Algebras of entire functions and representations of the twisted Heisenberg group

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