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In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations. Since around 1970, D-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial.

Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of D-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. This approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems (holonomic systems), and on the characteristic variety cut out by the symbols, which in the good case is a Lagrangian submanifold of the cotangent bundle of maximal dimension (involutive systems). The techniques were taken up from the side of the Grothendieck school by Zoghman Mebkhout, who obtained a general, derived category version of the Riemann–Hilbert correspondence in all dimensions.

Modules over the Weyl algebra

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The first case of algebraic D-modules are modules over the Weyl algebra An(K) over a field K of characteristic zero. It is the algebra consisting of polynomials in the following variables

x1, ..., xn, ∂1, ..., ∂n.

where the variables xi and ∂j separately commute with each other, and xi and ∂j commute for i ≠ j, but the commutator satisfies the relation

[∂i, xi] = ∂ixi − xi∂i = 1.

For any polynomial f(x1, ..., xn), this implies the relation

[∂i, f] = ∂f / ∂xi,

thereby relating the Weyl algebra to differential equations.

An (algebraic) D-module is, by definition, a left module over the ring An(K). Examples for D-modules include the Weyl algebra itself (acting on itself by left multiplication), the (commutative) polynomial ring K[x1, ..., xn], where xi acts by multiplication and ∂j acts by partial differentiation with respect to xj and, in a similar vein, the ring of holomorphic functions on Cn (functions of n complex variables.)

Given some differential operator P = an(x) ∂n + ... + a1(x) ∂1 + a0(x), where x is a complex variable, ai(x) are polynomials, the quotient module M = A1(C)/A1(C)P is closely linked to space of solutions of the differential equation

P f = 0,

where f is some holomorphic function in C, say. The vector space consisting of the solutions of that equation is given by the space of homomorphisms of D-modules .

D-modules on algebraic varieties

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The general theory of D-modules is developed on a smooth algebraic variety X defined over an algebraically closed field K of characteristic zero, such as K = C. The sheaf of differential operators DX is defined to be the OX-algebra generated by the vector fields on X, interpreted as derivations. A (left) DX-module M is an OX-module with a left action of DX on it. Giving such an action is equivalent to specifying a K-linear map

satisfying

(Leibniz rule)

Here f is a regular function on X, v and w are vector fields, , and [−, −] denotes the commutator. Therefore, if M is in addition a locally free OX-module, giving M a D-module structure is nothing else than equipping the vector bundle associated to M with a flat (or integrable) connection.[1]

As the ring DX is noncommutative, left and right D-modules have to be distinguished. However, the two notions can be exchanged, since there is an equivalence of categories between both types of modules, given by mapping a left module M to the tensor product M ⊗ ΩX, where ΩX is the line bundle given by the highest exterior power of differential 1-forms on X. This bundle has a natural right action determined by

ω ⋅ v := − Liev (ω),

where v is a differential operator of order one, that is to say a vector field, ω a n-form (n = dim X), and Lie denotes the Lie derivative.[2]

Locally, after choosing some system of coordinates x1, ..., xn (n = dim X) on X, which determine a basis ∂1, ..., ∂n of the tangent space of X, sections of DX can be uniquely represented as expressions

, where the are regular functions on X.

In particular, when X is the n-dimensional affine space, this DX is the Weyl algebra in n variables.

Many basic properties of D-modules are local and parallel the situation of coherent sheaves. This builds on the fact that DX is a locally free sheaf of OX-modules, albeit of infinite rank, as the above-mentioned OX-basis shows. A DX-module that is coherent as an OX-module can be shown to be necessarily locally free (of finite rank).

Functoriality

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D-modules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves. For a map f: X → Y of smooth varieties, the definitions are this:

DX→Y := OX ⊗f−1(OY) f−1(DY)

This is equipped with a left DX action in a way that emulates the chain rule, and with the natural right action of f−1(DY). The pullback is defined as

f∗(M) := DX→Y ⊗f−1(DY) f−1(M).

Here M is a left DY-module, while its pullback is a left module over X. This functor is right exact, its left derived functor is denoted Lf∗. Conversely, for a right DX-module N,

f∗(N) := f∗(N ⊗DX DX→Y)

is a right DY-module. Since this mixes the right exact tensor product with the left exact pushforward, it is common to set instead

f∗(N) := Rf∗(N ⊗LDX DX→Y).

Because of this, much of the theory of D-modules is developed using the full power of homological algebra, in particular derived categories.


Holonomic modules

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Holonomic modules over the Weyl algebra

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It can be shown that the Weyl algebra is a (left and right) Noetherian ring. Moreover, it is simple, that is to say, its only two-sided ideals are the zero ideal and the whole ring. These properties make the study of D-modules manageable. Notably, standard notions from commutative algebra such as Hilbert polynomial, multiplicity and length of modules carry over to D-modules. More precisely, DX is equipped with the Bernstein filtration, that is, the filtration such that FpAn(K) consists of K-linear combinations of differential operators xα∂β with |α| + |β| ≤ p (using multiindex notation). The associated graded ring is seen to be isomorphic to the polynomial ring in 2n indeterminates. In particular it is commutative.

Finitely generated D-modules M are endowed with so-called "good" filtrations F∗M, which are ones compatible with F∗An(K), essentially parallel to the situation of the Artin–Rees lemma. The Hilbert polynomial is defined to be the numerical polynomial that agrees with the function

n ↦ dimK FnM

for large n. The dimension d(M) of an An(K)-module M is defined to be the degree of the Hilbert polynomial. It is bounded by the Bernstein inequality

n ≤ d(M) ≤ 2n.

A module whose dimension attains the least possible value, n, is called holonomic.

The A1(K)-module M = A1(K)/A1(K)P (see above) is holonomic for any nonzero differential operator P, but a similar claim for higher-dimensional Weyl algebras does not hold.

