On the asymptotic Sn-structure of invariant differential operators on symplectic manifolds

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 Sn, which extends to an action of Sn+1. We study this structure viewing n as a parameter, in the sense of Deligneʼs 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=Cn of Sn+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 fractions 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 Sn+1 where n is viewed as a parameter.We show that the space of invariant operators of order 2m has polynomial dimension in n of degree equal to 2m, while the part not coming from Poisson polynomials has polynomial dimension of degree ⩽2m−3. We use this to compute asymptotics of the dimension of invariant operators. We also give new bounds on the order of invariant operators for a fixed n.We apply this to the Poisson and Hochschild homology associated to the singularity C2dn/Sn+1. Namely, the canonical surjection from HP0(OC2dn/Sn+1,OC2dn) to (the Brylinski spectral sequence in degree zero) restricts to an isomorphism in the aforementioned isotypic part h⊗⩽2d+1, and also in h⊗⩽4. We prove a partial converse. Finally, the kernel of the entire surjection has dimension on the order of times the dimension of the homology group.