\(D\)-modules, perverse sheaves, and representation theory. Translated from the Japanese by Kiyoshi Takeuchi. Expanded edition.

*(English)*Zbl 1136.14009
Progress in Mathematics 236. Basel: Birkhäuser (ISBN 978-0-8176-4363-8/hbk). x, 407 p. (2008).

The aim of this book is to give a comprehensive introduction to \(D\)-modules (mainly on algebraic varieties) and the application of this theory to representation theory. The exposition culminates with the proof of two important theorems, i.e. of the Riemann-Hilbert correspondence (in the first part) and the proof of the Kazhdan-Lusztig conjecture (in the second part). While there are some excellent texts on \(D\)-modules, this is the first systematic presentation on representation theory in connection with \(D\)-modules.

Since now the theory of \(D\)-modules has attained a high degree of maturity, the first part is an excellent introduction. The first chapter, Preliminary Notions, contains in approximatively 40 pages the basic notions and operations (inverse and direct images, tensor products, Kashiwara’s equivalence base change theorem for direct images). The second chapter presents the coherent \(D\)-modules. We remark the discussion concerning the relations among duality functions and inverse and direct images, etc.

In the third chapter, the authors consider the holonomic \(D\)-modules, the stability of holomonicity, adjunction formulas, finiteness property, minimal extensions (in particular it is proved that any simple object in the category of \(D_{X}\)-modules is a minimal extension of an integrable connection on a locally smooth subvariety). All this results are proved for \(X\) a smooth algebraic variety.

In Chapter 4, the authors consider analytic \(D_{X}\)-modules (i.e. \(D\)-modules on the underlying complex manifold). After a short survey in general theory of analytic \(D\)-modules, without proofs (for details the reader is send to J.-E. Björck’s “Analytic \(D\)-modules and Applications” [Mathematics and its Applications. 247. Dordrecht: Kluwer Academic Publishers (1993; Zbl 0805.32001)], and M. Kashiwara’s “\(D\)-modules and Microlocal calculus” [ Translations of Mathematical Monographs. 217. Providence, RI: American Mathematical Society (2003; Zbl 1017.32012)]), the authors introduce the solution complex and the de Rham functors, the Kashiwara version of the Cauchy-Kowalevski theorem, and prove Kashiwara’s constructibility theorem (and a shorter proof in the algebraic case due to Beilinson-Bernstein).

Chapter 5 presents the theory of meromorphic connections (both in the analytic and in the algebraic case, and the Riemann-Hilbert-Deligne correspondence for meromorphic regular integrable connections). In Chapter 6 the regular holonomic \(D\)-modules are introduced (using Bernstein’s definition) and some important result are proved (in the algebraic case):

(i) The duality function preserves regularity,

(ii) if \(f\) is a morphism of smooth algebraic varieties, the functor of direct image, direct image with compact supports, the inverse image, the adjunction functors preserve regularity,

(iii) the curve testing criterion.

Let \(D_{rh}^{b}(D_{X})\) the full subcategory of \(D_{h}^{b}(D_{X})\) whose objects \(M^{\bullet}\in D_{h}^{b}(D_{X})\) (= the full subcategory of \(D_{c}^{b}(D_{X})\) whose cohomology groups are holonomic), such that \(H^{i}(M^{\bullet})\) are regular holonomic. Here \(D_{c}^{b}(D_{X})\) is the full subcategory of \(D^{b}(D_{X})\) whose objects consist of bounded complexes of coherent \(D_{X}\)-modules. Then, in Chapter 7, the Riemann-Hilbert correspondence is examined very carefully. The original version [Z. Mebkhout, Compos. Math. 51, 51–62, 63–88 (1984; Zbl 0566.32021); M. Kashiwara, Sémin. Goulaouic-Schwartz 1979–1980, Équat. dériv. part., Exposé No. 19, 6 p. (1980; Zbl 0444.58014)], affirming that for a complex manifold the de Rham functor gives an equivalence of categories between \(D_{rh}^{b}(X)\) and \(D_{c}^{b}(X)\) (= the category of bounded complexes of sheaves with constructible cohomology) is not proved (The authors don’t mention the work of Z. Mebkhout [Sur le problème de Hilbert-Riemann, Complex analysis, microlocal calculus and relativistic quantum theory, Proc. Colloq., Les Houches 1979, Lect. Notes Phys. 126, 90–110 (1979; Zbl 0444.32003)].

