Lectures on the Orbit Method

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1 Lectures on the Orbit Method A. A. Kirillov Graduate Studies in Mathematics Volume 64 American Mathematical Society Providence, Rhode Island

2 Preface Introduction xv xvii Chapter 1. Geometry of Coadjoint Orbits 1 1. Basic definitions Coadjoint representation Canonical form GQ 4 2. Symplectic structure on coadjoint orbits The first (original) approach The second (Poisson) approach The third (symplectic reduction) approach Integrality condition Coadjoint invariant functions General properties of invariants Examples The moment map The universal property of coadjoint orbits Some particular cases Polarizations Elements of symplectic geometry Invariant polarizations on homogeneous symplectic manifolds 26 vn

3 viii Contents Chapter 2. Representations and Orbits of the Heisenberg Group Heisenberg Lie algebra and Heisenberg Lie group Some realizations Universal enveloping algebra U(t)) The Heisenberg Lie algebra as a contraction Canonical commutation relations Creation and annihilation operators Two-sided ideals in C/(h) H. Weyl reformulation of CCR The standard realization of CCR Other realizations of CCR Uniqueness theorem Representation theory for the Heisenberg group The unitary dual H The generalized characters of H The infinitesimal characters of H The tensor product of unirreps Coadjoint orbits of the Heisenberg group Description of coadjoint orbits Symplectic forms on orbits and the Poisson structure on h* Projections of coadjoint orbits Orbits and representations Restriction-induction principle and construction of unirreps Other rules of the User's Guide Polarizations Real polarizations Complex polarizations Discrete polarizations 69 Chapter 3. The Orbit Method for Nilpotent Lie Groups Generalities on nilpotent Lie groups Comments on the User's Guide The unitary dual 73

4 ix 2.2. The construction of unirreps Restriction-induction functors Generalized characters Infinitesimal characters Functional dimension Plancherel measure Worked-out examples The unitary dual Construction of unirreps Restriction functor Induction functor Decomposition of a tensor product of two unirreps Generalized characters Infinitesimal characters Functional dimension Plancherel measure Other examples Proofs Nilpotent groups with 1-dimensional center The main induction procedure The image of U(g) and the functional dimension The existence of generalized characters Homeomorphism of G and O(G) 106 Chapter 4. Solvable Lie Groups Exponential Lie groups Generalities Pukanszky condition Restriction-induction functors Generalized characters Infinitesimal characters General solvable Lie groups Tame and wild Lie groups Tame solvable Lie groups 123

5 3. Example: The diamond Lie algebra g The coadjoint orbits for g Representations corresponding to generic orbits Representations corresponding to cylindrical orbits Amendments to other rules Rules Rules 6,7, and Chapter 5. Compact Lie Groups Structure of semisimple compact Lie groups Compact and complex semisimple groups Classical and exceptional groups Coadjoint orbits for compact Lie groups Geometry of coadjoint orbits Topology of coadjoint orbits Orbits and representations Overlook Weights of a unirrep Functors Ind and Res '" Borel-Weil-Bott theorem The integral formula for characters Infinitesimal characters Intertwining operators 176 Chapter 6. Miscellaneous Semisimple groups Complex semisimple groups Real semisimple groups Lie groups of general type Poincare group Odd symplectic groups Beyond Lie groups Infinite-dimensional groups p-adic and adelic groups Finite groups 189

6 xi 3.4. Supergroups Why the orbit method works Mathematical argument Physical argument Byproducts and relations to other domains Moment map Integrable systems Some open problems and subjects for meditation Functional dimension Infinitesimal characters Multiplicities and geometry Complementary series Finite groups Infinite-dimensional groups 205 Appendix I. Abstract Nonsense Topology Topological spaces Metric spaces and metrizable topological spaces Language of categories Introduction to categories The use of categories Application: Homotopy groups Cohomology Generalities Group cohomology Lie algebra cohomology Cohomology of smooth manifolds 220 Appendix II. Smooth Manifolds Around the definition Smooth manifolds. Geometric approach Abstract smooth manifolds. Analytic approach Complex manifolds Algebraic approach 236

7 xii Contents 2. Geometry of manifolds Fiber bundles Geometric objects on manifolds Natural operations on geometric objects Integration on manifolds Symplectic and Poisson manifolds Symplectic manifolds Poisson manifolds Mathematical model of classical mechanics Symplectic reduction 265 Appendix III. Lie Groups and Homogeneous Manifolds Lie groups and Lie algebras Lie groups Lie algebras Five definitions of the functor Lie: G ~~> g Universal enveloping algebras Review of the set of Lie algebras Sources of Lie algebras The variety of structure constants Types of Lie algebras Semisimple Lie algebras Abstract root systems Lie algebra sl{2, C) Root system related to (g, f>) Real forms Homogeneous manifolds G-sets G-manifolds Geometric objects on homogeneous manifolds 325 Appendix IV. Elements of Functional Analysis Infinite-dimensional vector spaces Banach spaces Operators in Banach spaces 335

8 xiii 1.3. Vector integrals Hilbert spaces Operators in Hilbert spaces Types of bounded operators Hilbert-Schmidt and trace class operators Unbounded operators Spectral theory of self-adjoint operators Decompositions of Hilbert spaces Application to representation theory Mathematical model of quantum mechanics 355 Appendix V. Representation Theory Infinite-dimensional representations of Lie groups Generalities on unitary representations Unitary representations of Lie groups Infinitesimal characters Generalized and distributional characters Non-commutative Fourier transform Induced representations Induced representations of finite groups Induced representations of Lie groups ^representations of smooth G-manifolds Mackey Inducibility Criterion 389 References 395 Index 403

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