Regular Weyl systems and irreducibility

Fix a finite-dimensional real symplectic vector space $(V,\omega )$ and a real parameter $\lambda$. A Weyl system with parameter $\lambda$ on a nonzero complex Hilbert space $\mathcal{H}$ is a map $W:V\to U(\mathcal{H})$ such that $W(0)=I,W(v)W(w)=e^{-i\lambda \omega (v,w)\mathrm{/}2}W(v+w)(v,w\in V)\mathrm{.}$ It is regular when, for every $v\in V$, the map $t\mapsto W(tv)$ is strongly continuous on $\mathbb{R}$. It is irreducible when the only closed subspaces invariant under every $W(v)$ are $\{0\}$ and $\mathcal{H}$, as in Strongly continuous unitary representations, invariant linear subspaces and intertwiners.

Each $W(v)$ is unitary. Substituting $w=-v$ in the relation gives $W(v)W(-v)=I$, because $\omega (v,-v)=0$; hence $W(v{)}^{\ast }=W(-v)$. The scalar $\lambda$ is the central character parameter: the associated representation of The Heisenberg group of a finite-dimensional symplectic vector space acts on $(0,t)$ by $e^{i\lambda t}I$.