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.

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