higher-dimensional Weyl algebras does not hold. General definition As mentioned above, modules over the Weyl algebra correspond to D-modules on affine space. The Bernstein filtration not being available on DX for general varieties X, the definition is generalized to arbitrary affine smooth varieties X by means of order filtration on DX, defined by the order of differential operators. The associated graded ring gr DX is given by regular functions on the cotangent bundle T∗X. The characteristic variety is defined to be the subvariety of the cotangent bundle cut out by the radical of the annihilator of gr M, where again M is equipped with a suitable filtration (with respect to the order filtration on DX). As usual, the affine construction then glues to arbitrary varieties. The Bernstein inequality continues to hold for any (smooth) variety X. While the upper bound is an immediate consequence of the above interpretation of in terms of the cotangent bundle, the lower bound is more subtle. Properties and characterizations Holonomic modules have a tendency to behave like finite-dimensional vector spaces. For example, their length is finite. Also, M is holonomic if and only if all cohomology groups of the complex Li∗(M) are finite-dimensional K-vector spaces, where i is the closed immersion of any point of X. For any D-module M, the dual module is defined by Holonomic modules can also be characterized by a homological condition: M is holonomic if and only if D(M) is concentrated (seen as an object in the derived category of D-modules) in degree 0. This fact is a first glimpse of Verdier duality and the Riemann–Hilbert correspondence. It is proven by extending the homological study of regular rings (especially what is related to global homological dimension) to the filtered ring DX. Another characterization of holonomic modules is via symplectic geometry. The characteristic variety Ch(M) of any D-module M is, seen as a subvariety of the cotangent bundle T∗X of X, an involutive variety. The module is holonomic if and only if Ch(M) is Lagrangian. Applications One of the early applications of holonomic D-modules was the Bernstein–Sato polynomial. Kazhdan–Lusztig conjecture The Kazhdan–Lusztig conjecture was proved using D-modules. Riemann–Hilbert correspondence The Riemann–Hilbert correspondence establishes a link between certain D-modules and constructible sheaves. As such, it provided a motivation for introducing perverse sheaves. Geometric representation theory D-modules are also applied in geometric representation theory. A main result in this area is the Beilinson–Bernstein