Paul Sally

David A. Vogan

This book brings together five papers that have been influential in the study of Lie groups. Though published more than 20 years ago, these papers made fundamental contributions that deserve much broader exposure. In addition, the subsequent literature that has subsumed these papers cannot replace the originality and vitality they contain. The editors have provided a brief introduction to each paper, as well as a synopsis of the major developments which have occurred in the area covered by each paper. Included here are the doctoral theses of Arthur, Osborne, and Schmid. Arthur's thesis is closely related to Trombi's paper insofar as both deal with harmonic analysis on real semisimple Lie groups, and, in particular, analysis on the Schwartz space of Harish-Chandra. Arthur's thesis is concerned with the image under the Fourier transform of the Schwartz space of a semisimple Lie group of real rank one, while Trombi's paper provides an expository account of the harmonic analysis associated to the decomposition of the Schwartz space under the regular representation. In his thesis, Osborne extends the Atiyah-Bott fixed point theorem for elliptic complexes to obtain a fixed point formula for complexes that are not elliptic. Schmid proves a generalization of the Borel-Weil theorem concerning an explicit and geometric realization of the irreducible representations of a compact, connected semisimple Lie group. Langlands's fundamental paper provides a classification of irreducible, admissible representations of real reductive Lie groups.

Representation Theory and Harmonic Analysis on Semisimple Lie Groups

Anthony W. Knapp

David A. Vogan, Jr.

This book offers a systematic treatment--the first in book form--of the development and use of cohomological induction to construct unitary representations. George Mackey introduced induction in 1950 as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group. Later a parallel construction using complex analysis and its associated co-homology theories grew up as a result of work by Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups. The book, which is accessible to students beyond the first year of graduate school, will interest mathematicians and physicists who want to learn about and take advantage of the algebraic side of the representation theory of Lie groups. Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle," and four appendices on algebra and analysis.

Anthony W. Knapp and David A. Vogan, Jr

Cohomological Induction and Unitary Representations

Edward Frenkel

Masaki Kashiwara

Alain Valette

Nolan R. Wallach

Michael Cowling

Six leading experts lecture on a wide spectrum of recent results on the subject of the title. They present a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces, and recall the concept of amenability. They further illustrate how representation theory is related to quantum computing; and much more. Taken together, this volume provides both a solid reference and deep insights on current research activity.

Lectures given at the C.I.M.E. Summer School held in Venice, Italy, June 10-17, 2004

Representation Theory and Complex Analysis

Lectures Given at the C.I.M.E. Summer School Held in Venice, Italy, June 10-17, 2004

David A. Vogan

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