Holomorphy And Convexity In Lie Theory (de Gruyter Expositions In Mathematics) (english And German Edition)

Neeb (mathematics, Technische Univesitat, Darmstadt) explores abstract representation theory; convex geometry and representations of vector spaces; convex geometry of Lie algebras; highest weight representations of Lie algebras, Lie groups and Semigroups; and complex geometry in relation to representation theory. A. Abstract Representation Theory -- Chapter I. Reproducing Kernel Spaces 3 -- I.1. Operator-valued Positive Definite Kernels 3 -- I.2. The Cone Of Positive Definite Kernels 14 -- Chapter Ii. Representations Of Involutive Semigroups 20 -- Ii. 1. Involutive Semigroups 21 -- Ii. 2. Bounded Representations 24 -- Ii. 3. Hermitian Representations 29 -- Ii. 4. Representations On Reproducing Kernel Spaces 34 -- Chapter Iii. Positive Definite Functions On Involutive Semigroups 52 -- Iii. 1. Positive Definite Functions -- The Discrete Case 53 -- Iii. 2. Enveloping C*-algebras 68 -- Iii. 3. Multiplicity Free Representations 80 -- Chapter Iv. Continuous And Holomorphic Representations 99 -- Iv. 1. Continuous Representations And Positive Definite Functions 99 -- Iv. 2. Holomorphic Representations Of Involutive Semigroups 119 -- B. Convex Geometry And Representations Of Vector Spaces -- Chapter V. Convex Sets And Convex Functions 125 -- V.1. Convex Sets And Cones 126 -- V.2. Finite Reflection Groups And Convex Sets 138 -- V.3. Convex Functions And Fenchel Duality 147 -- V.4. Laplace Transforms 163 -- V.5. The Characteristic Function Of A Convex Set 174 -- Chapter Vi. Representations Of Cones And Tubes 184 -- Vi. 1. Commutative Representation Theory 185 -- Vi. 2. Representations Of Cones 195 -- Vi. 3. Holomorphic Representations Of Tubes 205 -- Vi. 4. Realization Of Cyclic Representations By Holomorphic Functions 209 -- Vi. 5. Holomorphic Extensions Of Unitary Representations 214 -- C. Convex Geometry Of Lie Algebras -- Chapter Vii. Convexity In Lie Algebras 221 -- Vii. 1. Compactly Embedded Cartan Subalgebras 222 -- Vii. 2. Root Decompositions 231 -- Vii. 3. Lie Algebras With Many Invariant Convex Sets 251 -- Chapter Viii. Convexity Theorems And Their Applications 265 -- Viii. 1. Admissible Coadjoint Orbits And Convexity Theorems 266 -- Viii. 2. The Structure Of Admissible Lie Algebras 292 -- Viii. 3. Invariant Elliptic Cones In Lie Algebras 306 -- D. Highest Weight Representations Of Lie Algebras, Lie Groups, And Semigroups -- Chapter Ix. Unitary Highest Weight Representations: Algebraic Theory 327 -- Ix. 1. Generalized Highest Weight Representations 328 -- Ix. 2. Positive Complex Polarizations 344 -- Ix. 3. Highest Weight Modules Of Finite-dimensional Lie Algebras 356 -- Ix. 4. The Metaplectic Factorization 361 -- Ix. 5. Unitary Highest Weight Representations Of Hermitian Lie Algebras 374 -- Chapter X. Unitary Highest Weight Representations: Analytic Theory 387 -- X.1. The Convex Moment Set Of A Unitary Representation 388 -- X.2. Irreducible Unitary Representations 394 -- X.3. The Metaplectic Representation And Its Applications 400 -- X.4. Special Properties Of Unitary Highest Weight Representations 411 -- X.5. Moment Sets For C*-algebras 419 -- X.6. Moment Sets For Group Representations 428 -- Chapter Xi. Complex Ol'shanskii Semigroups And Their Representations 442 -- Xi. 1. Lawson's Theorem On Ol'shanskii Semigroups 443 -- Xi. 2. Holomorphic Extension Of Unitary Representations 457 -- Xi. 3. Holomorphic Representations Of Ol'shanskii Semigroups 464 -- Xi. 4. Irreducible Holomorphic