XVI

Preface

well beyond the algebraic framework of earlier chapters. Thus Chapter 8

marks a shift toward the survey style used in the remainder of the book.

The chapters in Part II can to a large extent be read independently.

They supplement the more unified theme of Part I in a variety of ways,

often motivated by problems arising in Lie group representations. The book

ends with an introduction to the influential work of Beilinson, Ginzburg,

and Soergel on Koszul duality.

I have tried to keep prerequisites to a minimum. The reader needs to be

comfortable with the basic structure theory of semisimple Lie algebras over

C (summarized in Chapter 0) as well as with standard algebraic methods

including elementary homological algebra.

Exercises are scattered throughout the text (mainly in Part I) where I

thought they would do the most good. Some of the more straightforward

ones are used later in the development. At any rate, the most important

exercise for the reader is to engage actively with the ideas presented. Exam-

ples are also interspersed, though unfortunately it is difficult to gain much

direct insight from low rank cases of the sort which can be done by hand.

The deeper parts of the theory have required some imaginative leaps not

based on examples alone.

The substantial reference list includes all source material cited, together

with related books and survey articles. I have added a somewhat arbitrary

sample of other research papers to point the reader in directions such as

those sketched in Chapter 13. There is also a list of frequently used symbols,

most of which are introduced early in the book. Anyone who consults the

literature will encounter a wide array of notational choices; here I have tried

to keep things simple and consistent to the extent possible.

The mathematics presented here is not original, though parts of the

treatment may be. Many people have provided helpful feedback on ear-

lier versions of the chapters, including Troels Agerholm, Henning Ander-

sen, Brian Boe, Tom Braden, Jon Brundan, Walter Mazorchuk, Wolfgang

Soergel, Catharina Stropple, and Geordie Williamson. I am especially in-

debted to Jens Carsten Jantzen for his detailed suggestions at many stages

of the writing. His ideas have left a lasting imprint on the study of category

O. Naturally, the final choices made are my own responsibility. Corrections

and suggestions from readers are welcome.

J. E. Humphreys

February 2008

j eh@math.umass.edu