## Representations of Semisimple Lie Algebras in the BGG Category OThis 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 |

271 | |

Frequently Used Symbols | 283 |

### Common terms and phrases

A G A+ A G f A G h adjoint antidominant weight arbitrary BGG Reciprocity BGG resolution BGG Theorem Bruhat ordering central character Chapter cohomology composition factor multiplicities compute contravariant form coset defined denote dimension direct sum dominant element embedding exact functor example Exercise finite dimensional module fixed flag variety follows formal characters Grothendieck group highest weight module Homo implies indecomposable induction injective integral weights involving isomorphic Jantzen filtration Kazhdan-Lusztig Kazhdan-Lusztig Conjecture KL Conjecture KL polynomials Lemma Lie algebra Lie group linkage class Loewy M G O M(fi M(sa maximal vector notation parabolic Verma modules projective functors projective module proof Proposition prove regular scalar self-dual semisimple short exact sequence shows simple modules simple submodule socle standard filtration subalgebra subcategory subgroup submodule summand tilting module translation functors upper closure Verma module M(A wall-crossing functors weight spaces Weyl group