Following Beilinson-Bernstein, an algebraic version is given, with a sketch of proof (i.e. for the verification that the construction of the isomorphism coincides with the one induced by the de Rham functor, the reader is send to M. Saito [Publ. Res. Inst. Math. Sci. 26, No. 2, 221–333 (1990; Zbl 0727.14004)]. What is proved in detail is the following statement (Thm. 7.2-5): The de Rham functor induces an equivalence between \(\text{Mod}_{rh}(D_{X})\) and \(\text{Perv}(\mathbb{C}_{X})\) where \(\text{Mod}_{rh}(D_{X})\) is the full subcategory consisting of regular holonomic \(D_{X}\)-modules of \(\text{Mod}_{h}(D_{X})\) (= the category of holomonic \(D_{X}\)-modules) \(X\) being a smooth algebraic variety, and \(\text{Perv}(\mathbb{C}_{X})\) is the category of perverse sheaves. As an application the authors give a proof of the comparison theorem.

Chapter 8 considers in detail perverse sheaves and intersection cohomology groups, assuming the basic notions concerning constructible sheaves, introduced previously. This self-contained acount is specially interesting. It contains also the W. Borho-R. MacPherson theorem in [ Analysis and Topology on Singular Spaces, (Luminy 1981), Astéristique 101–102, 23–74 (1983; Zbl 0576.14046)] which allows to construct geometrically the representation of the Weyl groups of semi simple algebraic groups. This chapter contains a paragraph about Hodge modules. The second part of the book is dedicated to representation theory. In fact, the text is focused on explaining how the \(D\)-module theory play a crucial role in proving the celebrated conjecture of Kazhdan-Lusztig.

Chapter 9 summarizes (with no proofs) the basic notions of algebraic groups and Lie algebras; some examples are given in about 30 pages; one finds here many important results (such as Weyl’s character formula or the Borel-Weil-Bott theorem). The next chapter discuss conjugacy class in semi simple Lie algebras using the theory of invariant polynomials for the adjoint representation, and in particular a parametrization of conjugacy classes, used in the sequel in chapter 11, “Representation of Lie algebras and \(D\)-modules”, a proof of the Beilinson-Bernstein correspondence [A. Beilinson and J. Bernstein, C. R. Acad. Sci., Paris, Sér. I 292, 15–18 (1981; Zbl 0476.14019)] between representations of semisimple Lie algebras and \(D\)-modules on flag manifolds. In particular, in the classification of equivariant \(D\)-modules, the Riemann-Hilbert correspondence is used in an essential way.

Finally, in Chapter 12, the authors give an account of the Kazhdan-Lusztig conjecture. The strategy of proof of this conjecture (Thm. 12.2.7) is to use the Beilinson-Bernstein correspondence between \(g\)-modules (\(g\) being a Lie algebra) and \(D\)-modules on flag manifolds, and then the Riemann-Hilbert correspondence between \(D\)-modules on flag manifolds and perverse sheaves on flag manifolds. In \(\S\) 12.3 the \(D\)-modules associated to the higher weight modules are introduced, and, again the Riemann-Hilbert correspondence is used. Kazhdan-Lusztig predicted that the Kazhdan-Lusztig polynomials should be closely related to the geometry of Schubert varieties. The precise statement is given by Th. 12.2.5 (where the intersection cohomology complex of a certain Schubert variety appears). The proof of this theorem is given in the thirteenth chapter, “Hecke algebras and Hodge module”.

In this last chapter, the authors describe a geometric realization of the group algebras of Weyl groups (resp. Hecke algebras) via \(D\)-modules (resp. via Hodge modules). The proof of the calculation of intersection cohomology of Schubert varieties (i.e. Th. 12.2.5) is done using the theory of Hodge modules. The book has also 5 appendices (Algebraic Varieties, Derived categories and Derived Functions, Sheaves an functor in Derived Categories, Filtered Rings, Symplectic Geometry), of about 65 pages, in which the reader will find all technical ingredients used in the book.

