Representation Theory of Lie Groups: (IAS/Park City Mathematics Series)

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.

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