Generalized Fourier Transforms Associated with the Oscillator Representation

Abstract

By identifying the Fourier transform $$\mathcal {F}$$ with an operator in the oscillator representation of the metaplectic group $${\widetilde{Sp}(2n,\mathbb {R})}$$ , the twofold cover of the symplectic group $$Sp(2n,\mathbb {R})$$ , we study generalized Fourier transforms inspired by the work of De Bie, Oste and Van der Jeugt. We obtain several families of operators $$\mathcal {T}$$ ’s that have the important properties similar to $$\mathcal {F}$$ using various dual pair correspondences.