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Representations of Semisimple Lie Algebras in the BGG Category O.
・ISBN 978-1-4704-6326-7 paper US$ 89.00
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| 著者・編者 | Humphreys, James E, |
|---|---|
| シリーズ | Graduate Studies in Mathematics |
| 出版社 | (American Mathematical Society, US) |
| 出版年月 | 2008 |
| ページ数 | 289 pp. |
| 言語 | ENG |
| ニュース番号 | ＜M25-13806＞ |
解説
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 g over C. The setting is the module category O introduced by Bernstein-Gelfand-Gelfand, which includes all highest weight modules for 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 g. Basic techniques in category 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 O , projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson-Ginzburg-Soergel.
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 g. Basic techniques in category 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 O , projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson-Ginzburg-Soergel.