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Poisson traces, D-modules, and symplectic resolutions. (English) Zbl 1407.53098

In this paper, the authors give a survey on Poisson traces (or zeroth Poisson homology) developed in a series of their recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, the conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical \(D\)-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the \(D\)-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the \(D\)-module is the pushforward of the canonical \(D\)-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. One explains many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra.
The authors compute the \(D\)-module in the case of surfaces with isolated singularities and show that it is not always semisimple. They also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein-Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix one gives a brief recollection of the theory of \(D\)-modules on singular varieties that one requires.

MSC:

53D55 Deformation quantization, star products
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
32S20 Global theory of complex singularities; cohomological properties
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