# Poisson traces, D-modules, and symplectic resolutions

## TL;DR

This paper surveys the theory of Poisson traces and D-modules on singular Poisson varieties, exploring their properties, relationships to symplectic resolutions, and applications to quantizations and representation theory.

## Contribution

It introduces a canonical D-module approach to Poisson traces, proves finiteness results, and conjectures a deep link between Poisson traces and symplectic resolutions.

## Key findings

- Poisson traces are finite-dimensional for varieties with finitely many symplectic leaves.
- The D-module associated with Poisson traces is holonomic in key cases.
- Examples include symmetric powers of du Val singularities and Slodowy slices, with explicit computations.

## Abstract

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities, and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein-Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.