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Representation Theory of Lie Groups
Edited by: Jeffrey Adams, University of Maryland, College Park, MD, and David Vogan, Massachusetts Institute of Technology, Cambridge, MA
A co-publication of the AMS and IAS/Park City Mathematics Institute.
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IAS/Park City Mathematics Series
2000; 340 pp; hardcover
Volume: 8
ISBN-10: 0-8218-1941-0
ISBN-13: 978-0-8218-1941-8
List Price: US$57
Member Price: US$45.60
Order Code: PCMS/8
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This book contains written versions of the lectures given at the PCMI Graduate Summer School on the representation theory of Lie groups. The volume begins with lectures by A. Knapp and P. Trapa outlining the state of the subject around the year 1975, specifically, the fundamental results of Harish-Chandra on the general structure of infinite-dimensional representations and the Langlands classification.

Additional contributions outline developments in four of the most active areas of research over the past 20 years. The clearly written articles present results to date, as follows: R. Zierau and L. Barchini discuss the construction of representations on Dolbeault cohomology spaces. D. Vogan describes the status of the Kirillov-Kostant "philosophy of coadjoint orbits" for unitary representations. K. Vilonen presents recent advances in the Beilinson-Bernstein theory of "localization". And Jian-Shu Li covers Howe's theory of "dual reductive pairs".

Each contributor to the volume presents the topics in a unique, comprehensive, and accessible manner geared toward advanced graduate students and researchers. Students should have completed the standard introductory graduate courses for full comprehension of the work. The book would also serve well as a supplementary text for a course on introductory infinite-dimensional representation theory.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Readership

Graduate students and research mathematicians interested in representation theory specifically Lie groups and their representations.

Reviews

"Altogether, the volume brings a coherent description of an important and beautiful part of representation theory, which certainly will be of substantial use for postgraduate students and mathematicians interested in the area."

-- European Mathematical Society Newsletter

Table of Contents

A. W. Knapp and P. E. Trapa, Representations of semisimple Lie groups
  • Introduction
  • Some representations of \(SL(n,\mathbb{R})\)
  • Semsimple groups and structure theory
  • Introduction to representation theory
  • Cartan subalgebras and highest weights
  • Action by the Lie algebra
  • Cartan subgroups and global characters
  • Discrete series and asymptotics
  • Langlands classification
  • Bibliography
R. Zierau, Representations in Dolbeault cohomology
  • Introduction
  • Complex flag varieties and orbits under a real form
  • Open \(G_0\)-orbits
  • Examples, homogeneous bundles
  • Dolbeault cohomology, Bott-Borel-Weil theorem
  • Indefinite harmonic theory
  • Intertwining operators I
  • Intertwining operators II
  • The linear cycle space
  • Bibliography
L. Barchini, Unitary representations attached to elliptic orbits. A geometric approach
  • Introduction
  • Globalizations
  • Dolbeault cohomology and maximal globalization
  • \(L^2\)-cohomology and discrete series representations
  • Indefinite quantization
  • Bibliography
D. A. Vogan, Jr., The method of adjoint orbits for real reductive groups
  • Introduction
  • Some ideas from mathematical physics
  • The Jordan decomposition and three kinds of quantization
  • Complex polarizations
  • The Kostant-Sekiguchi correspondence
  • Quantizing the action of \(K\)
  • Associated graded modules
  • A good basis for associated graded modules
  • Proving unitarity
  • Exercises
  • Bibliography
K. Vilonen, Geometric methods in representation theory
  • Introduction
  • Overview
  • Derived categories of constructible sheaves
  • Equivariant derived categories
  • Functors to representations
  • Matsuki correspondence for sheaves
  • Characteristic cyles
  • The character formula
  • Microlocalization of Matsuki = Sekiguchi
  • Homological algebra (appendix by M. Hunziker)
  • Bibliography
Jian-Shu Li, Minimal representations and reductive dual pairs
  • Introduction
  • The oscillator representation
  • Models
  • Duality
  • Classification
  • Unitarity
  • Minimal representations of classical groups
  • Dual pairs in simple groups
  • Bibliography
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