Heisenberg group polarization and the fixed-central-character orbit

In coordinates from [F2], bilinearity gives $\omega ((q,p),(q^{\mathrm{\prime }},p^{\mathrm{\prime }}))=q\cdot p^{\mathrm{\prime }}-p\cdot q^{\mathrm{\prime }}$. For three vectors $v,w,u$,$\omega (v,w)+\omega (v+w,u)=\omega (v,w)+\omega (v,u)+\omega (w,u)=\omega (w,u)+\omega (v,w+u)\mathrm{.}$Thus $c(v,w)=-\frac{1}{2}\omega (v,w)$ satisfies the normalized 2-cocycle identity $c(v,w)+c(v+w,u)=c(w,u)+c(v,w+u)$. Substitution in the group law proves associativity; $\omega (v,0)=\omega (0,v)=0$ gives identity $(0,0)$, and $\omega (v,-v)=0$ gives inverse $(-v,-t)$. Addition and $\omega$ are continuous, so multiplication and inverse are continuous. The underlying finite-dimensional Euclidean space $V\times \mathbb{R}$ is locally compact and second countable.

An element $(v,t)$ commutes with every $(w,0)$ exactly when $\omega (v,w)=0$ for all $w$. Nondegeneracy in [F1] forces $v=0$, and each $(0,t)$ is central by the product formula. Hence the center is the displayed central coordinate. The projection $(v,t)\mapsto v$ is onto with kernel $\{(0,t):t\in \mathbb{R}\}$, proving exactness of the central extension in the definition.

For $q\in K$ and $(p,t)\in N$, direct multiplication gives $(q,0,0)(0,p,t)(-q,0,0)=(0,p,t-q\cdot p)$; thus ${\alpha }_q(p,t)=(p,t-q\cdot p)$. This is an automorphism because it is additive, has inverse ${\alpha }_{-q}$, and ${\alpha }_{q+q^{\mathrm{\prime }}}={\alpha }_q{\alpha }_{q^{\mathrm{\prime }}}$. Every group element has the unique factorization$(q,p,z)=(0,p,z-q\cdot p\mathrm{/}2)(q,0,0)\mathrm{.}$Therefore $(n,q)\mapsto n(q,0,0)$ is a bijection from the semidirect product of [F4] to $H(V,\omega )$; the conjugation computation makes it a homomorphism. Its coordinate formula and inverse are continuous, so it is a topological group isomorphism.

Apply [F3] to $N\cong {\mathbb{R}}^{n+1}$. Rescaling the coordinates by $2\pi$ gives every character exactly once as ${\chi }_{a,\mu }(p,t)=e^{i(a\cdot p+\mu t)}$. Since ${\alpha }_{-q}(p,t)=(p,t+q\cdot p)$, composition yields$q\cdot {\chi }_{a,\mu }={\chi }_{a+\mu q,\mu }\mathrm{.}$

The frequency parametrization in step 3.1 is a homeomorphism for the compact-open topology of [F5]. First consider ${\beta }_b(t)=e^{ibt}$ on $\mathbb{R}$. On any nonempty compact set $C\subset \mathbb{R}$, put $M={\max }_{t\in C}\mathrm{\mid }t\mathrm{\mid }$; the estimate $\mathrm{\mid }{e}^{ibt}-e^{i{b}_{0}t}\mathrm{\mid }\le \mathrm{\mid }b-b_0\mathrm{\mid }M$ proves continuity uniformly on $C$. Conversely, for any $\delta >0$, require uniform distance less than $1$ from ${\beta }_{b_0}$ on the compact interval $[-\pi \mathrm{/}\delta ,\pi \mathrm{/}\delta ]$. If $\mathrm{\mid }b-b_0\mathrm{\mid }\ge \delta$, then $t=\pi \mathrm{/}(b-b_0)$ lies in that interval and $\mathrm{\mid }{e}^{i(b-b_0)t}-1\mathrm{\mid }=2$, a contradiction. This compact-open neighborhood therefore forces $\mathrm{\mid }b-b_0\mathrm{\mid }<\delta$ and proves inverse continuity. The finite-product dual homeomorphism in [F3] now proves the full parameter map $(a,\mu )\mapsto {\chi }_{a,\mu }$ is a homeomorphism ${\mathbb{R}}^{n+1}\to \widehat{N}$; the rescaling by $2\pi$ is a Euclidean homeomorphism. Restricting to $\mu =\lambda$ gives the corresponding orbit subspace topology, including $n=0$, where the fixed-central-character orbit is a singleton.

When $\mu =\lambda \mathrm{\ne }0$, the map $q\mapsto a+\lambda q$ is a bijection of ${\mathbb{R}}^n$ with itself. It is continuous with continuous inverse $a\mapsto (a-a_0)\mathrm{/}\lambda$, so the orbit through $(a_0,\lambda )$ is all of ${\widehat{N}}_{\lambda }$ and its stabilizer is $\{0\}$. This proves the free transitive orbit and stabilizer claims.

Steps 1.1–5.1 establish the group, topological, central, semidirect, and dual-orbit assertions, including the zero-dimensional case $n=0$ where the Lagrangian subgroup $K$ is trivial.