导语

  

内容提要

  

    詹姆斯·汉弗莱斯著的《BGG范畴O中半单Lie代数的表示(英文版)(精)》是第一本介绍极其重要的Kazhdan-Lusztig猜想(1979年)有关工作的书，该猜想是关于C上半单Lie代数g的最高权单模的特征标的。这个架构是由Bernstein-Gelfand-Gelfand(BGG)引进的模范畴O，它包括了g的所有最高权模，如Verma模和有限维单模。这个范畴的类比在表示论的许多领域中已颇具影响。
    第I部分可用作自学或中级研究生课程一个学期的教材，附有大量的习题和例题。主要的预备知识是要求熟悉g的结构理论。书中讲述了范畴D中的基本技术，如BGG互反和Jantzen平移函子，最后以Kazhdan-Lusztig猜想证明的一个概述(归功于Beilinson-Bernstein和Brylinski-Kashiwara)结束。完整证明超出了本书范围，它需要深刻的几何方法：D-模和旗簇的反常层。
    第II部分介绍了当前研究中重要的相关专题：抛物范畴D、射影函子、斜模、扭变和完备函子，以及Beilinson-Ginzburg-Soergel的Koszul对偶定理。

Preface
Chapter 0.Review of Semisimple Lie Algebras
  §0.1.Cartan Decomposition
  §0.2.Root Systems
  §0.3.Weyl Groups
  §0.4.Chevalley-Bruhat Ordering of W
  §0.5.Universal Enveloping Algebras
  §0.6.Integral Weights
  §0.7.Representations
  §0.8.Finite Dimensional Modules
  §0.9.Simple Modules for s(2，C)
Part I.Highest Weight Modules
  Chapter 1.Category O：Basics
    §1.1.Axioms and Consequences
    §1.2.Highest Weight Modules
    §1.3.Verma Modules and Simple Modules
    §1.4.Maximal Vectors in Verma Modules
    §1.5.Example：s(2，C)
    §1.6.Finite Dimensional Modules
    §1.7.Action of the Center
    §1.8.Central Characters and Linked Weights
    §1.9.Harish-Chandra Homomorphism
    §1.10.Harish-Chandra's Theorem
    §1.11.Category O is Artinian
    §1.12.Subcategories OX
    §1.13.Blocks
    §1.14.Formal Characters of Finite Dimensional Modules
    §1.15.Formal Characters of Modules in O
    §1.16.Formal Characters of Verma Modules
    Notes
  Chapter 2. Characters of Finite Dimensional Modules
    §2.1.Summary of Prerequisites
    §2.2.F0rmal Characters Revisited
    §2.3.The Functions P and q
    §2.4.Formulas of Weyl and Kostant
    §2.5.Dimension Formula
    §2.6.Maximal Submodule of M(λ)，入∈Λ+
    §2.7.Related Topics
    Notes
  Chapter 3.Category O：Methods
    §3.1.Horn and Ext
    §3.2.Duality in O
    §3.3.Duals of Highest Weight Modules
    §3.4.The Reflection Group W[λ]
    §3.5.Dominant and Antidominant Weights
    §3.6.Tensoring Verma Modules with Finite Dimensional Modules
    §3.7.Standard Filtrations
    §3.8.Projectives in O
    §3.9.Indecomposable Projectives
    §3.10.Standard Filtrations of Projectives
    §3.11.BGG Reciprocity
    §3.12.Example：s(2，C)
    §3.13.Projective Generators and Finite Dimensional Algebras
    §3.14.Contravariant Forms
    §3.15.Universal Construction
    Notes
  Chapter 4.Highest Weight Modules I
  Chapter 5.Highest Weight Modules II
  Chapter 6.Extensions and resolutions
  Chapter 7.Translation Functors
  Chapter 8.Kazhdan-Lusztig Theory
Part II.Further Developments
  Chapter 9.Parabolic Versions of Category O
  Chapter 10.Projective Functors and Principal Series
  Chapter 11.Tilting Modules
  Chapter 12.Twisting and Completion Functors
  Chapter 13.Complements
Bibliography
Frequently Used Symbols
Index