Symmetry in Geometry and Analysis, Volume 2: Festschrift in Honor of Toshiyuki Kobayashi (Progress in Mathematics, 358)

Contents
Contents of Volume 1
Contents of Volume 3
The Source Operator Method: An Overview
	1 Introduction
	2 Background: Real Simple Jordan Algebra
	3 Functional Equation of Zeta Functions
	4 Construction of the Source Operator
	5 Construction of the Generalized Rankin–Cohen Operators
	6 Other Applications
		6.1 The Case of Differential Forms
		6.2 The Case of Spinors
		6.3 Other Examples
	References
Some Mixed Norm Bounds for the Spectral Projections of the Heisenberg Sublaplacian
	1 Introduction
	2 Some Context
	3 Notation and Preliminaries
		3.1 The Heisenberg Group
		3.2 The Sublaplacian L and the Twisted Laplacian L
		3.3 Laguerre Functions
	4 The Spectral Decomposition of L
	5 Bounding the Spectral Projections of the Twisted Laplacian
		5.1 Koch–Ricci Estimates
		5.2 Necessary Conditions
	6 An Example Involving Bigraded Spherical Harmonics
	7 Bounding the Spectral Projections of the Sublaplacian
	References
Four Variations on the Rankin–Cohen Brackets
	1 Introduction
	2 The Ω-Process and the Generalized Construction of Transvectants
	3 A Straightforward Construction (after V. Ovsienko an P. Redou)
	4 The Source Operator Method
	5 The Laplace Transform, the Holographic Transform, and the Rankin–Cohen Brackets
	6 Link with the Jacobi Polynomials
	References
Restricting Holomorphic Discrete Series Representations to a Compact Dual Pair
	1 Introduction
		1.1 Statement of the Results
	2 Preliminaries
		2.1 Euclidean Jordan Algebras
		2.2 L2-Model for Vector-Valued Holomorphic Discrete Series
	3 Restriction of a Holomorphic Discrete Series
		3.1 Stratification
		3.2 Branching to PSL2(R)
		3.3 Branching to PSL2(R)H
		3.4 The Case g=so(2,n+1)
	References
Nets of Standard Subspaces on Non-compactly Causal SymmetricSpaces
	1 Introduction
	2 Analytic and Hyperfunction Vectors
		2.1 The Space of Analytic Vectors
		2.2 Analytic Vectors for One-Parameter Groups
		2.3 Distribution Vectors for One-Parameter Groups
	3 Hyperfunction Boundary Values
	4 Reeh–Schlieder and Bisognano–Wichmann Property
		4.1 The Reeh–Schlieder Property
		4.2 The Bisognano–Wichmann Property
			4.2.1 A Realization as a Space of Holomorphic Functions
			4.2.2 Verification of the Bisognano-Wichmann Property
		4.3 Locality
	5 Covering Groups of SL2(R)
		5.1 Growth Estimates and Hypergeometric Functions
			5.1.1 The Hypergeometric Functions
			5.1.2 Spherical Functions
			5.1.3 χ-Spherical Functions
		5.2 More General Asymptotics
		5.3 An Application to Intersections of Standard Subspaces
		5.4 Positive Energy Representations of PSL2(R)
	6 Outlook: Beyond Linear Simple Lie Groups
		6.1 Casselman's Theorem for Non-linear Groups
		6.2 Growth Control
		6.3 Localization in Central Unitary Characters
	7 Appendix 1: An Automatic Continuity Theorem
	8 Appendix 2: Wedge Regions in Non-compactly Causal Spaces
	References
Heisenberg Parabolically Induced Representations of Hermitian Lie Groups, Part II: Next-to-Minimal Representations and Branching Rules
	1 Introduction
		1.1 Next-to-Minimal Representations
		1.2 Branching GSL(2,R)M
	2 Preliminaries
		2.1 Heisenberg Parabolically Induced Representations
		2.2 Intertwining Operators
		2.3 The Heisenberg Group Fourier Transform
		2.4 The Metaplectic Representation
		2.5 The Fourier Transform of Intertwining Operators
	3 Associated Varieties
		3.1 The Gelfand–Kirillov Dimension
		3.2 The Case g=so(2,n)
		3.3 The Case g=so*(2n)
		3.4 The Case g=e6(-14)
		3.5 The Case g=e7(-25)
		3.6 The Case g=su(p,q)
	4 Restriction from G to SL(2,R)M
		4.1 The Fourier Transformed Model of the Next-to-Minimal Representation
		4.2 The Case G=SO0(2,n)
		4.3 The Case G=E6(-14)
			4.3.1 The Subalgebra su(5, 1) e6(-14)
			4.3.2 Tensor Products of Highest and Lowest Weight Representations of su(5, 1)
	References
Quantum-Classical Correspondences for Locally Symmetric Spaces
	1 Introduction
		1.1 Some Motivating Examples
		1.2 Outline of the Program and its Purpose
	2 Locally Symmetric Spaces
		2.1 Symmetric Spaces
			2.1.1 Equivariant Differential Geometry on G/K
			2.1.2 Invariant Differential Operators
		2.2 Representation Theory and Harmonic Analysis
			2.2.1 Principal Series Representations for Minimal Parabolics
