Bernstein inequality and holonomic modules
Ivan Losev, Pavel Etingof
TL;DR
The paper develops a unified holonomic-module theory for filtered algebras with commutative associated graded and finitely many symplectic leaves, encompassing Uλ, quantizations of symplectic resolutions, quantum Hamiltonian reductions, and symplectic reflection algebras. It defines holonomic A-modules via isotropic associated varieties and proves a generalized Bernstein inequality, with equality for holonomic simples under finite leaf-fundamental-group conditions, as well as equi-dimensionality of V(M). It then shows holonomic modules have finite length in several key classes (finite-length regular bimodules or specific constructions), reducing Bernstein bounds to simple modules, and provides completion/ localisation tools to analyze ideals and supports. The results yield finite-length regular bimodules in important cases (Rational Cherednik algebras, quantized symplectic resolutions, and certain Hamiltonian reductions) and pave the way for a broader noncommutative-geometry framework connecting D-module ideas with quantum algebras. The Appendix with Etingof motivates the holonomic definition in the global-quantization setting and clarifies the isotropy/Holonomic correspondence in that context.
Abstract
In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that the generalized Bernstein inequality holds for simple modules and turns into equality for holonomic simples provided the algebraic fundamental groups of all leaves are finite. Under the same assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length or if the algebra in question is a quantum Hamiltonian reduction, then any holonomic module has finite length. This allows to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.