Algebras of entire functions and representations of the twisted Heisenberg group

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