Bernstein inequality and holonomic modules

TL;DR

This paper explores the representation theory of certain filtered algebras with symplectic structures, introducing holonomic modules, proving Bernstein inequalities, and analyzing their properties and implications in symplectic and quantum algebra contexts.

Contribution

It defines holonomic modules for these algebras and proves Bernstein inequalities and related properties, extending concepts from D-module theory to new algebraic settings.

Findings

Bernstein inequality holds for simple modules

Holonomic simples satisfy equality in Bernstein inequality

Regular bimodule has finite length in key cases

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.