Summing up, this is a valuable book, which puts in evidence important applications of \(D\)-module theory.

Since now the theory of \(D\)-modules has attained a high degree of maturity, the first part is an excellent introduction. The first chapter, Preliminary Notions, contains in approximatively 40 pages the basic notions and operations (inverse and direct images, tensor products, Kashiwara’s equivalence base change theorem for direct images). The second chapter presents the coherent \(D\)-modules. We remark the discussion concerning the relations among duality functions and inverse and direct images, etc.

In the third chapter, the authors consider the holonomic \(D\)-modules, the stability of holomonicity, adjunction formulas, finiteness property, minimal extensions (in particular it is proved that any simple object in the category of \(D_{X}\)-modules is a minimal extension of an integrable connection on a locally smooth subvariety). All this results are proved for \(X\) a smooth algebraic variety.

In Chapter 4, the authors consider analytic \(D_{X}\)-modules (i.e. \(D\)-modules on the underlying complex manifold). After a short survey in general theory of analytic \(D\)-modules, without proofs (for details the reader is send to J.-E. Björck’s “Analytic \(D\)-modules and Applications” [Mathematics and its Applications. 247. Dordrecht: Kluwer Academic Publishers (1993; Zbl 0805.32001)], and M. Kashiwara’s “\(D\)-modules and Microlocal calculus” [ Translations of Mathematical Monographs. 217. Providence, RI: American Mathematical Society (2003; Zbl 1017.32012)]), the authors introduce the solution complex and the de Rham functors, the Kashiwara version of the Cauchy-Kowalevski theorem, and prove Kashiwara’s constructibility theorem (and a shorter proof in the algebraic case due to Beilinson-Bernstein).

Chapter 5 presents the theory of meromorphic connections (both in the analytic and in the algebraic case, and the Riemann-Hilbert-Deligne correspondence for meromorphic regular integrable connections). In Chapter 6 the regular holonomic \(D\)-modules are introduced (using Bernstein’s definition) and some important result are proved (in the algebraic case):

(i) The duality function preserves regularity,

(ii) if \(f\) is a morphism of smooth algebraic varieties, the functor of direct image, direct image with compact supports, the inverse image, the adjunction functors preserve regularity,

(iii) the curve testing criterion.

Let \(D_{rh}^{b}(D_{X})\) the full subcategory of \(D_{h}^{b}(D_{X})\) whose objects \(M^{\bullet}\in D_{h}^{b}(D_{X})\) (= the full subcategory of \(D_{c}^{b}(D_{X})\) whose cohomology groups are holonomic), such that \(H^{i}(M^{\bullet})\) are regular holonomic. Here \(D_{c}^{b}(D_{X})\) is the full subcategory of \(D^{b}(D_{X})\) whose objects consist of bounded complexes of coherent \(D_{X}\)-modules. Then, in Chapter 7, the Riemann-Hilbert correspondence is examined very carefully. The original version [Z. Mebkhout, Compos. Math. 51, 51–62, 63–88 (1984; Zbl 0566.32021); M. Kashiwara, Sémin. Goulaouic-Schwartz 1979–1980, Équat. dériv. part., Exposé No. 19, 6 p. (1980; Zbl 0444.58014)], affirming that for a complex manifold the de Rham functor gives an equivalence of categories between \(D_{rh}^{b}(X)\) and \(D_{c}^{b}(X)\) (= the category of bounded complexes of sheaves with constructible cohomology) is not proved (The authors don’t mention the work of Z. Mebkhout [Sur le problème de Hilbert-Riemann, Complex analysis, microlocal calculus and relativistic quantum theory, Proc. Colloq., Les Houches 1979, Lect. Notes Phys. 126, 90–110 (1979; Zbl 0444.32003)].

