Structures symplectiques, structures complexes, spineurs symplectiques et transformation de Fourier

Abstract

Planck's constant is very useful in the development of the theory of symplectic Clifford algebras introduced by the author in 1977 [1,a], and to solve many connected problems for example the Poisson Lie algebra deformations [1,c]. In this paper we give a precise link between a complex structure J and the Fourier transform which is nothing but the natural left action of the covering J̃ of J in a symplectic convenient spinor space (modulo a constant factor). Thus Fourier transform becomes a geometric transformation separated from integration technics, good peculiarity for global problems. We explain nice algebraic properties of the Fourier transform taking them in the symplectic context with adapted metric in any signature. Some applications are given: Hermite's functions, Plancherel-Parseval's theorem, covariance problemes … . Our approach is particularly convenient for explain results in Maslov's theory [1,b] and the difficulties in defining a global Fourier transform over a symplectic manifold.