }_{-q}(n))$ by [F5]. Equality of the two multiplication operators and [F7] give$\sigma ({\alpha }_{-q}(n))={\chi }_{\lambda q,\lambda }(n)I_{K_0}\text{for almost every }q$. In the coordinates $N={\mathbb{R}}^n\times \mathbb{R}$, set $D_N={\mathbb{Q}}^n\times \mathbb{Q}$, a countable dense subset. Intersecting the corresponding conull sets gives one conull set $E\subset K$ on which the equality holds for every $n\in D_N$. The normalized measure class on $G\mathrm{/}N$ has full support by [F4], so $E$ is dense. For fixed $n\in D_N$, both sides are continuous in $q$; equality on $E$ extends to $q=0$, giving $\sigma (n)={\chi }_{0,\lambda }(n)I_{K_0}$. Strong continuity of $\sigma$ and continuity of ${\chi }_{0,\lambda }$ extend this equality from $D_N$ to all $N$.

If $K_0$ had a nonzero proper closed subspace $L$, then [F11] gives $K_0=L\oplus L^{\perp }$ with $L^{\perp }\mathrm{\ne }0$. Both subspaces are invariant under the scalar action $\sigma (n)={\chi }_{0,\lambda }(n)I$. In the induced model the sections valued in $L$ form a closed invariant subspace: the pointwise orthogonal projection onto $L$ is bounded, and the scalar fiber action preserves it. By [F4], choose a nonempty relatively compact open subset $U\subseteq G\mathrm{/}N$; full support and Radon finiteness give $0<\mu (U)<\mathrm{\infty }$. For nonzero $\mathrm{\ell }\in L$ and $r\in L^{\perp }$, the sections ${\mathbf{1}}_U\mathrm{\ell }$ and ${\mathbf{1}}_{U}r$ are nonzero elements of the two respective subspaces. This contradicts irreducibility of $\pi$. Thus $K_0$ is one-dimensional and $\sigma$ is equivalent to ${\chi }_{0,\lambda }$. The Schrödinger model in [F8] is strongly continuous and irreducible on the second-countable group $G$; choosing a nonzero vector and using a countable dense subset of $G$ as in step 2.1 shows that $L^2({\mathbb{R}}^n)$ is separable. Its position projections $P_0(E):=M_{{\mathbf{1}}_E}$ form a PVM by [F12], and covariance follows because translations shift these projections while modulation operators commute with them. Translations act transitively on $G\mathrm{/}N$, so $({\pi }_0,P_0)$ is a transitive system on the standard Borel quotient from [F10]. Mackey's theorem [F4] induces it from some strongly continuous ${\sigma }_0:N\to U(K_1)$. For $n=(0,p,t)\in N$, the explicit Schrödinger formula acts by multiplication by ${\chi }_{\lambda x,\lambda }(n)$. In the induced model, [F5] gives the fiber field ${\sigma }_0({\alpha }_{-x}(n))$. Since the intertwiner also identifies $P_0$ with multiplication by indicators, [F7] shows these fields agree almost everywhere. The dense-set and continuity argument of step 5.1 gives ${\sigma }_0(n)={\chi }_{0,\lambda }(n)I_{K_1}$ for all $n\in N$. The nonzero Schrödinger space forces $K_1\mathrm{\ne }0$; the same invariant-subspace argument used above for $K_0$, now using irreducibility from [F8], forces $\dim K_1=1$. Thus ${\sigma }_0$ is equivalent to the one-dimensional character ${\chi }_{0,\lambda }$.

The uniqueness clause [F6] now identifies the transitive systems of $\pi$ and the Schrödinger model, because their inducing representations are both equivalent to ${\chi }_{0,\lambda }$. Hence the underlying unitary representations are unitarily equivalent; restricting the intertwiner to $(v,0)$ gives unitary equivalence of the Weyl systems. Existence and irreducibility of the model are [F8], so this proves the unique equivalence class for the fixed $\lambda \mathrm{\ne }0$. The argument does not apply to zero central parameter or to infinite-dimensional phase space.