Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups: Second Edition

The book by Borel and Wallach is a classic treatment of the use of cohomology in representation theory, particularly in the setting of automorphic forms and discrete subgroups. The authors begin with general material, covering Lie algebra cohomology, as well as continuous and differentiable cohomology. Much of the machinery is designed for the study of the cohomology of locally symmetric spaces, realized as double coset spaces, where the quotient is by a maximal compact subgroup and by a discrete subgroup. Such spaces are central to applications to number theory and the study of automorphic forms. The authors give a careful presentation of relative Lie algebra cohomology of admissible and of unitary -modules. As part of the general development, the Langlands classification of irreducible admissible representations is given. Computations of important examples are another valuable part of the book. In the twenty years between the first and second editions of this work, there was immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. The second edition is a corrected and expanded version of the original, which was an important catalyst in the growth of the field. Besides the fundamental material on cohomology and discrete subgroups present in the first edition, this edition also contains expositions of some of the most important developments of the two intervening decades Ch. 0. Notation And Preliminaries -- Ch. I. Relative Lie Algebra Cohomology -- Ch. Ii. Scalar Product, Laplacian And Casimir Element -- Ch. Iii. Cohomology With Respect To An Induced Representation -- Ch. Iv. The Langlands Classification And Uniformly Bounded Representations -- Ch. V. Cohomology With Coefficients In Ii[subscript [infinity]](g) -- Ch. Vi. The Computation Of Certain Cohomology Groups -- Ch. Vii. Cohomology Of Discrete Subgroups And Lie Algebra Cohomology -- Ch. Viii. The Construction Of Certain Unitary Representations And The Computation Of The Corresponding Cohomology Groups -- Ch. Ix. Continuous Cohomology And Differentiable Cohomology -- Ch. X. Continuous And Differentiable Cohomology For Locally Compact Totally Disconnected Groups -- Ch. Xi. Cohomology With Coefficients In Ii[subscript [infinity]](g): The P-adic Case -- Ch. Xii. Differentiable Cohomology For Products Of Real Lie Groups And T.d. Groups -- Ch. Xiii. Cohomology Of Discrete Cocompact Subgroups -- Ch. Xiv. Non-cocompact S-arithmetic Subgroups. A. Borel, N. Wallach. Includes Bibliographical References (p. 253-258) And Index. Deals with two types of cohomology spaces pertaining to reductive Lie group G and a discrete cocompact subgroup. Material presented here naturally divides into two parts, one devoted mainly to real Lie groups, the other to locally compact totally disconnected groups, in particular reductive p-adic groups, or products of real Lie groups and totally disconnected groups. Each part in turn contains roughly three main items: general results on the cohomology used, specific results for cohomology and representations of reductive groups, and applications to discrete cocompact subgroups. This second edition reports on developments in the field since 1980. Annotation copyrighted by Book News, Inc., Portland, OR It has been nearly twenty years since the first edition of this work. In the intervening years, there has been immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. This second edition is a corrected and expanded version of the original, which was an important catalyst in the expansion of the field. Besides the fundamental material on cohomology and discrete subgroups present in the first edition, this edition also contains expositions of some of the most important developments of the last two decades. There has been immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. This second edition presents the fundamental material on cohomology and discrete subgroups, and also contains expositions of some of the most important developments over the years.