Representations of Semisimple Lie Algebras in the BGG Category (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 94)

Product Description 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. Review 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 About the Author James E. Humphreys: University of Massachusetts, Amherst, Amherst, MA