Softcover ISBN:  9781470463267 
Product Code:  GSM/94.S 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
Electronic ISBN:  9781470421205 
Product Code:  GSM/94.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 

Book DetailsGraduate Studies in MathematicsVolume: 94; 2008; 289 ppMSC: Primary 17; Secondary 20; 22;
This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra \(\mathfrak{g}\) over \(\mathbb {C}\). The setting is the module category \(\mathscr {O}\) introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for \(\mathfrak{g}\) such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory.
Part I can be used as a text for independent study or for a midlevel one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of \(\mathfrak{g}\). Basic techniques in category \(\mathscr {O}\) such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan–Lusztig Conjecture (due to Beilinson–Bernstein and Brylinski–Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: \(D\)modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category \(\mathscr {O}\), projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.ReadershipGraduate students and research mathematicians interested in Lie theory, and representation theory.

Table of Contents

Chapters

Chapter 0. Review of semisimple Lie algebras

Part 1. Highest weight modules

Chapter 1. Category $\mathcal {O}$: Basics

Chapter 2. Characters of finite dimensional modules

Chapter 3. Category $\mathcal {O}$: Methods

Chapter 4. Highest weight modules I

Chapter 5. Highest weight modules II

Chapter 6. Extensions and resolutions

Chapter 7. Translation functors

Chapter 8. KazhdanLusztig theory

Part 2. Further developments

Chapter 9. Parabolic versions of category $\mathcal {O}$

Chapter 10. Projective functors and principal series

Chapter 11. Tilting modules

Chapter 12. Twisting and completion functors

Chapter 13. Complements


Additional Material

Reviews

One of the goals Humphreys had in mind was to provide a textbook suitable for graduate students. This has been achieved by keeping prerequisites to a minimum, by careful dealing with technical parts of the proofs, and by offering a large number of exercises.
Mathematical Reviews


RequestsReview Copy – for reviewers who would like to review an AMS bookPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra \(\mathfrak{g}\) over \(\mathbb {C}\). The setting is the module category \(\mathscr {O}\) introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for \(\mathfrak{g}\) such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory.
Part I can be used as a text for independent study or for a midlevel one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of \(\mathfrak{g}\). Basic techniques in category \(\mathscr {O}\) such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan–Lusztig Conjecture (due to Beilinson–Bernstein and Brylinski–Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: \(D\)modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category \(\mathscr {O}\), projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.
Graduate students and research mathematicians interested in Lie theory, and representation theory.

Chapters

Chapter 0. Review of semisimple Lie algebras

Part 1. Highest weight modules

Chapter 1. Category $\mathcal {O}$: Basics

Chapter 2. Characters of finite dimensional modules

Chapter 3. Category $\mathcal {O}$: Methods

Chapter 4. Highest weight modules I

Chapter 5. Highest weight modules II

Chapter 6. Extensions and resolutions

Chapter 7. Translation functors

Chapter 8. KazhdanLusztig theory

Part 2. Further developments

Chapter 9. Parabolic versions of category $\mathcal {O}$

Chapter 10. Projective functors and principal series

Chapter 11. Tilting modules

Chapter 12. Twisting and completion functors

Chapter 13. Complements

One of the goals Humphreys had in mind was to provide a textbook suitable for graduate students. This has been achieved by keeping prerequisites to a minimum, by careful dealing with technical parts of the proofs, and by offering a large number of exercises.
Mathematical Reviews