Pavel Etingof and Travis Schedler with an Appendix by Ivan Losev

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 C∞-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. Table of