  We consider the space of polydifferential operators on n functions on
symplectic manifolds invariant under symplectic automorphisms, whose study was
initiated by Mathieu in 1995. Permutations of inputs yield an action of S_n,
which extends to an action of S_{n+1}. We study this structure viewing n as a
parameter, in the sense of Deligne's category. For manifolds of dimension 2d,
we show that the isotypic part of this space of <= 2d+1-th tensor powers of the
reflection representation h=C^n of S_{n+1} is spanned by Poisson polynomials.
We also prove a partial converse, and compute explicitly the isotypic part of
<= 4-th tensor powers of the reflection representation.
  We give generating functions for the isotypic parts corresponding to Young
diagrams which only differ in the length of the top row, and prove that they
are rational functions whose denominators are related to hook lengths of the
diagrams obtained by removing the top row. This also gives such a formula for
the same isotypic parts of induced representations from Z/(n+1) to S_{n+1}
where n is viewed as a parameter.
  We apply this to the Poisson and Hochschild homology associated to the
singularity C^{2dn}/S_{n+1}. Namely, the Brylinski spectral sequence from the
zeroth Poisson homology of the S_{n+1}-invariants of the n-th Weyl algebra of
C^{2d} with coefficients in the whole Weyl algebra degenerates in the 2d+1-th
tensor power of h, as well as its fourth tensor power. Furthermore, the kernel
of this spectral sequence has dimension on the order of 1/n^3 times the
dimension of the homology group.