General definition

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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 gr DX in terms of the cotangent bundle, the lower bound is more subtle.

Properties and characterizations

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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

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One of the early applications of holonomic D-modules was the Bernstein–Sato polynomial.

Kazhdan–Lusztig conjecture

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The Kazhdan–Lusztig conjecture was proved using D-modules.

Riemann–Hilbert correspondence

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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

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D-modules are also applied in geometric representation theory. A main result in this area is the Beilinson–Bernstein localization. It relates D-modules on flag varieties G/B to representations of the Lie algebra of a reductive group G. D-modules are also crucial in the formulation of the geometric Langlands program.

Notes

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Bibliography

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D-module

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In mathematics, particularly and , a D-module (or module over the ring of differential operators) is a quasicoherent sheaf of modules over the sheaf DX\mathcal{D}_X of differential operators on a smooth algebraic variety or XX, providing an algebraic framework to encode and solve systems of linear partial differential equations. The theory originated in the work of Japanese mathematician in the early 1960s, who outlined D-modules and holonomic systems in lectures at the as part of his broader vision for . This approach was systematized by Masaki Kashiwara in his 1970 thesis, which established foundational results on micro-local analysis and the structure of D-modules over complex manifolds. Independently, developed a parallel algebraic theory around 1971, focusing on rings of differential operators in the context of algebraic varieties over fields of characteristic zero. These contributions built on earlier ideas from on the definition of differential operators, integrating sheaf theory and . Key properties of D-modules include their Noetherian nature and finite global homological dimension, bounded by twice the dimension of the underlying space, which enables powerful tools like derived categories and functors such as direct and inverse images under morphisms. Central concepts are the characteristic variety, a subvariety of the encoding singularities, and holonomic D-modules, which have Lagrangian characteristic varieties and finite length in the category of D-modules, corresponding to regular holonomic systems of PDEs. Kashiwara's equivalence theorem states that restriction to closed subvarieties preserves the category of D-modules, facilitating local-to-global studies. D-modules have profound applications across , including the study of flat connections on vector bundles, where a connection corresponds to a D-module structure, and , computed via the de Rham complex of a D-module. In , the Beilinson-Bernstein localization theorem equates categories of Harish-Chandra modules for Lie algebras with certain D-modules on flag varieties, impacting the proof of the Kazhdan-Lusztig conjectures. They also bridge and , modeling quantum systems and integrable hierarchies through .

Algebraic foundations

The Weyl algebra

The Weyl algebra AnA_n over a field kk of characteristic zero is the associative kk-algebra generated by the elements x1,…,xnx_1, \dots, x_n and ∂1=∂∂x1,…,∂n=∂∂xn\partial_1 = \frac{\partial}{\partial x_1}, \dots, \partial_n = \frac{\partial}{\partial x_n} subject to the commutation relations [∂i,xj]=δij[\partial_i, x_j] = \delta_{ij} for i,j=1,…,ni,j = 1, \dots, n, where δij\delta_{ij} is the , and all other generators commute. This structure arises as the ring of algebraic differential operators on the Akn\mathbb{A}^n_k, where the xix_i act by multiplication and the ∂i\partial_i act by partial differentiation. The Weyl algebra AnA_n is a simple Noetherian domain with global homological dimension nn. The Weyl algebra can be constructed iteratively as an Ore extension: starting from the polynomial ring k[x1,…,xn]k[x_1, \dots, x_n], adjoin the derivations ∂i\partial_i one by one, each satisfying the Leibniz rule ∂i(xjf)=δijf+xj∂if\partial_i (x_j f) = \delta_{ij} f + x_j \partial_i f for f∈k[x1,…,xn]f \in k[x_1, \dots, x_n]. It possesses a universal property: given any kk-algebra BB containing an embedded copy of k[x1,…,xn]k[x_1, \dots, x_n] and equipped with kk-linear derivations δi:B→B\delta_i: B \to B extending the standard partial derivatives on the polynomials and satisfying the Leibniz rule, there exists a unique kk-algebra homomorphism An→BA_n \to B sending xi↦xix_i \mapsto x_i and ∂i↦δi\partial_i \mapsto \delta_i. For n=1n=1, the first Weyl algebra A1=k⟨x,∂⟩/(∂x−x∂−1)A_1 = k\langle x, \partial \rangle / (\partial x - x \partial - 1) is the ring of linear differential operators with polynomial coefficients on the affine line. It is a simple ring, possessing no nontrivial two-sided ideals. Its simple left modules have been classified by Block as consisting of certain weight modules (holonomic, with Gelfand-Kirillov dimension 1) and modules without weights (non-holonomic, with dimension 2), all infinite-dimensional over kk. The Weyl algebra AnA_n admits a filtration by order of operators, where the degree of a monomial xI∂Jx^I \partial^J (with multi-indices I,JI, J) is ∣J∣|J|, the total order of differentiation. The associated graded algebra grAn\mathrm{gr} A_n with respect to this is isomorphic to the commutative k[x1,…,xn,ξ1,…,ξn]k[x_1, \dots, x_n, \xi_1, \dots, \xi_n] in 2n2n variables, where ξi\xi_i represents the symbol of ∂i\partial_i. The Gelfand-Kirillov dimension of AnA_n is thus 2n2n. Bernstein's asserts that for any nonzero finitely generated left AnA_n-module MM, the dimension of its characteristic variety is at least nn. The global homological dimension of AnA_n is nn.