Representations 470 -- Xi. 5. Gelfand-raikov Theorems For Ol'shanskii Semigroups 476 -- Xi. 6. Decomposition And Characters Of Holomorphic Representations 477 -- Chapter Xii. Realization Of Highest Weight Representations On Complex Domains 493 -- Xii. 1. The Structure Of Groups Of Harish-chandra Type 494 -- Xii. 2. Representations Of Groups Of Harish-chandra Type 514 -- Xii. 3. The Compression Semigroup And Its Representations 524 -- Xii. 5. Hilbert Spaces Of Square Integrable Holomorphic Functions 538 -- E. Complex Geometry And Representation Theory -- Chapter Xiii. Complex And Convex Geometry Of Complex Semigroups 557 -- Xiii. 1. Locally Convex Functions And Local Recession Cones 559 -- Xiii. 2. Invariant Convex Sets And Functions In Lie Algebras 563 -- Xiii. 3. Calculations In Low-dimensional Cases 571 -- Xiii. 4. Biinvariant Plurisubharmonic Functions 576 -- Xiii. 5. Complex Semigroups And Stein Manifolds 586 -- Xiii. 6. Biinvariant Domains Of Holomorphy 595 -- Chapter Xiv. Biinvariant Hilbert Spaces And Hardy Spaces On Complex Semigroups 600 -- Xiv. 1. Biinvariant Hilbert Spaces 601 -- Xiv. 2. Hardy Spaces Defined By Sup-norms 608 -- Xiv. 3. Hardy Spaces Defined By Square Integrability 616 -- Xiv. 4. The Fine Structure Of Hardy Spaces 623 -- Chapter Xv. Coherent State Representations 645 -- Xv. 1. Complex Structures On Homogeneous Spaces 646 -- Xv. 2. Coherent State Representations 650 -- Xv. 3. Heisenberg's Uncertainty Principle And Coherent States 656 -- Appendix I. Bounded Operators On Hilbert Spaces 665 -- Appendix Ii. Spectral Measures And Unbounded Operators 677 -- Appendix Iii. Holomorphic Functions On Infinite-dimensional Spaces 686 -- Appendix Iv. Symplectic Geometry 694 -- Appendix V. Simple Modules Of P-length 2 705 -- Appendix Vi. Symplectic Modules Of Convex Type 715 -- Appendix Vii. Square Integrable Representations Of Locally Compact Groups 727 -- Appendix Viii. The Stone-von Neumann-mackey Theorem 742. By Karl-hermann Neeb. Includes Bibliographical References And Index. Preface Introduction A. Abstract Representation Theory Chapter I. Reproducing Kernel Spaces I.1. Operator-Valued Positive Definite Kernels I.2. The Cone of Positive Definite Kernels Chapter II. Representations of Involutive Semigroups II.1. Involutive Semigroups II.2. Bounded Representations II.3. Hermitian Representations II.4. Representations on Reproducing Kernel Spaces Chapter III. Positive Definite Functions on Involutive Semigroups III.1. Positive Definite Functions — the Discrete Case III.2. Enveloping C*-algebras III.3. Multiplicity Free Representations Chapter IV. Continuous and Holomorphic Representations IV.1. Continuous Representations and Positive Definite Functions IV.2. Holomorphic Representations of Involutive Semigroups B. Convex Geometry and Representations of Vector Spaces Chapter V. Convex Sets and Convex Functions V.1. Convex Sets and Cones V.2. Finite Reflection Groups and Convex Sets V.3. Convex Functions and Fenchel Duality V.4. Laplace Transforms V.5. The Characteristic Function of a Convex Set Chapter VI. Representations of Cones and Tubes VI.1. Commutative Representation Theory VI.2. Representations of Cones VI.3. Holomorphic Representations of Tubes VI.4. Realization of Cyclic Representations by Holomorphic Functions VI.5. Holomorphic Extensions of Unitary Representations C. Convex Geometry of Lie Algebras Chapter VII. Convexity in Lie Algebras VII.1. Compactly Embedded Cartan Subalgebras VII.2. Root Decompositions VII.3. Lie Algebras With Many Invariant Convex Sets Chapter