Poisson traces, D-modules, and symplectic resolutions

The space of Poisson traces on a Poisson algebra is the dual to the zeroth Poisson (or Lie) homology, i.e., the functionals which annihilate Poisson brackets. I will recall a systematic way of understanding these in terms of a canonical D-module on the spectrum of the Poisson algebra, whose space of solutions is the space of Poisson traces. When the variety has finitely many symplectic leaves, this D-module is holonomic, and as a corollary, the space of Poisson traces is finite-dimensional. In particular, this applies to quotients of a symplectic vector space (or symplectic variety) by a finite group of automorphisms. I will then describe a conjecture that states that, when the Poisson variety admits a symplectic resolution, the canonical D-module is obtained by pushing forward the canonical volume form from the resolution. This conjecture is satisfied in the case when the variety is a Slodowy slice of the nilpotent cone of a semisimple Lie algebra, equipped with the Springer resolution. This is joint work with Pavel Etingof.