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Dirac Operators in Representation Theory
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Dirac Operators in Representation Theory

Dirac Operators in Representation Theory

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

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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