Representation Theory and Harmonic Analysis on Semisimple Lie Groups

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Paul J. Sally (Jr.), David A. Vogan
American Mathematical Soc., 1989 - Mathematics - 350 pages
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.
 

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Contents

Introduction
1
Harmonic analysis of tempered distributions on semisimple Lie groups ofreal rank one
13
On the classification of irreducible representations of real algebraic groups
101
Lefschetz formulas on nonelliptic complexes
171
Homogeneous complex manifolds and representations of semisimple Liegroups
223
On HarishChandras theory of the Eisenstein integral for real semisimple Lie groups
287
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