Preface

Representation theory plays a central role in Lie theory and has developed in

numerous specialized directions over recent decades. Motivation comes from

many areas of mathematics and physics, notably the Langlands program.

The methods involved are also diverse, including fruitful interactions with

"modern" algebraic geometry. Here we focus primarily on algebraic methods

in the case of a semisimple Lie algebra g over C with universal enveloping

algebra U(g), where the prerequisites are relatively modest.

The category Mod?7(g) of all [/(g)-modules is much too large to be

understood algebraically. Fortunately, many interesting Lie group repre-

sentations can be studied effectively in terms of a more limited subcate-

gory where modules are subjected to appropriate finiteness conditions: the

BGG category O introduced in the early 1970s by Joseph Bernstein, Israel

Gelfand, and Sergei Gelfand. Their papers, stimulated in part by Verma's

1966 thesis [251], have led to far-reaching work involving a growing list of

researchers. In this book we discuss systematically the early work leading

to the Kazhdan-Lusztig Conjecture and its proof around 1980. This is at

the core of more recent developments, some of which we go on to introduce

in the later chapters. Taken on its own, the study of category O offers a

rewarding tour of the beautiful terrain that lies just beyond the classical

Cartan-Weyl theory of finite dimensional representations of g.

Part I (comprising Chapters 1-8) is written in textbook style, at the level

of a second year graduate course in a U.S. university. The emphasis here is

on highest weight modules, starting with Verma modules and culminating

in the determination of formal characters of simple highest weight modules

in the setting of the Kazhdan-Lusztig Conjecture (1979). The proof of this

conjecture requires sophisticated ideas from algebraic geometry which go

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