Title: Poisson traces, D-modules, and symplectic resolutions,
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.