Dirac Operators in Representation Theory [Hardcover]

This book presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. 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. The book is an excellent contribution to the mathematical literature of representation theory, and 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.

A comprehensive treatment of new ideas on Dirac operators and Dirac cohomology. Using Dirac operators as a unifying theme, the authors demonstrate how many of the important results in representation theory fit together when viewed from this perspective.

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 BottBorelWeil theorem and the AtiyahSchmid theorem

* Dirac cohomology, defined by Kostant's cubic Dirac operator, along with other closely related kinds of cohomology, such as lC$