Modules over the Weyl algebra

Modules over the Weyl algebra AnA_n are defined as left or right modules over this non-commutative ring, where AnA_n is generated by coordinate functions x1,…,xnx_1, \dots, x_n and partial derivatives ∂1,…,∂n\partial_1, \dots, \partial_n satisfying the commutation relations [xi,xj]=[∂i,∂j]=0[x_i, x_j] = [\partial_i, \partial_j] = 0 and [∂i,xj]=δij[\partial_i, x_j] = \delta_{ij}. Left AnA_n-modules correspond to systems of linear partial differential equations with coefficients acting on functions, while right modules arise in the study of distributions, such as the right module generated by the δ0\delta_0, which is isomorphic to An/(x1,…,xn)AnA_n / (x_1, \dots, x_n)A_n. Finitely generated modules over AnA_n admit good filtrations, which are exhaustive and separated compatible with the order filtration on AnA_n, ensuring that the associated graded module is finitely generated over the gr An≅k[x1,…,xn,ξ1,…,ξn]\mathrm{gr}\, A_n \cong k[x_1, \dots, x_n, \xi_1, \dots, \xi_n]. Regular holonomic modules over AnA_n are those finitely generated modules that admit a good filtration and have finite length in the of modules equipped with good filtrations. This finite length property implies that regular holonomic modules possess a finite with simple subquotients, and both submodules and quotients of such modules remain regular holonomic. The category of regular holonomic AnA_n-modules is artinian and noetherian, with the k[x1,…,xn]k[x_1, \dots, x_n] serving as a prototypical example of a regular holonomic module. Bernstein's inequality provides a fundamental bound on the dimension of modules over AnA_n, stating that for any nonzero finitely generated AnA_n-module MM, the dimension d(M)d(M), defined via the growth rate of the Hilbert polynomial of a good filtration on MM, satisfies d(M)≥nd(M) \geq n. This dimension relates directly to the growth of solutions to associated differential equations: in the analytic setting, the space of holomorphic solutions to a system corresponding to MM on a ball of radius rr has dimension growing asymptotically like rd(M)−nr^{d(M) - n}, so the inequality implies that solution spaces grow at least as slowly as a constant (for d(M)=nd(M) = n) but no faster than polynomially. Equality in Bernstein's inequality holds precisely for regular holonomic modules, linking algebraic dimension to the moderate growth of solutions characteristic of regular singularities. Simple modules over the Weyl algebra exhibit diverse behaviors, particularly in low dimensions. For n=1n=1, the first Weyl algebra A1A_1, the simple modules are classified by R. E. Block into two types: the holonomic modules, which correspond to cyclic modules generated by exponential solutions (all infinite-dimensional over the base field), and the infinite-dimensional non-holonomic modules, parameterized by similarity classes of matrices under the action of C×⋉GL2(C)\mathbb{C}^\times \ltimes \mathrm{GL}_2(\mathbb{C}). These non-holonomic simples have dimension d(M)>1d(M) > 1 and arise as torsion-free modules over the center; A1A_1 admits no finite-dimensional representations due to the relations among its generators. In higher dimensions, classification remains partial, but simple holonomic modules are indecomposable and play a role in composition series of regular holonomic modules. Extensions to twisted modules over the Weyl algebra incorporate additional structure, such as an involution or twisting , leading to twisted generalized Weyl algebras where the commutation relations are modified by a ring and derivation. Modules with an involution, often anti-involutions preserving the , allow for *-representations that are bounded or unbounded, preserving key properties like while adapting to quantum or deformed settings. These twisted constructions maintain noetherianity and provide classifications of simple weight modules under torsion-free conditions, extending the of standard Weyl modules.

D-modules on varieties

Definition and construction

Let XX be a smooth algebraic variety over a field kk of characteristic zero. The sheaf of differential operators DX\mathcal{D}_X on XX is the sheaf of kk-algebras generated over OX\mathcal{O}_X by the sheaf of derivations ΘX=\Derk(OX,OX)\Theta_X = \Der_k(\mathcal{O}_X, \mathcal{O}_X). It admits a natural by order of operators: FpDX\mathcal{F}^p\mathcal{D}_X is the subsheaf generated by products of at most pp derivations, so that FpDX⋅FqDX⊆Fp+qDX\mathcal{F}^p\mathcal{D}_X \cdot \mathcal{F}^q\mathcal{D}_X \subseteq \mathcal{F}^{p+q}\mathcal{D}_X. The associated graded sheaf is \grDX=⨁p≥0FpDX/Fp+1DX≅\SymOX(ΘX)\gr \mathcal{D}_X = \bigoplus_{p \geq 0} \mathcal{F}^p\mathcal{D}_X / \mathcal{F}^{p+1}\mathcal{D}_X \cong \Sym_{\mathcal{O}_X}(\Theta_X), the on the tangent sheaf, which identifies with the structure sheaf OT∗X\mathcal{O}_{T^*X} of the T∗XT^*X. Locally on , DX\mathcal{D}_X is the sheaf associated to the Weyl . A (left) DX\mathcal{D}_X-module is a quasi-coherent sheaf M\mathcal{M} of OX\mathcal{O}_X-modules equipped with a compatible left DX\mathcal{D}_X-action, meaning that for sections f∈OX(U)f \in \mathcal{O}_X(U), ξ∈ΘX(U)\xi \in \Theta_X(U), and m∈M(U)m \in \mathcal{M}(U), the action satisfies f⋅m=fmf \cdot m = f m and ξ⋅m=ξ(m)\xi \cdot m = \xi(m), with higher-order operators defined inductively via the Leibniz rule ξ⋅(P⋅m)=(ξ⋅P)⋅m+P⋅(ξ⋅m)\xi \cdot (P \cdot m) = (\xi \cdot P) \cdot m + P \cdot (\xi \cdot m) for P∈DXP \in \mathcal{D}_X of order at least one. Right DX\mathcal{D}_X-modules are defined analogously, with an anti-isomorphism DX≅tDX\mathcal{D}_X \cong {}^t\mathcal{D}_X (via adjoint action) providing an equivalence between the categories of left and right modules. A standard construction of left DX\mathcal{D}_X-modules proceeds via quantization, starting from a quasi-coherent OX\mathcal{O}_X-module E\mathcal{E} (such as the sheaf of sections of a ) and extending scalars along the inclusion OX↪DX\mathcal{O}_X \hookrightarrow \mathcal{D}_X to form DX⊗OXE\mathcal{D}_X \otimes_{\mathcal{O}_X} \mathcal{E}, where DX\mathcal{D}_X acts on the left factor. This yields the trivial or free DX\mathcal{D}_X-module induced by E\mathcal{E}, with the original OX\mathcal{O}_X-action recovered via the augmentation DX→OX\mathcal{D}_X \to \mathcal{O}_X. Basic examples include the structure sheaf OX\mathcal{O}_X itself, made into a left DX\mathcal{D}_X-module by the canonical action of multiplication and derivations. Another is the delta module δp\delta_p at a closed point p∈Xp \in X, defined locally as DX/DX⋅mp\mathcal{D}_X / \mathcal{D}_X \cdot \mathfrak{m}_p (where mp\mathfrak{m}_p is the sheaf at pp), a simple left DX\mathcal{D}_X-module with support {p}\{p\} that models the Dirac delta distribution. Coherence for DX\mathcal{D}_X-modules requires that M\mathcal{M} be locally finitely generated as a DX\mathcal{D}_X-module, ensuring finite-dimensional solution spaces in the analytic topology; however, the full category \Mod(DX)\Mod(\mathcal{D}_X) consists of all quasi-coherent left DX\mathcal{D}_X-modules, without finite generation. The theory of DX\mathcal{D}_X-modules, introduced by M. Sato and developed by M. Kashiwara, generalizes the affine case of modules over the Weyl algebra to the geometric setting of sheaves on varieties.

