To every Poisson algebraic variety X over an algebraically closed field of
characteristic zero, we canonically attach a right D-module M(X) on X. If X is
affine, solutions of M(X) in the space of algebraic distributions on X are
Poisson traces on X, i.e., distributions invariant under Hamiltonian flows.
When X has finitely many symplectic leaves, we prove that M(X) is holonomic.
Thus, when X is affine and has finitely many symplectic leaves, the space of
Poisson traces on X is finite-dimensional. As an application, we deduce that
noncommutative filtered algebras whose associated graded algebras are
coordinate rings of Poisson varieties with finitely many symplectic leaves have
finitely many irreducible representations. The appendix, by Ivan Losev,
strengthens this to show that in such algebras, there are finitely many prime
ideals, and they are all primitive.

We also describe explicitly (in the settings of affine varieties and compact
smooth manifolds) the space of Poisson traces on X when X=V/G, where V is
symplectic and G is a finite group acting faithfully on V. In particular, we
show that this space is finite-dimensional.