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On the twisted Fock spaces $ \mathcal{F}^\lambda(\C^{2n}) $ we consider afamily of unitary operators $\rho_\lambda(a,b) $ indexed by $ (a,b) \in \C^n\times \C^n.$ The composition formula for $ \rho_\lambda(a,b) \circ\rho_\lambda(a^\prime,b^\prime) $ leads us to a group $ \He^n_\lambda(\C) $which contains two copies of the Heisenberg group $ \He^n.$ The operators $\rho_\lambda(a,b) $ lift to $ \He_\lambda^n(\C) $ providing an irreducibleunitary representation. However, its restriction to $ \He^n_\lambda(\R) $ isnot irreducible.

Algebras of entire functions and representations of the twisted Heisenberg group