IAS/Park City Mathematics Series 2000; 340 pp; hardcover Volume: 8 ISBN10: 0821819410 ISBN13: 9780821819418 List Price: US$57 Member Price: US$45.60 Order Code: PCMS/8
 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 HarishChandra on the general structure of infinitedimensional 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 KirillovKostant "philosophy of coadjoint orbits" for unitary representations. K. Vilonen presents recent advances in the BeilinsonBernstein theory of "localization". And JianShu 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 infinitedimensional representation theory. Titles in this series are copublished 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, BottBorelWeil 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 KostantSekiguchi 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
JianShu 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
