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

TL;DR

This paper studies the structure of invariant differential operators on symplectic manifolds, revealing their representation-theoretic properties, explicit computations, and applications to Poisson and Hochschild homology, with a focus on asymptotic behavior as n grows.

Contribution

It introduces a new perspective on the S_n-structure of invariant differential operators using Deligne's category, providing explicit formulas and asymptotic analysis.

Findings

Isotypic parts are spanned by Poisson polynomials for certain tensor powers.

Generated functions for Young diagram representations are rational with hook-length denominators.

The Brylinski spectral sequence degenerates at a specific tensor power, with the kernel's dimension decreasing as 1/n^3.

Abstract

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.