Functoriality

Functoriality of D\mathcal{D}-modules under morphisms of varieties is governed by a series of categorical functors that allow for the transfer of modules between different spaces, facilitating computations, gluing constructions, and compatibility with derived operations. For a morphism f:X→Yf: X \to Y of smooth varieties, the direct image functor f+:Mod(DX)→Mod(DY)f_+: \mathrm{Mod}(\mathcal{D}_X) \to \mathrm{Mod}(\mathcal{D}_Y) is defined by f+M=Rf∗(DY←X⊗DXLM)f_+ \mathcal{M} = Rf_* (\mathcal{D}_{Y \leftarrow X} \otimes_{\mathcal{D}_X}^{\mathbb{L}} \mathcal{M}), where DY←X\mathcal{D}_{Y \leftarrow X} is the (DY,DX)(\mathcal{D}_Y, \mathcal{D}_X)-bimodule of relative differential operators (right DX\mathcal{D}_X-, left DY\mathcal{D}_Y-action), and Rf∗Rf_* denotes the derived pushforward. This functor is left exact and preserves coherence when ff is proper on the support of M\mathcal{M}. Its right adjoint is the extraordinary inverse image f!:Mod(DY)→Mod(DX)f^!: \mathrm{Mod}(\mathcal{D}_Y) \to \mathrm{Mod}(\mathcal{D}_X), given by f!N=DX(f+DYN)f^! \mathcal{N} = D_X (f_+ D_Y \mathcal{N}), where DD denotes the duality functor on the derived category. When ff is proper, the pair (f+,f!)(f_+, f^!) forms an adjoint pair, enabling Verdier duality compatibilities. The f∗:Mod(DY)→Mod(DX)f^*: \mathrm{Mod}(\mathcal{D}_Y) \to \mathrm{Mod}(\mathcal{D}_X) is defined as f∗N=DX/Y⊗f−1DYLf−1Nf^* \mathcal{N} = \mathcal{D}_{X / Y} \otimes_{f^{-1} \mathcal{D}_Y}^{\mathbb{L}} f^{-1} \mathcal{N}, where DX/Y\mathcal{D}_{X / Y} is the (DX,f−1DY)(\mathcal{D}_X, f^{-1} \mathcal{D}_Y)-bimodule obtained by transferring the structure via ff (with appropriate left/right actions to preserve left modules). For smooth morphisms ff, this is and preserves holonomicity, with the explicit formula involving s: f∗N≅jX/Y∗(f−1N)f^* \mathcal{N} \cong j_{X/Y}^* (f^{-1} \mathcal{N}), where jX/Yj_{X/Y} is the relative . The characteristic variety satisfies ch(f∗N)⊂fd−1fπ∗ch(N)\mathrm{ch}(f^* \mathcal{N}) \subset f_d^{-1} f_\pi^* \mathrm{ch}(\mathcal{N}), ensuring control over singularities. The extraordinary direct image f!:Mod(DX)→Mod(DY)f_!: \mathrm{Mod}(\mathcal{D}_X) \to \mathrm{Mod}(\mathcal{D}_Y) is the left to f!f^!, defined as f!M=DY(f!DXM)f_! \mathcal{M} = D_Y (f^! D_X \mathcal{M}). It plays a crucial role in the study of nearby and vanishing cycles for stratified morphisms, where the vanishing cycle functor ϕf\phi_f and nearby cycle functor ψf\psi_f on D\mathcal{D}-modules relate to f!f_! via the Kashiwara-Malgrange VV-filtration on the direct image under a . Specifically, for a function f:X→A1f: X \to \mathbb{A}^1, the vanishing cycles ϕfM\phi_f \mathcal{M} fit into a distinguished triangle involving f!Mf_! \mathcal{M} and the specialization, capturing the behavior at the critical locus f−1(0)f^{-1}(0). In the bounded derived category Db(DX)D^b(\mathcal{D}_X), these functors exhibit strong compatibilities. The inverse image is compatible with tensor products: f∗(N1⊗DYN2)≅f∗N1⊗DXf∗N2f^*( \mathcal{N}_1 \otimes_{\mathcal{D}_Y} \mathcal{N}_2 ) \cong f^* \mathcal{N}_1 \otimes_{\mathcal{D}_X} f^* \mathcal{N}_2, preserving the monoidal structure. For internal Hom, there is a natural transformation f∗RHomDY(N1,N2)→RHomDX(f∗N1,f∗N2)f^* R\mathrm{Hom}_{\mathcal{D}_Y}(\mathcal{N}_1, \mathcal{N}_2) \to R\mathrm{Hom}_{\mathcal{D}_X}(f^* \mathcal{N}_1, f^* \mathcal{N}_2), which becomes an isomorphism under suitable coherence assumptions. Direct images commute with tensor in the sense that f+(M1⊗DXM2)≅f+M1⊗DYf+M2f_+ (\mathcal{M}_1 \otimes_{\mathcal{D}_X} \mathcal{M}_2) \cong f_+ \mathcal{M}_1 \otimes_{\mathcal{D}_Y} f_+ \mathcal{M}_2 when ff is proper and the supports align. These properties underpin the six-functor formalism for D\mathcal{D}-modules, analogous to that for sheaves. For étale morphisms f:X→Yf: X \to Y, Kashiwara's equivalence establishes an isomorphism between the category of DX\mathcal{D}_X-modules and those of DY\mathcal{D}_Y via f∗f^*, preserving the bounded derived category of coherent modules, which under the Riemann-Hilbert correspondence corresponds to an equivalence between perverse sheaves on XX and YY. This follows from the fact that étale maps preserve the differential structure, making f+f_+ and f∗f^* mutually inverse up to shift.

