Synopsis "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 vogans 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 kostants 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.
