- ホーム
- > 洋書
- > 英文書
- > Science / Mathematics
Full Description
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.
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