Holonomic D-modules

General definition

In the of DD-modules, a coherent DXD_X-module MM on a smooth variety XX of nn is defined to be holonomic if the of its characteristic variety Ch(M)\mathrm{Ch}(M) is equal to nn. The characteristic variety Ch(M)\mathrm{Ch}(M) is the support in the T∗XT^*X of the associated graded module grFM\mathrm{gr}_F M with respect to a good filtration FF on MM, and this condition captures modules of minimal complexity among coherent DXD_X-modules. This originates from the independent works of Kashiwara in the analytic setting and Bernstein in the algebraic setting over the Weyl algebra. An equivalent characterization arises from the solution complex: for a holonomic DXD_X-module MM, the groups Hi(DR(M))\mathbb{H}^i(\mathrm{DR}(M)) are finite-dimensional vector spaces. This finiteness reflects the controlled growth of solutions to the associated system of partial differential equations and generalizes the finite-dimensional solution spaces for ordinary differential equations with regular singularities. Representative examples of holonomic DXD_X-modules include the structure sheaf OX\mathcal{O}_X, whose characteristic variety is the zero section of T∗XT^*X (of dimension nn), and its twists OX⊗L\mathcal{O}_X \otimes L by invertible sheaves LL (which preserve the characteristic variety up to ). A of a non-holonomic coherent DXD_X-module is DXD_X itself (the free rank-one module), whose characteristic variety is the entire T∗XT^*X (of dimension 2n>n2n > n for n>0n > 0). Similarly, sheaves of higher rank, such as free modules over the viewed via the embedding into DXD_X, exhibit characteristic varieties of dimension greater than nn. The category Hol(DX)\mathrm{Hol}(D_X) of holonomic DXD_X-modules forms an abelian of the category of coherent DXD_X-modules, closed under extensions, kernels, and cokernels, and it admits a well-behaved tensor structure. In the specific case of modules over the Weyl algebra (corresponding to X=AnX = \mathbb{A}^n), holonomic modules are precisely the regular holonomic modules.

Properties and characterizations

Holonomic DXD_X-modules on a smooth variety XX of dimension nn are characterized by several key algebraic and geometric properties that distinguish them from more general coherent DXD_X-modules. A fundamental result is that a coherent DXD_X-module MM is holonomic if and only if its de Rham complex DR(M)=M⊗DXLΩX∙\mathrm{DR}(M) = M \otimes^{\mathrm{L}}_{D_X} \Omega_X^\bullet has finite-dimensional stalks for its sheaves Hi(DR(M))xH^i(\mathrm{DR}(M))_x at every point x∈Xx \in X. This finiteness condition reflects the "finite-dimensional solution space" nature of holonomic systems of partial differential equations. Another central characterization involves duality. For a holonomic DXD_X-module MM, there is a natural HomDX(M,DX)≅RHhomDX(M,ωX)\mathrm{Hom}_{D_X}(M, D_X) \cong \mathrm{RHhom}_{D_X}(M, \omega_X), where ωX\omega_X is the dualizing complex on XX, typically ΩX∙\Omega_X^\bullet for smooth XX. This duality ensures that holonomic modules are self-dual up to shift and plays a crucial role in preserving holonomicity under dualizing functors. The characteristic variety Ch(M)⊂[T∗X](/page/T−X)\mathrm{Ch}(M) \subset [T^*X](/page/T-X) and microsupport SS(M)⊂[T∗X](/page/T−X)\mathrm{SS}(M) \subset [T^*X](/page/T-X) of a holonomic DXD_X-module MM are Lagrangian subvarieties, meaning they have pure nn and are involutive with respect to the canonical symplectic structure on T∗XT^*X. This purity of dimension follows from the definition of holonomicity and Gabber's theorem on the involutivity of characteristic varieties for coherent DD-modules, which implies that components of Ch(M)\mathrm{Ch}(M) cannot have dimension strictly less than nn unless M=0M = 0. Regarding filtrations, every coherent DXD_X-module admits a good filtration {FkM}\{F_k M\}, where the associated graded grFM\mathrm{gr}_F M is finitely generated as a grDX\mathrm{gr} D_X-module. A coherent DXD_X-module MM is holonomic if and only if, for any such good filtration, the support of grFM\mathrm{gr}_F M has dimension nn. In the analytic setting, there is an equivalence between regular holonomic DXD_X-modules and tempered holonomic DXD_X-modules. Specifically, a holonomic DXD_X-module on a complex analytic manifold is regular if and only if all its holomorphic solutions are tempered distributions, meaning the solution sheaf consists of tempered functions with respect to a Whitney stratification of XX. This equivalence underscores the connection between regularity conditions and growth estimates in microlocal analysis.

