Introduction to the Second Edition
This second edition includes a number of corrections, minor changes or ampli-
fications to the original text, as well as some further material that reports on later
The numbering in the first edition has been maintained. The new additions
have been inserted either at the beginning or the end of a paragraph, or a chapter.
This explains some numbering that is a bit unusual: In section 3 of Chapter 0, in
particular, there is a subsection 3.0 (which has subsections). The main new topics
I, §8, which gives a construction, in the framework of this book, of the Zucker-
man functors and describes their main properties.
II, §10 provides sharp bounds, case by case, for the vanishing theorems, due to
Enright, Kumaresan, Parthasarathy, Vogan-Zuckerman, which in many cases are
improvements of the ones given originally.
VI, §0 introduces the translation functors and their relationship with relative
Lie algebra cohomology.
VI, §5 is devoted to the Vogan-Zuckerman theorem, which describes
(F, V), where V runs through the irreducible unitary (g, if)-modules and
F through the finite dimensional irreducible (g, if)-modules.
XIII, §4 studies the cohomology of an 5-arithmetic subgroup of G with coeffi-
cients in a rational G-module.
Moreover, a new Chapter XIV has been added. It outlines how the main results
proved in Chapters VII, VIII and XIII for the cohomology of discrete cocompact
subgroups extend to general 5-arithmetic subgroups of semisimple algebraic groups
over number fields.
It has been almost 20 years since the publication of the original version of
this book. During that time the methods of homological algebra have become
increasingly important in the construction of admissible representations and in the
study of arithmetic groups. Although some of the original material in this book has
been superseded, it is still a useful reference. We thank the American Mathematical
Society, in particular S. Gelfand, for having encouraged us to publish this second
edition. The authors would also like to thank the editorial staff for an extremely
helpful and thorough reading of the manuscript.
A. BOREL, N. WALLACH
THE INSTITUTE FOR ADVANCED STUDY, Princeton, NJ 08540
UNIVERSITY OF CALIFORNIA, San Diego, La Jolla, CA 92014