Dirac Operators in Representation Theory / Mathematics: Theory & Applications (PDF)
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...
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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.
- Autoren: Jing-Song Huang , Pavle Pandzic
- 2007, 2006, 200 Seiten, Englisch
- Verlag: Springer-Verlag GmbH
- ISBN-10: 0817644938
- ISBN-13: 9780817644932
- Erscheinungsdatum: 27.05.2007
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- Dateiformat: PDF
- Größe: 1.72 MB
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- Vorlesefunktion
This book contains a more detailed explanation of the results from several recent papers of the authors. The book is aimed at a somewhat broader audience. Clifford algebras are presented rather thoroughly. Some basics of Lie groups and their representations are mostly relegated to earlier literature. There is a good introduction to the so-called cohomological induction, which is short but still gives the main ideas of some parts of the proofs. – MathSciNet
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