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OverviewThis 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. Full Product DetailsAuthor: James E. HumphreysPublisher: American Mathematical Society Imprint: American Mathematical Society Edition: UK ed. Volume: No. 94 Weight: 0.710kg ISBN: 9780821846780ISBN 10: 0821846787 Pages: 289 Publication Date: 30 July 2008 Audience: Professional and scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: Temporarily unavailable  The supplier advises that this item is temporarily unavailable. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out to you. Table of ContentsReview 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.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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