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# Dirac Operators in Representation Theory

Jing-Song Huang, Pavle Pandzic

200 pages, parution le 31/03/2005

## 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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## Caractéristiques techniques

 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

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