D-modules perverse sheaves and representation theory pdf

There is a result of gri ths which says that if we have a oneparameter family of smooth varieties f. This will lead us to the notion of perverse sheaves. The nal goal of the course is the riemannhilbert correspondence, which shows that a category of dmodules encodes topological information about the underlying space. As such, it provided a motivation for introducing perverse sheaves. That new branch may be called geometric representation theory.

Newest perversesheaves questions feed to subscribe to this rss feed, copy and. Preface dmodules, perverse sheaves, and representation theory is a greatly expanded translation of the japanese edition entitled d kagun to daisugun dmodules and algebraic groups which was published by springerverlag tokyo, 1995. Key to dmodules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects dmodules to representation theory and other areas. In terms of prior knowledge, in the first half of the course just basic undergraduate algebra will be needed. It also nds fruitful applications in other areas, such as algebraic geometry, hodge theory and representation theory. Dmodules, perverse sheaves, and representation theory, and a primer of algebraic dmodules.

However, to make the above relations precise, we need to use a generalization of perverse sheaves. It doesnt treat perverse sheaves, but it has a good overview of stratified spaces in the first few chapter and develops constructible sheaves and the six functors in chapter v. As beilinsonbernstein put it in their seminal paper bebe, the discovery of dmodules and of perverse sheaves has made representation theory, to a large extent, part of algebraic geometry. Dmodules are also applied in geometric representation theory. A bit more reader friendly than kashiwaraschapira is borels intersection cohomology. Systematic and carefully written, this is a unique and essential textbook at the graduate level for classroom use or selfstudy. Hecke eigensheaves and the geometric langlands conjecture 45 4. Tanisaki, dmodules, perverse sheaves, and representation theory, birkh auser, 2008. Algebraic geometry, representation theory and topology of singular spaces. Dmodules continues to be an active area of stimulating research in such mathematical areas as algebra, analysis, differential equations, and representation theory. The following example can serve as a test case for the. The techniques developed are quite general and can be. Hecke algebras and kazhdanlusztig polynomials 10 4.

Sorry, we are unable to provide the full text but you may find it at the following locations. Tanisaki, dmodules, perverse sheaves, and representation theory. Newest dmodules questions mathematics stack exchange. On the other hand, the intersection cohomology complex see below is a perverse sheaf, regardless of the singularities of x. Translated from the 1995 japanese edition by takeuchi. A perverse sheaf is an object c of the bounded derived category of sheaves with constructible cohomology on. Nikolaev lecture 1 10 january 2012 0 references the original references on the subject are unreadable. Dmodules or coherent sheaves on some related variety. For many applications in representation theory, perverse sheaves can be treated as a black box, a category with certain formal properties. Dmodules, perverse sheaves, and representation theory by ryoshi hotta, takeuchim kiyoshi and toshiyuki tanisaki no static citation data no static citation data cite.

Dmodules, perverse sheaves, and representation theory is a greatly expanded translation of the japanese edition entitled d kagun to daisugun dmodules and algebraic groups which was published by springerverlag tokyo, 1995. Key to dmodules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects dmodules to representation theory and other areas of mathematics. Contents preface v introduction 1 part i d modules and perverse sheaves 1 preliminary notions 15 1. The authors begin by recalling the main results of the theory of indsheaves and subanalytic sheaves, explaining in detail the operations on dmodules and their tempered holomorphic solutions. Request pdf on mirabolic dmodules let an algebraic group g act on x, a connected algebraic manifold, with finitely many orbits. The main application of dmodules explained in this essay is to hodge theory. Regular holonomic dmodules and the riemannhilbert correspondence 7 3. This book, the first devoted specifically to holonomic dmodules, provides a unified treatment of both regular and irregular dmodules. Geometric representation theory studies representations of various symmetry objects like algebraic groups, hecke algebras, quantum groups, quivers etc. Oblomkov the cohomology ring of certain compactified jacobians. The course will essentially cover the rst eight chapters of htt, minimizing however the analytic.

However, more recently some relations between modular perverse sheaves, with coe cients in. Pdf we prove that the length function for perverse sheaves and algebraic regular holonomic dmodules on a smooth complex algebraic variety y is an. Dmodules continues to be an active area of stimulating research in such mathematical areas as algebra, analysis, differential equations, and representation theory key to dmodules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects dmodules to representation theory and other areas of mathematics. The riemannhilbert correspondence establishes a link between certain dmodules and constructible sheaves. Hitchin type moduli stacks in automorphic representation theory. D modules perverse sheaves and representation theory 1st. Dmodules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Tanisaki, dmodules, perverse sheaves, and representation theory, progress in mathematics 236, birkhauser. Key to dmodules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects dmodules to representation theory. The extension of two perverse sheaves is a perverse sheaf. D modules perverse sheaves and representation theory 1st edition recognizing the pretentiousness ways to acquire this book d modules perverse sheaves and representation theory 1st edition is additionally useful. Tanisaki dmodules, perverse sheaves, and representation theory a. The eld grew from a sequence of signi cant discoveries of connections among dmodules, constructible sheaves, and representation theory, including the microlocal studies of dmodules, the riemannhilbert correspondence, the theory of perverse sheaves, and the.

Perverse sheaves, and representation theory progress in mathematics book 236 ryoshi hotta. Key to dmodules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects d modules to representation theory and other areas of mathematics. Dmodules, perverse sheaves, and representation theory. Finally, if time allows, we will discuss applications of dmodules in representation theory. A main result in this area is the beilinsonbernstein localization. Key to dmodules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects dmodules to representation. Dmodule theory and its typical applications to representation theory as we believe that this is a nice way to understand the whole subject. In particular, we show that their solution complexes are monodromic. The relation between perverse sheaves and ls gln x is even more immediate. You might also try dimcas sheaves in topology, which does talk about perverse. I am going to take part in a seminar on dmodule and applications, the textbooks that will be used are. Mathematical tripos part iii lecture courses in 20192020.

Symplectic approaches in geometric representation theory. My thesis, some microlocal aspects of perverse coherent sheaves and equivariant dmodules. This work examines in detail the foundations of dmodule theory and its intersection with perverse sheaves and representation theory. This approach was highly successful in representation theory. Cn for a nite set, then the cohomology of the bres hwx. Online access exact functors on perverse coherent sheaves. Part i is devoted to dmodule theory, placing special emphasis on holonomic modules and constructible sheaves. Hilbert correspondence gives us a deep relationship between perverse sheaves and regular holonomic dmodules. Perverse sheaves, and representation theory translated by kiyoshi takeuchi birkhauser boston basel berlin. Finally, we will discuss applications of dmodules to representation theory, namely the beilinsonbernstein localization theorem. Among applications of perverse sheaf methods to representation theory that. We discuss bfunctions, and study the riemannhilbert correspondence between holonomic dmodules and perverse sheaves. While the course contains many abstract notions and constructions, the style.

Key to dmodules, perverse sheaves, and dmodules theory is the authors essential algebraicanalytic approach to the theory, which they have a large number of practice problems to help understand computations with perverse sheaves. You have remained in right site to start getting this info. Dmodules, perverse sheaves, and representation theory by ryoshi hotta. Most results obtained in geometric representation theory applied to categories of representations over a eld of characteristic zero.

Dennis gaitsgory, nick rozenblyum, crystals and dmodules, pure and applied mathematics quarterly volume 10 2014 number 1 arxiv. Regular and irregular holonomic dmodules cambridge core. Hochschild cohomology of torusequivariant dmodules. Dmodules, perverse sheaves, and representation theory core. Lusztig zmgraded lie algebras and perverse sheaves, iv. On some topological properties of fourier transforms of. Zhang shtukas and the taylor expansion of lfunctions ii.