Representations of semisimple Lie algebras in the BGG category O
著者
書誌事項
Representations of semisimple Lie algebras in the BGG category O
（Graduate studies in mathematics, v. 94）
American Mathematical Society, c2008
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注記
Includes bibliographical reference (p. 271-282) and index
内容説明・目次
内容説明
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.
目次
Review of semisimple Lie algebras Highest weight modules: Category $\mathcal{O}$: Basics Characters of finite dimensional modules Category $\mathcal{O}$: Methods Highest weight modules I Highest weight modules II Extensions and resolutions Translation functors Kazhdan-Lusztig theory Further developments: Parabolic versions of category $\mathcal{O}$ Projective functors and principal series Tilting modules Twisting and completion functors Complements Bibliography Frequently used symbols Index.
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