Holonomic modules over the Weyl algebra

Holonomic modules over the Weyl algebra AnA_n, the ring of differential operators on An\mathbb{A}^n, are finitely generated left (or right) AnA_n-modules MM satisfying Bernstein's inequality with equality, namely the Gelfand-Kirillov dimension GKdim(M)=n\mathrm{GKdim}(M) = n or equivalently the dimension d(M)=nd(M) = n from the Bernstein filtration. This condition ensures that the characteristic variety Ch(M)\mathrm{Ch}(M) is Lagrangian, of dimension nn, distinguishing holonomic modules from those with larger growth rates. The associated Bernstein polynomial bM(s)b_M(s), which annihilates powers of a generator in the filtered setting, has degree at most nn for holonomic MM, reflecting the bounded complexity of the module's annihilator ideal. Irreducible holonomic modules over AnA_n are precisely the cyclic modules generated by delta distributions supported at points in An\mathbb{A}^n. For instance, the module An⋅δaA_n \cdot \delta_a generated by the Dirac delta at a∈Ana \in \mathbb{A}^n is simple and holonomic, with characteristic variety the conormal bundle to the point {a}\{a\}. More generally, such modules arise as minimal extensions of integrable connections on open subsets, and all holonomic modules have finite length with simple factors of this form. Holonomic modules over AnA_n correspond to systems of linear partial differential equations whose solution spaces are finite-dimensional on Zariski-open subsets of An\mathbb{A}^n. For example, the hypergeometric differential equations, encoded by holonomic ideals in AnA_n, admit solutions spanning vector spaces of dimension equal to the rank of the module, such as the classical Gauss hypergeometric functions satisfying second-order equations. A holonomic AnA_n-module MM has regular singularities in the sense of Malgrange if and only if the generalized eigenspaces of the Euler operator (or a suitable derivation) on the localized module M[f−1]M[f^{-1}] are finite-dimensional for hypersurface complements. This condition ensures that the module's singularities are tame, with the nearby cycles functor yielding finite-dimensional invariants. In contrast, non-holonomic modules over AnA_n, such as those with GKdim(M)>n\mathrm{GKdim}(M) > n, include infinite-dimensional representation spaces like the simple quotients An/aAnA_n / a A_n where aa is a cyclic maximal right ideal of codimension one, yielding GKdim=2n−1\mathrm{GKdim} = 2n-1. Stafford constructed the first explicit examples of such irreducible non-holonomic modules, while Bernstein and Lunts later produced infinite families via generic elements in filtered components. These modules exhibit unbounded growth and infinite-dimensional solution spaces, contrasting sharply with the finite-dimensionality of holonomic cases.

Applications

Riemann–Hilbert correspondence

The Riemann–Hilbert correspondence provides a profound link between and by establishing an between regular holonomic DX\mathcal{D}_X-modules on a smooth complex XX and perverse sheaves on the underlying real manifold RXR X endowed with its classical . Formulated and proved by Beilinson, Bernstein, and Deligne, this correspondence equates the bounded derived category of holonomic DX\mathcal{D}_X-modules, denoted Hol⁡(DX)\operatorname{Hol}(\mathcal{D}_X), with the derived category of perverse sheaves, denoted Perv⁡(RX)\operatorname{Perv}(R X). The theorem builds on earlier analytic work by Kashiwara, who established the equivalence for complex analytic manifolds using microlocal methods. Central to the correspondence is the solution functor Sol⁡(M)=RHom⁡DX(M,OX)\operatorname{Sol}(M) = \mathbb{R}\operatorname{Hom}_{\mathcal{D}_X}(M, \mathcal{O}_X), which associates to a holonomic DX\mathcal{D}_X-module MM a complex of sheaves of holomorphic functions solving the corresponding ; under the equivalence, this yields a on RXR X. The inverse functor is constructed via the image under the inclusion of RXR X into the complex analytic space XanX^{\mathrm{an}}, composed with a shift and adjustment to ensure perversity. This pair of functors induces a triangulated equivalence, preserving key structures such as DX\mathcal{D}_X-module duality and Verdier duality on . In the analytic setting, the proof relies on microlocalization, which refines the characteristic variety of a D\mathcal{D}-module to a Lagrangian of the , and the , an autoequivalence of the category of D\mathcal{D}-modules that interchanges regular singularities with essential ones. Kashiwara's approach uses these tools to show that the solution complex of a holonomic module has constructible cohomology sheaves, fitting precisely into the perverse t-structure after shifting by the . The algebraic version by Beilinson, Bernstein, and Deligne adapts this via analytification and compatibility with stratifications, ensuring the equivalence holds without relying on for smooth XX. A key application is the algebraic de Rham theorem for holonomic modules: for a holonomic DX\mathcal{D}_X-module MM, the hypercohomology H∗(X,DR⁡(M))\mathbb{H}^*(X, \operatorname{DR}(M)) computes the de Rham cohomology, and by the correspondence, this equals the hypercohomology of the associated perverse sheaf Sol⁡(M)\operatorname{Sol}(M) on RXR X, providing topological invariants from differential equations. This enables computations of global sections and Ext groups in Hol⁡(DX)\operatorname{Hol}(\mathcal{D}_X) via sheaf cohomology on RXR X. The correspondence extends to singular varieties by embedding XX into a smooth ambient variety YY and restricting modules along the inclusion, preserving holonomicity and perversity under the direct and inverse images. This reduction allows the full theory to apply beyond smooth settings, with the equivalence holding relative to the stratification induced by the embedding.

