| Preface |
|
v | |
| Introduction |
|
xiii | |
| A. Abstract Representation Theory |
|
|
Reproducing Kernel Spaces |
|
|
3 | (17) |
|
Operator-Valued Positive Definite Kernels |
|
|
3 | (11) |
|
The Cone of Positive Definite Kernels |
|
|
14 | (6) |
|
Representations of Involutive Semigroups |
|
|
20 | (32) |
|
|
|
21 | (3) |
|
|
|
24 | (5) |
|
Hermitian Representations |
|
|
29 | (5) |
|
Representations on Reproducing Kernel Spaces |
|
|
34 | (18) |
|
Positive Definite Functions on Involutive Semigroups |
|
|
52 | (47) |
|
Positive Definite Functions-the Discrete Case |
|
|
53 | (15) |
|
|
|
68 | (12) |
|
Multiplicity Free Representations |
|
|
80 | (19) |
|
Continuous and Holomorphic Representations |
|
|
99 | (26) |
|
Continuous Representations and Positive Definite Functions |
|
|
99 | (20) |
|
Holomorphic Representations of Involutive Semigroups |
|
|
119 | (6) |
| B. Convex Geometry and Representations of Vector Spaces |
|
|
Convex Sets and Convex Functions |
|
|
125 | (59) |
|
|
|
126 | (12) |
|
Finite Reflection Groups and Convex Sets |
|
|
138 | (9) |
|
Convex Functions and Fenchel Duality |
|
|
147 | (16) |
|
|
|
163 | (11) |
|
The Characteristic Function of a Convex Set |
|
|
174 | (10) |
|
Representations of Cones and Tubes |
|
|
184 | (37) |
|
Commutative Representation Theory |
|
|
185 | (10) |
|
|
|
195 | (10) |
|
Holomorphic Representations of Tubes |
|
|
205 | (4) |
|
Realization of Cyclic Representations by Holomorphic Functions |
|
|
209 | (5) |
|
Holomorphic Extensions of Unitary Representations |
|
|
214 | (7) |
| C. Convex Geometry of Lie Algebras |
|
|
Convexity in Lie Algebras |
|
|
221 | (44) |
|
Compactly Embedded Cartan Subalgebras |
|
|
222 | (9) |
|
|
|
231 | (20) |
|
Lie Algebras With Many Invariant Convex Sets |
|
|
251 | (14) |
|
Convexity Theorems and Their Applications |
|
|
265 | (62) |
|
Admissible Coadjoint Orbits and Convexity Theorems |
|
|
266 | (26) |
|
The Structure of Admissible Lie Algebras |
|
|
292 | (14) |
|
Invariant Elliptic Cones in Lie Algebras |
|
|
306 | (21) |
| D. Highest Weight Representations of Lie Algebras, Lie Groups, and Semigroups |
|
|
Unitary Highest Weight Representations: Algebraic Theory |
|
|
327 | (60) |
|
Generalized Highest Weight Representations |
|
|
328 | (16) |
|
Positive Complex Polarizations |
|
|
344 | (12) |
|
Highest Weight Modules of Finite-Dimensional Lie Algebras |
|
|
356 | (5) |
|
The Metaplectic Factorization |
|
|
361 | (13) |
|
Unitary Highest Weight Representations of Hermitian Lie Algebras |
|
|
374 | (13) |
|
Unitary Highes Weight Representations: Analytic Theory |
|
|
387 | (55) |
|
The Convex Moment Set of a Unitary Representation |
|
|
388 | (6) |
|
Irreducible Unitary Representations |
|
|
394 | (6) |
|
The Metaplectic Representation and Its Applications |
|
|
400 | (11) |
|
Special Properties of Unitary Highest Weight Representations |
|
|
411 | (8) |
|
Moment Sets for C*-algebras |
|
|
419 | (9) |
|
Moment Sets for Group Representations |
|
|
428 | (14) |
|
Complex Ol'shanskii Semigroups and Their Representations |
|
|
442 | (51) |
|
Lawson's Theorem on Ol'shanskii Semigroups |
|
|
443 | (14) |
|
Holomorphic Extension of Unitary Representations |
|
|
457 | (7) |
|
Holomorphic Representations of Ol'shanskii Semigroups |
|
|
464 | (6) |
|
Irreducible Holomorphic Representations |
|
|
470 | (6) |
|
Gelfand-Raikov Theorems for Ol'shanskii Semigroups |
|
|
476 | (1) |
|
Decomposition and Characters of Holomorphic Representations |
|
|
477 | (16) |
|
Realization of Highest Weight Representations on Complex Domains |
|
|
493 | (64) |
|
The Structure of Groups of Harish-Chandra Type |
|
|
494 | (20) |
|
Representations of Groups of Harish-Chandra Type |
|
|
514 | (10) |
|
The Compression Semigroup and Its Representations |
|
|
524 | (6) |
|
|
|
530 | (8) |
|
Hilbert Spaces of Square Integrable Holomorphic Functions |
|
|
538 | (19) |
| E. Complex Geometry and Representation Theory |
|
|
Complex and Convex Geometry of Complex Semigroups |
|
|
557 | (43) |
|
Locally Convex Functions and Local Recession Cones |
|
|
559 | (4) |
|
Invariant Convex Sets and Functions in Lie Algebras |
|
|
563 | (8) |
|
Calculations in Low-Dimensional Cases |
|
|
571 | (5) |
|
Biinvariant Plurisubharmonic Functions |
|
|
576 | (10) |
|
Complex Semigroups and Stein Manifolds |
|
|
586 | (9) |
|
Biinvariant Domains of Holomorphy |
|
|
595 | (5) |
|
Biinvariant Hilbert Spaces and Hardy Spaces on Complex Semigroups |
|
|
600 | (45) |
|
Biinvariant Hilbert Spaces |
|
|
601 | (7) |
|
Hardy Spaces Defined by Sup-Norms |
|
|
608 | (8) |
|
Hardy Spaces Defined by Square Integrability |
|
|
616 | (7) |
|
The Fine Structure of Hardy Spaces |
|
|
623 | (22) |
|
Coherent State Representations |
|
|
645 | (106) |
|
Complex Structures on Homogeneous Spaces |
|
|
646 | (4) |
|
Coherent State Representations |
|
|
650 | (6) |
|
Heisenberg's Uncertainty Principle and Coherent States |
|
|
656 | (9) |
| Appendices |
|
|
Appendix I. Bounded Operators on Hilbert Spaces |
|
|
665 | (12) |
|
Appendix II. Spectral Measures and Unbounded Operators |
|
|
677 | (9) |
|
Appendix III. Holomorphic Functions on Infinite-Dimensional Spaces |
|
|
686 | (8) |
|
Appendix IV. Symplectic Geometry |
|
|
694 | (11) |
|
Appendix V. Simple Modules of p-Length 2 |
|
|
705 | (10) |
|
Appendix VI. Symplectic Modules of Convex Type |
|
|
715 | (12) |
|
Appendix VII. Square Integrable Representations of Locally Compact Groups |
|
|
727 | (15) |
|
Appendix VIII. The Stone--von Neumann--Mackey Theorem |
|
|
742 | (9) |
| Bibliography |
|
751 | (16) |
| List of Symbols |
|
767 | (4) |
| Index |
|
771 | |
Rohkem infot De Gruyter e-raamatute kohta