Algebras of entire functions and representations of the twisted Heisenberg group

On the twisted Fock spaces ${\mathcal{F}}^{\lambda }({\text{C}}^{2n})$ we consider a family of unitary operators ${\rho }_{\lambda }(a,b)$ indexed by $(a,b)\in {\text{C}}^n\times {\text{C}}^n\mathrm{.}$ The composition formula for ${\rho }_{\lambda }(a,b)\circ {\rho }_{\lambda }(a^{\mathrm{\prime }},b^{\mathrm{\prime }})$ leads us to a group ${\text{He}}_{\lambda }^n(\text{C})$ which contains two copies of the Heisenberg group ${\text{He}}^n\mathrm{.}$ The operators ${\rho }_{\lambda }(a,b)$ lift to ${\text{He}}_{\lambda }^n(\text{C})$ providing an irreducible unitary representation. However, its restriction to ${\text{He}}_{\lambda }^n(\mathbb{R})$ is not irreducible.