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Dirac operators are widely used in physics and in the mathematical areas of differential geometry and group-theoretic settings, in particular, in 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. This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. The early chapters give background material and lead up to a proof of Vogan’s conjecture on Dirac cohomology which illuminates the algebraic nature of Dirac operators. This proof is then used to obtain simple proofs of many classical theorems such as the Bott–Borel–Weil theorem and the Atiyah–Schmid theorem. The Dirac cohomology, defined by Kostant’s cubic Dirac operator, is closely related to other Lie algebra cohomologies, such as n-cohomology and (g,K)-cohomology. Via an approach similar to the proof of Vogan’s conjecture for the half Dirac operators, the authors present a new proof of the Casselman– Osburne theorem on Lie algebra cohomology. Other topics deal with the multiplicity of automorphic forms, the connection of Dirac operators to an equivariant cohomology and to K-theory. The exposition is systematic and self-contained and will be of interest to researchers and graduate students in representation theory, differential geometry, and physics.
