Representation Theory of Solvable Lie Groups and Related Topics (Springer Monographs in Mathematics)

The purpose of the book is to discuss the latest advances in the theory of unitary representations and harmonic analysis for solvable Lie groups. The orbit method created by Kirillov is the most powerful tool to build the ground frame of these theories. Many problems are studied in the nilpotent case, but several obstacles arise when encompassing exponentially solvable settings. The book offers the most recent solutions to a number of open questions that arose over the last decades, presents the newest related results, and offers an alluring platform for progressing in this research area. The book is unique in the literature for which the readership extends to graduate students, researchers, and beginners in the fields of harmonic analysis on solvable homogeneous spaces. Preface 6 Nomenclature 9 Contents 11 1 Branching Laws and the Multiplicity Function of Unitary Representations of Exponential Solvable Lie Groups 16 1.1 Introduction 16 1.2 Generalities and Notations 17 1.2.1 Coexponential Bases 17 1.2.2 Modular Functions and Quotient Measures 18 1.2.3 Induced Representations 20 1.2.4 Polarizations 20 1.2.5 Orbit Theory 21 1.2.6 Branching Laws: Induced Representations 21 1.2.7 Restrictions 22 1.3 Pseudo-Algebraic Geometry 22 1.3.1 Pseudo-Algebraic Sets 22 1.3.2 Semi-analytic Sets 25 1.3.3 Structure of Coadjoint Orbits 25 Saturated Orbits with Respect to an Ideal of Codimension One 26 1.4 Up-Down Representations of Exponential Solvable Lie Groups 26 1.4.1 Disintegration of Up-Down Representations 27 1.4.2 The Multiplicity Function of Up-DownRepresentations 30 The Case of Normal Subgroups 36 1.5 Down-Up Representations 39 1.5.1 The Down-Up Formula 40 1.5.2 The Down-Up Multiplicity Formula 43 1.5.3 Examples 45 1.5.4 The Case of Exponential Solvable Groups 48 1.6 The Multiplicity Function of Monomial Representations 52 2 Intertwining Operators for Irreducible Representations of an Exponential Solvable Lie Group 58 2.1 Introduction 58 2.2 A Trace Relation 58 2.3 Relations Between Two Polarizations 67 2.4 Vergne Polarizations 76 2.5 The General Case 79 2.6 A Local Result 82 2.7 The Case Where h1 + h2 Is a Subalgebra 108 2.8 The Key Point Is the Convergence 111 3 Intertwining Operators of Induced Representations and Restrictions of Representations of Exponential SolvableLie Groups 121 3.1 Introduction 121 3.2 Intertwining Operators of Induced Representations of Nilpotent Lie Groups 121 3.2.1 Generalities 123 Notation and Backgrounds 123 3.2.2 Disintegration of Monomial Representations 124 Choice of Measures 124 A Canonical Disintegration Formula 126 3.2.3 Construction of the Intertwining Operator 131 Construction of Polarizations and Malcev Bases 131 Step 0 132 Step s 132 Step s+1 133 3.2.4 Examples 151 Rational Disintegration of L2 (G) 154 3.3 The Case of Exponential Solvable Groups 155 3.3.1 A Base Space of the Disintegration of Induced Representations 156 3.3.2 Construction of the Intertwining Operator 158 Several Constructions 158 3.3.3 The Inverse Operator 165 3.3.4 A Rational Disintegration of L2(G) for an Exponential Solvable Lie GroupG 170 3.3.5 Examples 171 3.4 Intertwining of Representations Induced from Maximal Subgroups of Exponential Solvable Lie Groups 174 3.5 Intertwining Operators of the Restriction of Representations of Nilpotent Lie Groups 185 3.5.1 Double-Coset Space 185 The Set of Double Cosets 187 Description of HG B 187 3.5.2 A Measure on HG B 196 3.5.3 A Concrete Intertwining Operator 203 3.6 Disintegrating Tensor Products of Irreducible Representations of Nilpotent Lie Groups 205 3.6.1 A Concrete Intertwining Operator for Tensor Products of Unitary Representations 205 3.6.2 A Concrete Example 211 Disintegration of π1π2 212 Disintegration of π3π4 212 3.6.3 Criteria for Irreducibility of Tensor Products 213 3.7 Intertwining of Quasi-Regular Representations of Nilmanifolds 216 3.7.1 Rational Structures and Uniform Subgroups 217 Fundamental Domains for Uniform Subgroups 219 3.7.2 Intertwining Operators 220 3.7.3 On the Multiplicity Formula 230 3.7.4 Primary Projections 233 3.7.5 Characterization of Two-Step Nilmanifolds with Equivalent Quasi-Regular Representations 235 3.7.6 Decomposition of the Quasi-Regular Representation R 236 3.7.7 Intertwining Operators 239 4 Variants of Plancherel Formulas for Monomial Representations of Exponential Solvable Lie Groups 249 4.1 Layout of the Problems 249 4.2 The Penney-Plancherel Formula for Nilpotent Lie Groups 250 4.2.1 