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Cohomological induction and unitary representations. (English) Zbl 0863.22011
Princeton Mathematical Series. 45. Princeton, NJ: Univ. Press. xvii, 948 p. (1995).
This book is a thorough and excellent presentation of the “cohomological” approach to the construction and classification of irreducible representations of semisimple real Lie groups, which originated in the work of Gregg Zuckerman. The authors’ perspective is that this approach to constructing representations is based on complex analysis, as opposed to the Mackey induction theory which is based on real analysis. This being said, however, the actual techniques are essentially algebraic in nature, and the complex analysis is mostly there for background and motivation. While Mackey induction always provides spaces with inner products, the spaces constructed by algebraic methods do not have obvious inner products, and constructing inner products and proving unitarity becomes one of the main technical issues in this approach. The book is structured around five major theorems: the Duality Theorem, the Irreducibility Theorem, the Signature Theorem, the Unitarizability Theorem, and the Transfer Theorem. Together, these results allow cohomological induction to be used to construct irreducible representations, to identify ones which are unitary, and to relate the results to those of other approaches, specifically the Langlands classification. It is intended for readers who already know about elementary Lie theory, universal enveloping algebras, and the abstract representation theory of compact groups. The reader is also expected to be able to work with distributions on manifolds and with homological algebra, but the required material is provided in appendices. Later chapters assume the Cartan-Weyl theory for semisimple algebras and compact connected groups, basic facts about real forms and parabolic subalgebras, and some use of spectral sequences (the latter are conveniently presented in another appendix). The book begins with a thirty-eight page introduction which gives a very clear entry to these ideas and their history and development.
Reviewer: J.Repka (Toronto)

22E46 Semisimple Lie groups and their representations
22-02 Research exposition (monographs, survey articles) pertaining to topological groups