Representations of Semisimple Lie Algebras in the BGG Category O

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American Mathematical Soc., 2008 - Mathematics - 289 pages
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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 manyareas of representation theory.Part I can be used as a text for independent study or for a mid-level 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 beyondthe 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 Koszulduality theorem of Beilinson-Ginzburg-Soergel.
  

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Contents

Basics
13
Contents
19
Characters of Finite Dimensional Modules
37
Methods
47
Notes
71
Highest Weight Modules II
93
Extensions and Resolutions
107
Contents
110
KazhdanLusztig Theory
153
Parabolic Versions of Category O
181
Projective Functors and Principal Series
207
Tilting Modules
223
Twisting and Completion Functors
235
Complements
251
Bibliography
271
Frequently Used Symbols
283

Translation Functors
129

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About the author (2008)

James E. Humphreys was born in Erie, Pennsylvania, and received his A.B. from Oberlin College, 1961, and his Ph.D. from Yale University, 1966. He has taught at the University of Oregon, Courant Institute (NYU), and the University of Massachusetts at Amherst (now retired). He visits IAS Princeton, Rutgers. He is the author of several graduate texts and monographs.

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