Kazhdan–Lusztig conjecture

The Kazhdan–Lusztig conjecture, formulated in 1979, concerns the structure of representations of semisimple complex Lie algebras g\mathfrak{g}. It states that the character of an irreducible highest weight module L(λ)L(\lambda) can be expressed as an alternating sum over characters of Verma modules Δ(μ)\Delta(\mu) with coefficients given by evaluations of the Kazhdan–Lusztig polynomials Pμ,λ(q)P_{\mu,\lambda}(q) at q=1q=1, and these coefficients are positive integers. These polynomials, defined combinatorially via a recursive procedure on the Weyl group WW, admit a geometric realization as the Poincaré polynomials associated to the stalks of intersection cohomology sheaves on Schubert varieties Xw⊂G/BX_w \subset G/B, the flag variety of a semisimple complex Lie group GG. Specifically, for y≤wy \leq w in the Bruhat order, Py,w(q)P_{y,w}(q) equals ∑i(−1)idim⁡Hi+dim⁡Xw−l(w)+l(y)(i∗ICw)\sum_i (-1)^i \dim H^{i + \dim X_w - l(w) + l(y)}(i^* \mathrm{IC}_w), where ICw\mathrm{IC}_w is the intersection cohomology complex on XwX_w and i:Xy↪Xwi: X_y \hookrightarrow X_w is the inclusion. The Beilinson–Bernstein localization theorem provides the geometric framework for proving the using D-modules. It establishes that, for a regular integral central character , the category of U(g\mathfrak{g})-modules with infinitesimal character is equivalent to the category of quasi-coherent -twisted D-modules on the flag variety X=G/BX = G/B. The equivalence is realized by the global sections functor Γ(X,−\Gamma(X, -: Dλ_{\lambda}-mod →\to U(g\mathfrak{g})λ_{\lambda}-mod, which is exact and t-Exact for the standard t-structure on derived categories, with inverse given by induction from the structure sheaf twisted by the corresponding to λ\lambda. Under this equivalence, Verma modules Δ(ν)\Delta(\nu) localize to the direct image j!∗Oλj_{!*} \mathcal{O}_{\lambda} along the open Schubert cell, yielding holonomic D-modules supported on Schubert variety closures. The proof of the Kazhdan–Lusztig conjecture via this localization exploits the holonomic nature of these D-modules. The localized Verma module Δ(λ)∧\Delta(\lambda)^\wedge has a filtration whose successive quotients are direct images of intersection cohomology sheaves, which are pure perverse sheaves of weight equal to their dimension. Purity ensures that the Euler characteristic computations in the derived category yield non-negative coefficients for the simple modules in the composition series of Δ(λ)\Delta(\lambda), matching the positivity predicted by the . Since holonomic D-modules have finite-dimensional global sections and the localization is fully faithful, the character formula follows from the geometric multiplicities given by dim⁡HomDλ(ICμ,Δ(λ)∧[l(λ)−l(μ)])\dim \mathrm{Hom}_{D_\lambda}(\mathrm{IC}_\mu, \Delta(\lambda)^\wedge [l(\lambda) - l(\mu)]), confirming the Kazhdan–Lusztig polynomials as the precise multiplicity polynomials. An independent proof using similar holonomic systems with regular singularities was given concurrently. In the recursive formulation of the Kazhdan–Lusztig polynomials, the Py,w(q)P_{y,w}(q) satisfy a relation involving the R-polynomials Ry,w(q)R_{y,w}(q), which are explicitly determined by the Coxeter presentation and satisfy Ry,w(q)=ql(w)−l(y)Rw−1,y−1(q−1)R_{y,w}(q) = q^{l(w)-l(y)} R_{w^{-1},y^{-1}}(q^{-1}) for y<wy < w. These R-polynomials capture the "geometric" part of the recursion and can be computed using D-module invariants on varieties, including local singularity data via Bernstein-Sato polynomials associated to the defining ideals of Schubert varieties, which bound the degrees and provide for the local contributing to the polynomial coefficients. The D-module approach to the Kazhdan–Lusztig conjecture has inspired generalizations. In the setting of Hecke algebras with unequal parameters, analogous positivity conjectures for generalized KL polynomials have been formulated and partially resolved using modified localization functors or bimodule categories, though full geometric proofs remain open in positive characteristic. For quantum groups Uq(g)U_q(\mathfrak{g}), Lusztig's canonical basis provides a q-deformation where structure constants involve deformed KL polynomials, with D-module techniques adapted via crystal bases and quantum flag varieties to establish similar purity and positivity results.

