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.

Inhalt
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.