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 ρ : GL(n) → GL(V). Locally, let T(V) be the space of sections of the tensor bundle with fiber V over a sufficiently small open set U ⊂ M, in other words, T(V) is the space of tensor fields of type V on M on which the group Diff(M) of diffeomorphisms of M naturally acts. Elsewhere, the author classified the Diff(M)-invariant differential operators D : T(V1) ⨂ T(V2) → T(V3) for irreducible fibers with lowest weight. Here the result is generalized to bilinear operators invariant with respect to the group Diffω(M) of symplectomorphisms of the symplectic manifold (M,ω). We classify all first order invariant operators; the list of other operators is conjectural. Among the new operators we mention a 2nd order one which determins an “algebra” structure on the space of metrics (symmetric forms) on M.