  Let $M$ be an $n$-dimensional manifold, $V$ the space of a representation
$\rho: GL(n)\longrightarrow GL(V)$. Locally, let $T(V)$ be the space of
sections of the tensor bundle with fiber $V$ over a sufficiently small open set
$U\subset M$, 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 $D: T(V_{1})\otimes T(V_{2})\longrightarrow T(V_{3})$ for irreducible
fibers with lowest weight. Here the result is generalized to bilinear operators
invariant with respect to the group $\Diff_{\omega}(M)$ of symplectomorphisms
of the symplectic manifold $(M, \omega)$. 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 $M$.