Geometric representation theory

D-modules play a central role in geometric , providing tools to study representations of algebraic groups through sheaf-theoretic constructions on geometric spaces such as varieties and moduli stacks. Their holonomic property ensures finite-dimensional solution spaces, facilitating equivalences between categories of modules and sheaves on singular varieties. The Springer resolution establishes a geometric framework for understanding nilpotent orbits in the of a reductive group, where intersection cohomology (IC) sheaves on the closures of these orbits arise as holonomic D-modules. Specifically, the Springer resolution n~→n\tilde{\mathfrak{n}} \to \mathfrak{n} of the nilpotent cone n\mathfrak{n} allows the IC sheaves to be realized as pushforwards of equivariant sheaves from the smooth resolved space, endowing them with a natural D-module structure that captures the representation-theoretic data of the orbits. This construction, detailed in the work of Frenkel and Gaitsgory, links these sheaves to critical-level modules for affine Kac-Moody algebras, enabling the study of local systems on nilpotent varieties via D-module equivalences. Furthermore, in the context of character sheaves, the IC sheaves on distinguished nilpotent orbits serve as cuspidal objects, with their singular support confined to the nilpotent cone, as extended by recent modular analyses. In the geometric , D-modules on moduli stacks of flat bundles provide a categorical framework for the correspondence between representations of the Langlands dual group and automorphic sheaves. The moduli stack Bun_G of G-bundles on a , compactified via Drinfeld constructions, supports twisted D-modules that encode Hecke eigensheaves, with the flat bundles corresponding to de Rham local systems realized through pushforwards from the Betti side. Yang's for the Borel compactification using D-modules on smooth stacks like Bun_{K,B} establishes equivalences that resolve strata via Bott-Samelson varieties, facilitating the by linking Whittaker categories to representations. This approach, building on Beilinson-Drinfeld's oper framework, ensures that the D-modules remain coherent and equivariant under the action of the Langlands dual group. Categorical actions in are realized through Soergel bimodules, which can be interpreted geometrically as D-modules on flag varieties, providing a bridge between parabolic induction and geometric Satake. On the flag variety G/B, Soergel bimodules correspond to parity sheaves or Braden-MacPherson sheaves via moment graphs, inducing actions of Hecke categories on categories of representations. Fiebig's work demonstrates that these bimodules equate to D-modules supported on Schubert cells, allowing the translation of multiplicity conjectures into geometric problems solvable by global sections functors from the flag variety to the BGG category O. This equivalence preserves the monoidal structure, enabling categorical Kac-Moody actions that deform the standard representation categories. Quantization of cohomology via D-modules yields geometric Eisenstein series and Whittaker models, deforming classical automorphic forms into sheaf-theoretic objects on Bun_G. The quantum geometric Langlands conjecture posits an equivalence DMod_κ(Bun_G) ≅ L_κ DMod_{-κ̂}(Bun_{\hat{G}}), where κ is the level, and Whittaker models arise as coefficients extracting nilpotent singular support data from cuspidal D-modules. Recent proofs establish non-vanishing of these quantum Whittaker coefficients for adjoint-type groups, using microlocal geometry on Zastava spaces to show conservativity of localization functors for tempered sheaves. Geometric Eisenstein series, constructed as Hecke integrals of these D-modules, provide functorial maps between blocks, linking to spherical varieties in the Langlands program. Post-2000 developments have reformulated the Satake equivalence using D-modules on affine Grassmannians, enhancing the geometric Satake isomorphism with microlocal and factorization properties. The affine Grassmannian Gr_G, as an ind-scheme quotient LG / L^+G, supports perverse D-modules whose convolution algebra realizes the Hecke category, equivalent to representations of the dual group \hat{G}. Zhu's lectures detail the ind-projective structure and factorization via Beilinson-Drinfeld Grassmannians, proving the equivalence Sat_G ≃ Rep(\hat{G}) through hypercohomology functors. Mirković and Vilonen's 2007 proof extends this to general coefficients, constructing weight functors that identify the dual group scheme via Tannakian reconstruction, with applications to conformal blocks and uniformization of moduli spaces. Recent extensions to Kac-Moody groups incorporate twisted D-modules on determinant line bundles, yielding uniform proofs of the equivalence in positive characteristic.

References

  1. http://www-math.mit.edu/~etingof/dmodwien.pdf
  2. https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf
  3. https://www.cambridge.org/core/books/primer-of-algebraic-dmodules/rings-of-differential-operators/F87CB114734430D663D39697B5F2273B
  4. https://eamonqg.github.io/diff_ops.pdf
  5. https://www.math.uzh.ch/en/fileadmin/user/brodmann/publikation/B_WADM_Short.IRIN.2016.07.15.pdf
  6. https://www.ams.org/proc/1974-045-03/S0002-9939-1974-0382358-X/
  7. https://link.springer.com/article/10.1007/s10468-014-9499-2
  8. https://mathweb.ucsd.edu/~nwallach/WeylAlgebra.pdf
  9. https://d-nb.info/987679023/34
  10. https://www.math.stonybrook.edu/~cschnell/pdf/notes/d-modules.pdf
  11. https://www.cambridge.org/core/books/primer-of-algebraic-dmodules/modules-over-the-weyl-algebra/6E207DE5500181DC9848A12C8DA8B2F1
  12. https://people.mpim-bonn.mpg.de/gaitsgde/grad_2009/Dmod_brav.pdf
  13. https://ferngaston.wordpress.com/2016/06/30/representations-of-the-weyl-algebra/
  14. https://core.ac.uk/download/pdf/82513638.pdf
  15. https://arxiv.org/pdf/1504.05361
  16. https://link.springer.com/article/10.1023/A:1015669525867
  17. https://publications.lib.chalmers.se/records/fulltext/11261/11261.pdf
  18. https://joshuamundinger.github.io/assets/notes/d-modules-lecture-2.pdf
  19. https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/Dmod.pdf
  20. https://arxiv.org/pdf/1507.00118.pdf
  21. https://arxiv.org/abs/1505.01788
  22. https://gauss.math.yale.edu/~il282/Ginzburg_D_mod.pdf
  23. https://www.ihes.fr/~celliott/D_modules.pdf
  24. https://www.math.utah.edu/~milicic/Eprints/dmodules.pdf
  25. https://arxiv.org/pdf/1507.00118
  26. https://www.worldscientific.com/doi/10.1142/9789814273244_0002
  27. https://math.uchicago.edu/~sfilip/public_files/hypergeometric_lectures.pdf
  28. https://doi.org/10.1007/BF01388528
  29. https://arxiv.org/pdf/1602.07382
  30. https://arxiv.org/pdf/math/0508382
  31. https://arxiv.org/pdf/2407.14678
  32. https://arxiv.org/pdf/2104.09862
  33. https://arxiv.org/pdf/2508.19624
  34. https://arxiv.org/pdf/1308.2873
  35. https://arxiv.org/pdf/1701.00560
  36. https://arxiv.org/pdf/2508.19058
  37. https://arxiv.org/pdf/2111.03004
  38. https://arxiv.org/pdf/1603.05593
  39. https://arxiv.org/pdf/1703.07288
  40. https://arxiv.org/pdf/2406.05521
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