Representations of Semisimple Lie Algebras in the BGG Category O

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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 many areas 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 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.

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Product Details

  • ISBN-13: 9780821846780
  • Publisher: American Mathematical Society
  • Publication date: 8/17/2008
  • Series: Graduate Studies in Mathematics Series , #94
  • Pages: 289
  • Product dimensions: 7.10 (w) x 10.20 (h) x 0.80 (d)

Table of Contents

Ch. 0 Review of Semisimple Lie Algebras 1

Pt. 1 Highest Weight Modules

Ch. 1 Category [O]: Basics 13

Ch. 2 Characters of Finite Dimensional Modules 37

Ch. 3 Category [O]: Methods 47

Ch. 4 Highest Weight Modules I 73

Ch. 5 Highest Weight Modules II 93

Ch. 6 Extensions and Resolutions 107

Ch. 7 Translation Functors 129

Ch. 8 Kazhdan-Lusztig Theory 153

Pt. II Further Developments

Ch. 9 Parabolic Versions of Category [O] 181

Ch. 10 Projective Functors and Principal Series 207

Ch. 11 Tilting Modules 223

Ch. 12 Twisting and Completion Functors 235

Ch. 13 Complements 251

Bibliography 271

Frequently Used Symbols 283

Index 287

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