  We systematically develop a transform of the Fourier-Mukai type for sheaves
on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One
obtains a bijective correspondence between unitary local systems supported on
Lagrangian submanifolds of $X$ and holomorphic vector bundles with compatible
unitary connections supported on complex submanifolds of the relative Jacobian
of $X$ (suitable conditions being verified on both sides).