Following Beilinson-Bernstein, an algebraic version is given, with a sketch of proof (i.e. for the verification that the construction of the isomorphism coincides with the one induced by the de Rham functor, the reader is send to M. Saito [Publ. Res. Inst. Math. Sci. 26, No. 2, 221–333 (1990; Zbl 0727.14004)]. What is proved in detail is the following statement (Thm. 7.2-5): The de Rham functor induces an equivalence between \(\text{Mod}_{rh}(D_{X})\) and \(\text{Perv}(\mathbb{C}_{X})\) where \(\text{Mod}_{rh}(D_{X})\) is the full subcategory consisting of regular holonomic \(D_{X}\)-modules of \(\text{Mod}_{h}(D_{X})\) (= the category of holomonic \(D_{X}\)-modules) \(X\) being a smooth algebraic variety, and \(\text{Perv}(\mathbb{C}_{X})\) is the category of perverse sheaves. As an application the authors give a proof of the comparison theorem.

Chapter 8 considers in detail perverse sheaves and intersection cohomology groups, assuming the basic notions concerning constructible sheaves, introduced previously. This self-contained acount is specially interesting. It contains also the W. Borho-R. MacPherson theorem in [ Analysis and Topology on Singular Spaces, (Luminy 1981), Astéristique 101–102, 23–74 (1983; Zbl 0576.14046)] which allows to construct geometrically the representation of the Weyl groups of semi simple algebraic groups. This chapter contains a paragraph about Hodge modules. The second part of the book is dedicated to representation theory. In fact, the text is focused on explaining how the \(D\)-module theory play a crucial role in proving the celebrated conjecture of Kazhdan-Lusztig.

Chapter 9 summarizes (with no proofs) the basic notions of algebraic groups and Lie algebras; some examples are given in about 30 pages; one finds here many important results (such as Weyl’s character formula or the Borel-Weil-Bott theorem). The next chapter discuss conjugacy class in semi simple Lie algebras using the theory of invariant polynomials for the adjoint representation, and in particular a parametrization of conjugacy classes, used in the sequel in chapter 11, “Representation of Lie algebras and \(D\)-modules”, a proof of the Beilinson-Bernstein correspondence [A. Beilinson and J. Bernstein, C. R. Acad. Sci., Paris, Sér. I 292, 15–18 (1981; Zbl 0476.14019)] between representations of semisimple Lie algebras and \(D\)-modules on flag manifolds. In particular, in the classification of equivariant \(D\)-modules, the Riemann-Hilbert correspondence is used in an essential way.

Finally, in Chapter 12, the authors give an account of the Kazhdan-Lusztig conjecture. The strategy of proof of this conjecture (Thm. 12.2.7) is to use the Beilinson-Bernstein correspondence between \(g\)-modules (\(g\) being a Lie algebra) and \(D\)-modules on flag manifolds, and then the Riemann-Hilbert correspondence between \(D\)-modules on flag manifolds and perverse sheaves on flag manifolds. In \(\S\) 12.3 the \(D\)-modules associated to the higher weight modules are introduced, and, again the Riemann-Hilbert correspondence is used. Kazhdan-Lusztig predicted that the Kazhdan-Lusztig polynomials should be closely related to the geometry of Schubert varieties. The precise statement is given by Th. 12.2.5 (where the intersection cohomology complex of a certain Schubert variety appears). The proof of this theorem is given in the thirteenth chapter, “Hecke algebras and Hodge module”.

In this last chapter, the authors describe a geometric realization of the group algebras of Weyl groups (resp. Hecke algebras) via \(D\)-modules (resp. via Hodge modules). The proof of the calculation of intersection cohomology of Schubert varieties (i.e. Th. 12.2.5) is done using the theory of Hodge modules. The book has also 5 appendices (Algebraic Varieties, Derived categories and Derived Functions, Sheaves an functor in Derived Categories, Filtered Rings, Symplectic Geometry), of about 65 pages, in which the reader will find all technical ingredients used in the book.

Summing up, this is a valuable book, which puts in evidence important applications of \(D\)-module theory.

Reviewer: Gheorghe Gussi (Bucureşti)

##### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

22E46 | Semisimple Lie groups and their representations |

14F40 | de Rham cohomology and algebraic geometry |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |