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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.

Representations of groups. --- Dirac equation. --- Differential operators. --- Operators, Differential --- Differential equations --- Operator theory --- Differential equations, Partial --- Quantum field theory --- Wave equation --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Topological Groups. --- Group theory. --- Global differential geometry. --- Operator theory. --- Mathematical physics. --- Topological Groups, Lie Groups. --- Group Theory and Generalizations. --- Differential Geometry. --- Operator Theory. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Functional analysis --- Geometry, Differential --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Groups, Topological --- Continuous groups --- Mathematics --- Topological groups. --- Lie groups. --- Differential geometry. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Differential geometry --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups