A main result in this area is the beilinsonbernstein localization. Dmodules, perverse sheaves, and representation theory. Goal to a complex semisimple lie algebra g, one can attach its ag variety b. It relates d modules on flag varieties g b to representations of the lie algebra g \displaystyle \mathfrak g of a reductive group g. D modules continues to be an active area of stimulating research in such mathematical areas as algebra, analysis, differential equations, and representation theory. Introduction to algebraic d modules pavel etingof abstract. Introduction to d modules and representation theory jonathan wang contents 1.
Suppose that x specais a ne and note that any system of linear partial di erential equations with coe cients in agives rise to a cyclic d module on x. We emphasize the portions of the theory which provide some relevant geometric context for the topic of the second half of this essay, the beilinsonbernstein localization theorem. Finally, if time allows, we will discuss applications of dmodules in representation theory, namely the. Algebraic dmodules and representation theory of semisimple lie groups dragan mili. Dmodules on the affine flag variety and representations of. Algebraic geometry, representation theory and topology of singular spaces. The category d x of left d modules on x is the abelian category of sheaves of left d x modules which are quasicoherent as o x modules. Representation theory and dmodules on flag varieties numdam. We also establish an equivalence between a certain subcategory of. This book, the first devoted specifically to holonomic d modules, provides a unified treatment of both regular and irregular d modules. Key to d modules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects d modules to representation theory. Takeuchi, and toshiyuki tanisaki, translated by kiyoshi.
Dmodules, geometric representation theory, and arithmetic. I am going to take part in a seminar on d module and applications, the textbooks that will be used are. Introduction to dmodules and representation theory contents. The aim of this talk is to explain how to establish the theory for arbitrary. In this section, we introduce and develop the theory of dmodules on smooth algebraic varieties. Introduction to d modules in representation theory pavle pand zi c university of zagreb and university of freiburg 35th winter school \geometry and physics. A geometricallyoriented treatment of the subject is very timely and has long been desired, especially since the discovery of d modules in the early 1980s and the quiver approach to quantum groups in the early 1990s. Localization and representation theory of reductive lie groups. Oct 11, 2019 mit graduate seminar on dmodules and perverse sheaves.
We prove a localizationtype result, which establishes an equivalence between certain subcategories on both sides. We begin with defining some basic functors on dmodules, introduce the notion of. 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. In particular, we discuss the theory of localization of modules overthe envelop. Sabbah, dmodules coherents et holonomes, travaux en cours, hermann, paris 1993.
Here i mean by the representation theory the one for real semisimple lie group. Xis the borel subalgebra bx which corresponds to the point x. D modules, perverse sheaves, and representation theory, and a primer of algebraic d modules. An important role in this theory will be played by bernsteins inequality, which is an algebraic incarnation of the heisenberg uncertainty principle in quantum mechanics. Systems of linear pdes arise as an example when the variety is a ne nspace. Milicic, lectures on the algebraic theory of dmodules. Dmodules, perverse sheaves, and representation theory by ryoshi hotta 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.
Let us see what verma modules look like explicitly in the simplest case of g sl 2. Dmodules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Algebraic dmodules and representation theory of semisimple. This classic monograph provides an overview of modern advances in representation theory from a geometric standpoint. 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. Dmodules on smooth algebraic varieties in this section, we introduce and develop the theory of dmodules on smooth algebraic varieties. Introduction to algebraic dmodules pavel etingof abstract. Dmodules and representation theory of lie groups numdam. For a sheaf of rings a on a scheme x, we let moda denote the abelian category of sheaves.
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. Dmodules, perverse sheaves, and representation theory, by ryoshi. Dmodules, representation theory, and quantum groups. Dmodules, perverse sheaves, and representation theory ryoshi. Poissonde rham homology, d modules on poisson varieties, and conjectures iii. D modules are useful as a means of applying the methods of homological algebra and sheaf theory to the study of analytic systems of partial differential equations. Dmodules are also applied in geometric representation theory. Applications to representation theory and generalizations. D modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Dmodules, representation theory, and quantum groups lectures given at the 2nd session of the centro internazionale matematico estivo c. It estab lishes a bridge b etween analytic objects regula r holonomic d modules and geo metric ones con.
A nonvanishing theorem for cohomology of d modules 26 5. The idea is that the action of the differential operation given by a vector field. The theory of d modules is a rich area of study combining ideas from algebra and differential equations, and it has significant applications to diverse areas such as singularity theory and representation theory. Key to dmodules, perverse sheaves, and representation theory is the authors. For the new english edition, the two authors of the original book, r. Introduction to dmodules in representation theory pavle pand zi c university of zagreb and university of freiburg 35th winter school \geometry and physics srn, january 2015. We discuss bfunctions, and study the riemannhilbert correspondence between holonomic dmodules and perverse sheaves. 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. Hecke eigensheaves and the geometric langlands conjecture 45 4. Finally, if time allows, we will discuss applications of d modules in representation theory, namely the.
This talk also leans toward the viewpoint of representation theory, as the author is primarily interested in the use of algebraic d modules to study the representation theory of complex semisimple lie algebras. 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 theory of dmodules can be used to help us generalise this to explain what. 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. Dmodules, perverse sheaves, and representation theory, by ryoshi hotta, kiyoshi. The rst is a classical result of serre which allows for translation between algebraic geometry and commutative algebra. Representation theory and complex geometry modern birkhauser. In this talk, i will motivate the study of d modules by describing two similar theorems. D modules perverse sheaves and representation theory 1st edition. Poisson varieties, symplectic resolutions, and dmodules. Apr 16, 2014 one of the hightlights of the theory of dmodules is the riemannhilbert co rrespondence. Newest dmodules questions mathematics stack exchange.
While d modules have played a major role in the representation theory of real reductive groups for a long time, the use of appropriate differential operators on padic spaces to study representations of padic reductive groups is a more recent development. Let kbe an algebraically closed eld of characteristic zero. Dmodules, geometric representation theory, and arithmetic applications 48 december 2017 university of oxford abstracts tomoyuki abe university of tokyo title. It is often the case in representation theory that in order to gain a good understanding of a category of modules of some sort, one has to reinterpret it in more geometric terms, by which we mean either as the category of dmodules on an algebraic variety, or as the category of quasicoherent sheaves on some usually, di. D module theory is essentially the algebraic study of systems of linear partial differential equations. Dmodules, perverse sheaves, and representation theory is a greatly expanded translation of the japanese edition entitled d kagun. This book introduces d modules and their applications avoiding all unnecessary oversophistication.
This expository paper represents an introduction to some aspects of the current research in representation theory of semisimple lie groups. This work examines in detail the foundations of d module theory and its intersection with perverse sheaves and representation theory. Introduction to dmodules and representation theory 3 in general, o x y does not equal o x ko y because x ydoes not have the product topology. Dmodules, perverse sheaves, and representation theory by ryoshi hotta. D modules, perverse sheaves, and representation theory is a greatly expanded translation of the japanese edition entitled d kagun to daisugun d modules and algebraic groups which was published by springerverlag tokyo, 1995. The kernel of this morphism is a ghomogeneous vector bundle b over x. We will rst develop the theory of dmodules on the a ne space, and prove bernsteins theorem using this theory. Systematic and carefully written, this is a unique and essential textbook at the graduate level for classroom use or selfstudy. These are notes of my minicourse at the workshop geometry and representation theory, vienna, january 2017. Key to d modules, perverse sheaves, and representation theory is the authors essential algebraicanalytic approach to the theory, which connects d modules to representation. Gaitsgorys lecture notes on geometric representation theory. Dmodules, perverse sheaves, and representation theory book. 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.
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