On bilinear invariant differential operators acting on tensor fields on the symplectic manifold

Let M be an n-dimensional manifold, V the space of a representation rho : G L( n) --> GL ( V). Locally, let T(V) be the space of sections of the tensor bundle with ber V over a sufficiently small open set U subset of M, in oth er words, T(V) is the space of tensor fields of type V on M on which the gr oup Diff(M) of diffeomorphisms of M naturally acts. Elsewhere, the author c lassified the Diff(M)-invariant differential operators D : T(V-1) circle ti mes T(V-2) --> T(V-3) for irreducible fibers with lowest weight. Here the r esult is generalized to bilinear operators invariant with respect to the gr oup Diff(omega) ( M) of symplectomorphisms of the symplectic manifold ( M,o mega). We classify all rst order invariant operators; the list of other ope rators is conjectural. Among the new operators we mention a 2nd order one w hich determines an algebra structure on the space of metrics (symmetric for ms) on M.