Dirac Operators in Representation Theory

Jing-Song Huang, Pavle Pandzic - Collection Mathematics: Theory & Applications

200 pages, parution le 31/03/2005

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Résumé

This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective.

Key topics covered include:

Proof of Vogan's conjecture on Dirac cohomology
Simple proofs of many classical theorems, such as the Bott-Borel-Weil theorem and the Atiyah-Schmid theorem
Dirac cohomology, defined by Kostant's cubic Dirac operator, along with other closely related kinds of cohomology, such as n-cohomology and (g,K)-cohomology
Cohomological parabolic induction and $A_q(\lambda)$ modules
Discrete series theory, characters, existence and exhaustion
Sharpening of the Langlands formula on multiplicity of automorphic forms, with applications
Dirac cohomology for Lie superalgebras

An excellent contribution to the mathematical literature of representation theory, this self-contained exposition offers a systematic examination and panoramic view of the subject. The material will be of interest to researchers and graduate students in representation theory, differential geometry, and physics.

Sommaire

Preface
Lie Groups, Lie Algebras and Representations
Clifford Algebras and Spinors
Dirac Operators in the Algebraic Setting
A Generalized Bott-Borel-Weil Theorem
Cohomological Induction
Properties of Cohomologically Induced Modules
Discrete Series
Dimensions of Spaces of Automorphic Forms
Dirac Operators and Nilpotent Lie Algebra Cohomology
Dirac Cohomology for Lie Superalgebras
References
Index

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	PAPIER
Éditeur(s)	Birkhäuser
Auteur(s)	Jing-Song Huang, Pavle Pandzic
Collection	Mathematics: Theory & Applications
Parution	31/03/2005
Nb. de pages	200
Format	16 x 24
Couverture	Relié
Poids	490g
Intérieur	Noir et Blanc
EAN13	9780817632182
ISBN13	978-0-8176-3218-2