Representation Theory of Solvable Lie Groups and Related Topics

Preface
Nomenclature
Contents
1 Branching Laws and the Multiplicity Function of Unitary Representations of Exponential Solvable Lie Groups
	1.1 Introduction
	1.2 Generalities and Notations
		1.2.1 Coexponential Bases
		1.2.2 Modular Functions and Quotient Measures
		1.2.3 Induced Representations
		1.2.4 Polarizations
		1.2.5 Orbit Theory
		1.2.6 Branching Laws: Induced Representations
		1.2.7 Restrictions
	1.3 Pseudo-Algebraic Geometry
		1.3.1 Pseudo-Algebraic Sets
		1.3.2 Semi-analytic Sets
		1.3.3 Structure of Coadjoint Orbits
			Saturated Orbits with Respect to an Ideal of Codimension One
	1.4 Up-Down Representations of Exponential Solvable Lie Groups
		1.4.1 Disintegration of Up-Down Representations
		1.4.2 The Multiplicity Function of Up-DownRepresentations
			The Case of Normal Subgroups
	1.5 Down-Up Representations
		1.5.1 The Down-Up Formula
		1.5.2 The Down-Up Multiplicity Formula
		1.5.3 Examples
		1.5.4 The Case of Exponential Solvable Groups
	1.6 The Multiplicity Function of Monomial Representations
2 Intertwining Operators for Irreducible Representations of an Exponential Solvable Lie Group
	2.1 Introduction
	2.2 A Trace Relation
	2.3 Relations Between Two Polarizations
	2.4 Vergne Polarizations
	2.5 The General Case
	2.6 A Local Result
	2.7 The Case Where h1 + h2 Is a Subalgebra
	2.8 The Key Point Is the Convergence
3 Intertwining Operators of Induced Representations and Restrictions of Representations of Exponential SolvableLie Groups
	3.1 Introduction
	3.2 Intertwining Operators of Induced Representations of Nilpotent Lie Groups
		3.2.1 Generalities
			Notation and Backgrounds
		3.2.2 Disintegration of Monomial Representations
			Choice of Measures
			A Canonical Disintegration Formula
		3.2.3 Construction of the Intertwining Operator
			Construction of Polarizations and Malcev Bases
			Step 0
			Step s
			Step s+1
		3.2.4 Examples
			Rational Disintegration of L2 (G)
	3.3 The Case of Exponential Solvable Groups
		3.3.1 A Base Space of the Disintegration of Induced Representations
		3.3.2 Construction of the Intertwining Operator
			Several Constructions
		3.3.3 The Inverse Operator
		3.3.4 A Rational Disintegration of L2(G) for an Exponential Solvable Lie GroupG
		3.3.5 Examples
	3.4 Intertwining of Representations Induced from Maximal Subgroups of Exponential Solvable Lie Groups
	3.5 Intertwining Operators of the Restriction of Representations of Nilpotent Lie Groups
		3.5.1 Double-Coset Space
			The Set of Double Cosets
			Description of HG B
		3.5.2 A Measure on HG B
		3.5.3 A Concrete Intertwining Operator
	3.6 Disintegrating Tensor Products of Irreducible Representations of Nilpotent Lie Groups
		3.6.1 A Concrete Intertwining Operator for Tensor Products of Unitary Representations
		3.6.2 A Concrete Example
			Disintegration of π1π2
			Disintegration of π3π4
		3.6.3 Criteria for Irreducibility of Tensor Products
	3.7 Intertwining of Quasi-Regular Representations of Nilmanifolds
		3.7.1 Rational Structures and Uniform Subgroups
			Fundamental Domains for Uniform Subgroups
		3.7.2 Intertwining Operators
		3.7.3 On the Multiplicity Formula
		3.7.4 Primary Projections
		3.7.5 Characterization of Two-Step Nilmanifolds with Equivalent Quasi-Regular Representations
		3.7.6 Decomposition of the Quasi-Regular Representation R
		3.7.7 Intertwining Operators
4 Variants of Plancherel Formulas for Monomial Representations of Exponential Solvable Lie Groups
	4.1 Layout of the Problems
	4.2 The Penney-Plancherel Formula for Nilpotent Lie Groups
		4.2.1 Tempered Distributions of Positive Type
		4.2.2 Well-Adapted Bases
	4.3 The Plancherel-Bonnet Formula for Normal Inducing Subgroups of Exponential Solvable Lie Groups
		4.3.1 G-equivariant Projections
		4.3.2 Sobolev Spaces
		4.3.3 Sobolev Spaces and Monomial Representations
		4.3.4 Polarizations
		4.3.5 Decomposition of Measures
		4.3.6 The Bonnet-Plancherel Formula