Tempered Distributions of Positive Type 251 4.2.2 Well-Adapted Bases 253 4.3 The Plancherel-Bonnet Formula for Normal Inducing Subgroups of Exponential Solvable Lie Groups 257 4.3.1 G-equivariant Projections 258 4.3.2 Sobolev Spaces 260 4.3.3 Sobolev Spaces and Monomial Representations 262 4.3.4 Polarizations 264 4.3.5 Decomposition of Measures 265 4.3.6 The Bonnet-Plancherel Formula 268 4.3.7 A Variant of Penney's Plancherel Formula 275 4.4 The Penney-Plancherel Formula for Finite-Multiplicity Restrictions of Nilpotent Lie Groups 279 4.4.1 On Restrictions of Unitary Representations 280 4.4.2 The Plancherel and Penney-Plancherel Formulas 281 4.4.3 Examples 282 4.4.4 Proof of the Main Results 287 Proof of Theorem 4.4.3 287 Proof of Theorem 4.4.4 290 4.4.5 The Case of Normal Subgroups 291 4.4.6 An Intertwining Operator 292 5 Polynomial Conjectures 295 5.1 Introduction 295 5.2 The Case of Induced Representations 296 5.2.1 Towards the Conjecture 299 5.2.2 Special Cases 306 5.3 The Case of Restricted Representations 323 5.3.1 Frobenius Vectors 325 5.3.2 The Function PW on Ω(π) 328 5.3.3 Further Study of the Conjecture 334 5.3.4 Case 1. h n"0365n 341 5.3.5 Case 2. hn"0365n 341 5.3.6 Examples 355 6 Holomorphically Induced Representations of Solvable Lie Groups 364 6.1 Introduction 364 6.2 Intertwining Operators 364 6.2.1 Nilpotent Lie Groups and Maslov Index 365 6.2.2 Study of Connected Solvable Lie Groups 376 6.2.3 Explicit Expression of Intertwining Operators 383 6.2.4 Examples 390 6.3 Real Polarizations 391 6.3.1 Preliminaries 392 6.3.2 Irreducibility and Equivalence 399 7 Monomial Representations of Discrete Type of Exponential Solvable Lie Groups 413 7.1 Introduction 413 7.2 Preliminaries 413 7.3 Monomial Representations of Discrete Type 415 7.3.1 Generic and Strongly Generic Elements 417 7.3.2 A Basis for h/(h b) 424 7.4 A Convergence Proof 435 7.5 The Concrete Plancherel Formula 442 7.6 Invariant Differential Operators 456 7.7 Polarizations 461 8 Bounded Irreducible Representations 464 8.1 Introduction 464 8.2 Simple Modules of Banach Algebras 466 8.2.1 Elementary Definitions 466 8.2.2 The Spectrum in Banach Algebras 473 8.2.3 Simple Modules and the Spectrum 476 8.2.4 Construction of Simple Modules 478 8.2.5 Simple A -modules and Simple p A p-modules 481 Case of Boidol's Group 489 8.3 Irreducible Banach-Space Representations and Projections 498 8.3.1 Submodules of an Irreducible Module 498 8.3.2 Minimal Norm and Extension Norms 501 8.3.3 Topologically Simple Norms 503 8.4 Restricting and Extending Ideals 505 8.4.1 Definitions 505 8.4.2 Description of Extended and Restricted Ideals 507 8.5 Polynomial Growth and Functional Calculus 510 8.5.1 Definitions and Elementary Properties 510 8.5.2 Principles of Functional Calculus 515 8.5.3 Estimate for "026B30D u(nf)"026B30D ω 516 8.5.4 Properties of Functional Calculus 516 8.5.5 Computation of the Bound Used in FunctionalCalculus 517 8.6 Simple Modules of L1(G) for Nilpotent Lie Groups 522 8.7 Fell's Topology on Prim(G) and the Wiener Property 530 8.8 Variable Groups 536 8.8.1 Kirillov's Conjecture for Nilpotent Lie Groups 542 8.8.2 Coefficients of Monomial Representations 546 8.9 D-prime Ideals in the Schwartz Algebra of a Nilpotent Lie Group 550 8.9.1 Exponential Actions 550 8.9.2 Proof of Theorem 8.9.6 552 8.10 A Retract Theorem 556 8.10.1 Smooth Kernels 560 8.10.2 A Retract Theorem for Exponential Orbits in a Nilpotent Lie Group's Spectrum 565 8.10.3 An Application 569 8.11 Bounded Irreducible Representations of G 575 8.11.1 G -prime Ideals 575 8.11.2 The Representation πγ=indH G γ 578 8.11.3 An Example: Representations on Mixed Lp -spaces 580 Definitions 580 Representations on Mixed Lp -spaces 581 8.11.4 The Spaces ES 584 Spaces of Kernel Functions 584 A Retract 585 8.12 Using Projections in L1(G)/kerL1(G)(πγ) 587 8.12.1 The Weight ω 590 8.12.2 The Algebras (pλL1(G)pλ)/kerL1(G)(πλ) and L1(S,ω) 591 8.12.3 Conclusion: Two Problems to Solve 593 8.13 Irreducible Representations of L1(S ,ω) 594 8.13.1 Characters and Other Examples of Irreducible Representations 594 8.13.2 Estimating the Weight ω 595 Preparations 595 The Estimates 597 8.14 Classifications of Bounded Irreducible G -modules: The Main Theorem 601 8.14.1 Relationships Between Kernels 602 8.15 Characterization of Simple Modules of L1(G) 603 8.15.1 A Family of Simple Modules 603 8.15.2 A Character 603 8.15.3 Analysis of Simple L1(G)-modules 606 8.15.4 Equivalence of Two Simple L1(G)-modules 608 8.15.5 Symmetric Group Algebras 611 8.15.6 Final Remarks 612 Bibliography 613 Index 619