Symplectic Structures, Complex, Structures, Symplectic Spinors and the Fourier Transform

Abstract

We have directed our attention to the problems raised by the Fourier transform on curved symplectic manifolds. The generalization of the remarkable tool offered by this transform in Rⁿ to such manifolds is awkward and leads to the difficulty known as the ‘Maslov class’ or ‘index’. It appears that a geometrization of this transform is required before it can be turned into a global tool. In the process, the complex structure J adapted to the symplectic form and to an associated pseudo-riemannian form will prove to be closely linked with the Fourier transform, which can be identified, up to some constant factor, with the lifting J of J to a symplectic spin group. Then the geometrization can be carried out at once, if a few conditions, which hold in all usual cases, are satified; the geometric Fourier transform is the natural action of J on the symplectic spinors.