		4.3.7 A Variant of Penney\'s Plancherel Formula
	4.4 The Penney-Plancherel Formula for Finite-Multiplicity Restrictions of Nilpotent Lie Groups
		4.4.1 On Restrictions of Unitary Representations
		4.4.2 The Plancherel and Penney-Plancherel Formulas
		4.4.3 Examples
		4.4.4 Proof of the Main Results
			Proof of Theorem 4.4.3
			Proof of Theorem 4.4.4
		4.4.5 The Case of Normal Subgroups
		4.4.6 An Intertwining Operator
5 Polynomial Conjectures
	5.1 Introduction
	5.2 The Case of Induced Representations
		5.2.1 Towards the Conjecture
		5.2.2 Special Cases
	5.3 The Case of Restricted Representations
		5.3.1 Frobenius Vectors
		5.3.2 The Function PW on Ω(π)
		5.3.3 Further Study of the Conjecture
		5.3.4 Case 1. h n\"0365n
		5.3.5 Case 2. hn\"0365n
		5.3.6 Examples
6 Holomorphically Induced Representations of Solvable Lie Groups
	6.1 Introduction
	6.2 Intertwining Operators
		6.2.1 Nilpotent Lie Groups and Maslov Index
		6.2.2 Study of Connected Solvable Lie Groups
		6.2.3 Explicit Expression of Intertwining Operators
		6.2.4 Examples
	6.3 Real Polarizations
		6.3.1 Preliminaries
		6.3.2 Irreducibility and Equivalence
7 Monomial Representations of Discrete Type of Exponential Solvable Lie Groups
	7.1 Introduction
	7.2 Preliminaries
	7.3 Monomial Representations of Discrete Type
		7.3.1 Generic and Strongly Generic Elements
		7.3.2 A Basis for h/(h b)
	7.4 A Convergence Proof
	7.5 The Concrete Plancherel Formula
	7.6 Invariant Differential Operators
	7.7 Polarizations
8 Bounded Irreducible Representations
	8.1 Introduction
	8.2 Simple Modules of Banach Algebras
		8.2.1 Elementary Definitions
		8.2.2 The Spectrum in Banach Algebras
		8.2.3 Simple Modules and the Spectrum
		8.2.4 Construction of Simple Modules
		8.2.5 Simple A -modules and Simple p A p-modules
			Case of Boidol\'s Group
	8.3 Irreducible Banach-Space Representations and Projections
		8.3.1 Submodules of an Irreducible Module
		8.3.2 Minimal Norm and Extension Norms
		8.3.3 Topologically Simple Norms
	8.4 Restricting and Extending Ideals
		8.4.1 Definitions
		8.4.2 Description of Extended and Restricted Ideals
	8.5 Polynomial Growth and Functional Calculus
		8.5.1 Definitions and Elementary Properties
		8.5.2 Principles of Functional Calculus
		8.5.3 Estimate for \"026B30D u(nf)\"026B30D ω
		8.5.4 Properties of Functional Calculus
		8.5.5 Computation of the Bound Used in FunctionalCalculus
	8.6 Simple Modules of L1(G)  for Nilpotent Lie Groups
	8.7 Fell\'s Topology on Prim(G) and the Wiener Property
	8.8 Variable Groups
		8.8.1 Kirillov\'s Conjecture for Nilpotent Lie Groups
		8.8.2 Coefficients of Monomial Representations
	8.9  D-prime Ideals in the Schwartz Algebra of a Nilpotent Lie Group
		8.9.1 Exponential Actions
		8.9.2 Proof of Theorem 8.9.6
	8.10 A Retract Theorem
		8.10.1 Smooth Kernels
		8.10.2 A Retract Theorem for Exponential Orbits in a Nilpotent Lie Group\'s Spectrum
		8.10.3 An Application
	8.11 Bounded Irreducible Representations of G
		8.11.1 G -prime Ideals
		8.11.2 The Representation πγ=indH G γ
		8.11.3 An Example: Representations on Mixed Lp -spaces
			Definitions
			Representations on Mixed Lp -spaces
		8.11.4 The Spaces ES
			Spaces of Kernel Functions
			A Retract
	8.12 Using Projections in L1(G)/kerL1(G)(πγ)
		8.12.1 The Weight ω
		8.12.2 The Algebras (pλL1(G)pλ)/kerL1(G)(πλ) and L1(S,ω)
		8.12.3 Conclusion: Two Problems to Solve
	8.13 Irreducible Representations of L1(S ,ω)
		8.13.1 Characters and Other Examples of Irreducible Representations
		8.13.2 Estimating the Weight ω
			Preparations
			The Estimates
	8.14 Classifications of Bounded Irreducible G -modules: The Main Theorem
		8.14.1 Relationships Between Kernels
	8.15 Characterization of Simple Modules of  L1(G)
		8.15.1 A Family of Simple Modules
		8.15.2 A Character
		8.15.3  Analysis of Simple L1(G)-modules
		8.15.4 Equivalence of Two Simple L1(G)-modules
		8.15.5 Symmetric Group Algebras
		8.15.6 Final Remarks
Bibliography
Index