Dirac Operators in Representation Theory (Mathematics: Theory & Applications)

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. Cover......Page 1 Dirac Operators in Representation Theory......Page 4 Copyright - ISBN: 0817632182......Page 5 Preface......Page 6 Contents......Page 10 1.1 Lie groups and algebras......Page 14 1.2 Finite-dimensional representations......Page 22 1.3 Infinite-dimensional representations......Page 34 1.4 Infinitesimal characters......Page 39 1.5 Tensor products of representations......Page 43 2.1 Real Clifford algebras......Page 46 2.2 Complex Clifford algebras and spin modules......Page 53 2.3 Spin representations of Lie groups and algebras......Page 58 3.1 Dirac operators......Page 70 3.2 Dirac cohomology and Vogan's conjecture......Page 74 3.3 A differential on (U(g) ⊗ C(p))^K......Page 78 3.4 The homomorphism ζ......Page 83 3.5 An extension of Parthasarathy's Dirac inequality......Page 84 4.1 Kostant cubic Dirac operators......Page 86 4.2 Dirac cohomology of finite-dimensional representations......Page 89 4.3 Characters......Page 91 4.4 A generalized Weyl character formula......Page 94 4.5 A generalized Bott-Borel-Weil theorem......Page 95 5.1 Overview......Page 98 5.2 Some generalities about adjoint functors......Page 102 5.3 Homological algebra of Harish-Chandra modules......Page 108 5.4 Zuckerman functors......Page 114 5.5 Bernstein functors......Page 119 6.1 Duality theorems......Page 128 6.2 Infinitesimal character, K-types and vanishing......Page 134 6.3 Irreducibility and unitarity......Page 140 6.4 A[sub(q)](λ) modules......Page 142 6.5 Unitary modules with strongly regular infinitesimal character......Page 145 7. Discrete Series......Page 146 7.1 L[sup(2)]-index theorem......Page 147 7.2 Existence of discrete series......Page 150 7.3 Global characters......Page 151 7.4 Exhaustion of discrete series......Page 153 8.1 Hirzebruch proportionality principle......Page 158 8.2 Dimensions of spaces of automorphic forms......Page 160 8.3 Dirac cohomology and (g, K)-cohomology......Page 161 8.4 Cohomology of discrete subgroups......Page 163 9. Dirac Operators and Nilpotent Lie Algebra Cohomology......Page 166 9.1 u-homology and \bar{u}-cohomology differentials......Page 167 9.2 Hodge decomposition in the finite-dimensional case......Page 171 9.3 Hodge decomposition for p[sup(-)] - cohomology in the unitary case......Page 173 9.4 Calculating Dirac cohomology in stages......Page 175 9.5 Hodge decomposition for \bar{u}-cohomology in the unitary case......Page 181 9.6 Homological properties of Dirac cohomology......Page 185 10.1 Lie superalgebras of Riemannian type......Page 190 10.2 Dirac operator for (g, g0)......Page 196 10.3 Analog of Vogan's conjecture......Page 198 10.4 Dirac cohomology for Lie superalgebras......Page 200 References......Page 206 C......Page 210 P......Page 211 Z......Page 212 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 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.