
Preface  Preview Material  Table of Contents  Supplementary Material 
Graduate Studies in Mathematics 2008; 289 pp; hardcover Volume: 94 ISBN10: 0821846787 ISBN13: 9780821846780 List Price: US$61 Member Price: US$48.80 Order Code: GSM/94 See also: Lectures on Quantum Groups  Jens Carsten Jantzen Enveloping Algebras  Jacques Dixmier Lectures on the Orbit Method  A A Kirillov Geometric Representation Theory and Extended Affine Lie Algebras  Erhard Neher, Alistair Savage and Weiqiang Wang  This is the first textbook treatment of work leading to the landmark 1979 KazhdanLusztig 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 BernsteinGelfandGelfand, 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 midlevel 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 KazhdanLusztig Conjecture (due to BeilinsonBernstein and BrylinskiKashiwara). 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 BeilinsonGinzburgSoergel. Readership Graduate students and research mathematicians interested in Lie theory, and representation theory. Reviews "One of the goals Humphreys had in mind was to provide a textbook suitable for graduate students. This has been achieved by keeping prerequisites to a minimum, by careful dealing with technical parts of the proofs, and by offering a large number of exercises."  Mathematical Reviews 


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