  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.
