POISSON TRACES AND D-MODULES ON POISSON VARIETIES
Name
Etingof GAFA 2010.pdf
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Author(s)
Etingof, Pavel I.
Schedler, Travis
Date Issued
October 2010
Journal
Geometric and Functional Analysis
Publisher
Springer
Citation
Etingof, Pavel, and Travis Schedler. “Poisson Traces and D-Modules on Poisson Varieties.” Geometric And Functional Analysis 20.4 (2010) : 958-987-987. Copyright © 2010, Springer
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00039-010-0085-4
