文档介绍：Journal of Nonlinear Mathematical Physics 2001, , N 1, 31–37 Letter
On Bilinear Invariant Differential Operators
Acting on Tensor Fields on the Symplectic
Manifold
Pavel GROZMAN
Department of Mathematics, University of Stockholm
Roslagsv. 101, Kr¨aftriket hus 6, S-106 91, Stockholm, Sweden
E-mail: ******@
Received May 6, 2000; Revised August 30, 2000; Accepted October 3, 2000
Abstract
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.
Let ρ be a representation of the group Sp(2m; R) field of type ρ on
a2m-dimension