SYMPLECTIC REDUCTION OF A LAGRANGIAN FOLIATION IN A NEIGHBORHOOD OF A COMPACT LEAF

Let (M, OMEGA, L) be a 2 n-symplectic manifold endowed with a Lagrangian foliation, and (L0, del0) a compact leaf with its flat affine Weinstein's connection. One assumes that the holonomy of L0 is linearisable. One shows that if T(k) acts by translations on (L0, del0), then this action extends in a neighborhood of L0 as an hamiltonian action along the leaves. Application is given to the problem of symplectic linearisation of C in a neighborhood of L0.