POISSON TRACES AND D-MODULES ON POISSON VARIETIES

Abstract

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 flow. 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. More generally, to any morphism [phi]: X → Y and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M [subscript phi] (X,N)M(XN) on X, and prove that it is holonomic if X has finitely many symplectic leaves, [phi] is finite, and N is coherent. As an application, we deduce that noncommutative filtered algebras, for which the associated graded algebra is finite over its center whose spectrum has finitely many symplectic leaves, have finitely many irreducible finite-dimensional 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. This includes symplectic reflection algebras. Furthermore, we describe explicitly (in the settings of affine varieties and compact C ∞-manifolds [C superscript infinity symbol -manifolds]) the finite-dimensional space of Poisson traces on X when X = V/G, where V is symplectic and G is a finite group acting faithfully on V.