# Dirac Operators in Representation Theory

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.

less209 pages; ISBN 9780817644932

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Title: Dirac Operators in Representation Theory

Author: Jing-Song Huang; Pavle Pandzic

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- Contents

- Academic > Mathematics > Differential equations > Lie groups
- Academic > Mathematics > Algebra > Representations of groups
- Academic > Mathematics > Algebra > Nonassociative rings
- Academic > Mathematics > Algebra > Lie algebras
- Academic > Mathematics > General > Vinogradov, Ivan Matveevich, 1891-
- Academic > Mathematics > General > Mathematics; Periodicals
- Academic > Mathematics > General > Mathematics
- Academic > Mathematics > Analysis
- Academic > Physics > Physics; Dictionaries
- Academic > Physics > General
- Mathematics > Algebra
- Mathematics > Geometry
- Science > Physics

### In the press

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