Algebras of entire functions and representations of the twisted Heisenberg group

Abstract

On the twisted Fock spaces $ \mathcal{F}^λ(\C^{2n}) $ we consider a family 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_λ(\C) $ which contains two copies of the Heisenberg group $ \He^n.$ The operators ${\rho }_{\lambda }(a,b)$ lift to $ \He_λ^n(\C) $ providing an irreducible unitary representation. However, its restriction to $ \He^n_λ(\R) $ is not irreducible.

13 pages