VIII. Convexity Theorems and Their Applications VIII.1. Admissible Coadjoint Orbits and Convexity Theorems VIII.2. The Structure of Admissible Lie Algebras VIII.3. Invariant Elliptic Cones in Lie Algebras D. Highest Weight Representations of Lie Algebras, Lie Groups, and Semigroups Chapter IX. Unitary Highest Weight Representations: Algebraic Theory IX.1. Generalized Highest Weight Representations IX.2. Positive Complex Polarizations IX.3. Highest Weight Modules of Finite-Dimensional Lie Algebras IX.4. The Metaplectic Factorization IX.5. Unitary Highest Weight Representations of Hermitian Lie Algebras Chapter X. Unitary Highest Weight Representations: Analytic Theory X.1. The Convex Moment Set of a Unitary Representation X.2. Irreducible Unitary Representations X.3. The Metaplectic Representation and Its Applications X.4. Special Properties of Unitary Highest Weight Representations X.5. Moment Sets for C*-algebras X.6. Moment Sets for Group Representations Chapter XI. Complex Ol’shanskiĭ Semigroups and Their Representations XI.1. Lawson’s Theorem on Ol’shanskiĭ Semigroups XI.2. Holomorphic Extension of Unitary Representations XI.3. Holomorphic Representations of Ol’shanskiĭ Semigroups XI.4. Irreducible Holomorphic Representations XI.5. Gelfand-Raïkov Theorems for Ol’shanskiĭ Semigroups XI.6. Decomposition and Characters of Holomorphic Representations Chapter XII. Realization of Highest Weight Representations on Complex Domains XII.1. The Structure of Groups of Harish-Chandra Type XII.2. Representations of Groups of Harish-Chandra Type XII.3. The Compression Semigroup and Its Representations XII.4. Examples XII.5. Hilbert Spaces of Square Integrable Holomorphic Functions E. Complex Geometry and Representation Theory Chapter XIII. Complex and Convex Geometry of Complex Semigroups XIII.1. Locally Convex Functions and Local Recession Cones XIII.2. Invariant Convex Sets and Functions in Lie Algebras XIII.3. Calculations in Low-Dimensional Cases XIII.4. Biinvariant Plurisubharmonic Functions XIII.5. Complex Semigroups and Stein Manifolds XIII.6. Biinvariant Domains of Holomorphy Chapter XIV. Biinvariant Hilbert Spaces and Hardy Spaces on Complex Semigroups XIV.1. Biinvariant Hilbert Spaces XIV.2. Hardy Spaces Defined by Sup-Norms XIV.3. Hardy Spaces Defined by Square Integrability XIV.4. The Fine Structure of Hardy Spaces Chapter XV. Coherent State Representations XV.1. Complex Structures on Homogeneous Spaces XV.2. Coherent State Representations XV.3. Heisenberg’s Uncertainty Principle and Coherent States Appendices Appendix I. Bounded Operators on Hilbert Spaces Appendix II. Spectral Measures and Unbounded Operators Appendix III. Holomorphic Functions on Infinite-Dimensional Spaces Appendix IV. Symplectic Geometry Appendix V. Simple Modules of p-Length 2 Appendix VI. Symplectic Modules of Convex Type Appendix VII. Square Integrable Representations of Locally Compact Groups Appendix VIII. The Stone – von Neumann-Mackey Theorem Bibliography List of Symbols Index

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Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urba?ski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

This text is a treatment of the relationship between unitary representations of lie groups, holomorphic representations of complex semigroups and the complex and convex geometry of adjoint and coadjoint orbits. "This monograph is a remarkable collection of facts, theorems and theories concerning such subjects as holomorphy and convexity in Lie theory." Zentralblatt für Mathematik