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

relevant developments.

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

are:

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

Ext*

K

(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been superseded, it is still a useful reference.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

1999

THE INSTITUTE FOR ADVANCED STUDY, Princeton, NJ 08540

UNIVERSITY OF CALIFORNIA, San Diego, La Jolla, CA 92014

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