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.
Author(s): Jing-Song Huang, Pavle Pandzic
Series: Mathematics: Theory & Applications
Edition: 1
Publisher: Birkhäuser Boston
Year: 2006
Language: English
Pages: 212
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