			2.2.2 Intertwining Operators
			2.2.3 -Cohomology
	3 Dynamical Systems
		3.1 Classical Dynamics
			3.1.1 Anosov Flows
			3.1.2 Symplectic Geometry and Hamiltonian Actions
			3.1.3 Geodesic and Weyl Chamber Flow
		3.2 Quantum Dynamics
	4 Quantum-Classical Correspondences
		4.1 Quantum Invariants
			4.1.1 Joint Spectrum of the Algebra of Invariant Differential Operators in the Cocompact Case
			4.1.2 Laplace Resonances for Noncompact Locally Symmetric Spaces
			4.1.3 Laplace Resonances vs. Scattering Poles
			4.1.4 Microlocal Lifts, Wigner and Patterson-Sullivan Distributions
		4.2 Classical Invariants
			4.2.1 Ruelle-Taylor Resonances for Compact Riemannian Manifolds
			4.2.2 Ruelle Resonances for Noncompact Locally Hyperbolic Spaces
			4.2.3 -Invariant Distributions for Principal Series Representations
			4.2.4 Dynamical Zeta Functions
			4.2.5 Transfer Operators
		4.3 Correspondences
			4.3.1 The Compact Case for Generic Spectral Parameters
			4.3.2 Results for Noncompact Locally Symmetric Spaces
			4.3.3 Exceptional Spectral Parameters
			4.3.4 Applications
	5 Quotients of Trees and Affine Buildings
		5.1 Graphs of Bounded Degree
		5.2 Affine Buildings
	6 Open Problems
		6.1 Quantum Resonances in Higher Rank
		6.2 Ruelle-Taylor Resonances
		6.3 Dynamical Zeta Functions, -Cohomology, and Transfer Operators
		6.4 Horocycle Flow
		6.5 Patterson-Sullivan Distributions
		6.6 Singular Situations
		6.7 Graphs and Buildings
	References
Classification of K-type Formulas for the Heisenberg Ultrahyperbolic Operator s for SL̃(3,R) and Tridiagonal Determinants for Local Heun Functions
	1 Introduction
		1.1 Heisenberg Ultrahyperbolic Operator s
		1.2 K-Type Formulas
		1.3 Hypergeometric and Heun's Differential Equations
		1.4 Sequences {Pk(x;y)}k=0∞ and {Qk(x;y)}k=0∞ of Tridiagonal Determinants
		1.5 Organization
	2 Peter–Weyl Theorem for the Space of K-finite Solutions
		2.1 General Framework
		2.2 Peter–Weyl Theorem for Sol(u; λ)(ξ)K
	3 Specialization to (SL̃(3,R),B)
		3.1 Notation and Normalizations
		3.2 Two Identifications for so(3,C) sl(2,C)
			3.2.1 Identification k sl(2,C) via ΩI
			3.2.2 Identification k sl(2,C) via ΩII
		3.3 A Realization of the Subgroup M=ZK(a0)
		3.4 Irreducible Representations Irr(K) of K
		3.5 Irreducible Representations Irr(M) of M
		3.6 Peter–Weyl Theorem for Sol(u; λ)(σ)K with Polynomial Realization
		3.7 Recipe for Determining the K-type Formula for Sol(u; λ)(σ)K
	4 Heisenberg Ultrahyperbolic Operator for SL̃(3,R)
		4.1 Heisenberg Ultrahyperbolic Operator R(Ds)
		4.2 Relationship Between dπnI(Ds) and dπnII(Ds)
		4.3 Differential Equation dπnJ(Ds)f(t)=0 for J {I, II}
			4.3.1 Differential Equation dπnI(Ds)f(t)=0
			4.3.2 Differential Equation dπnII(Ds)f(t)=0
		4.4 Recipe for the K-type Decomposition of Sol(s;σ)K
	5 Hypergeometric Model dπnII(Ds)f(t)=0
		5.1 The Classification of SolII(s;n)
		5.2 The M-representations on SolII(s;n)
			5.2.1 Case 1
			5.2.2 Case 2a
			5.2.3 Case 2b
			5.2.4 Case 2c
		5.3 The Classification of HomM(SolII(s;n),σ)
	6 Heun Model dπnI(Ds)f(t)=0
		6.1 Relationships Between a[s;n](t), b[s;n](t), c[s;n](t) and u[s;n](t), v[s;n](t)
	7 Sequences {Pk(x;y)}k=0∞ and {Qk(x;y)}k=0∞ of Tridiagonal Determinants
		7.1 Sequences {Pk(x;y)}k=0∞ and {Qk(x;y)}k=0∞ of Tridiagonal Determinants
		7.2 Generating Functions of {Pk(x;n)}k=0∞ and {Qk(x;n)}k=0∞
		7.3 Factorization Formulas of P[n+22](x;n) and Q[n+12](x;n)
			7.3.1 Factorization Formulas of Pn+22(x;n) and Qn2(x;n) for n 2Z≥0
			7.3.2 Factorization Formula of Pn+12(x;n) for n 1+2Z≥0
		7.4 Functional Equations of {Pk(x;y)}k=0∞ and {Qk(x;y)}k=0∞
			7.4.1 Functional Equation of {Pk(x;y)}k=0∞ for n 2Z≥0
			7.4.2 Functional Equation of {Qk(x;y)}k=0∞ for n 2(1+Z≥0)
			7.4.3 Functional Equations of {Pk(x;y)}k=0∞ and {Qk(x;y)}k=0∞ for n 1+2Z≥0
	8 Appendix 1: Local Heun Functions
		8.1 General Facts
		8.2 The Local Solutions u[s;n](t) and v[s;n](t)
	9 Appendix 2: Functional Equation of Cayley Continuants and Krawtchouk Polynomials
		9.1 Functional Equation of Cayley Continuants {Cayk(x;y)}k=0∞
		9.2 Functional Equation of Krawtchouk Polynomials {Kk(x;y)}k=0∞
		9.3 Relationship with Jacobi Polynomials Pk(α, β)(z)
	References
Gauss–Berezin Integral Operators, Spinors over Orthosymplectic Supergroups, and Lagrangian Super-Grassmannians
	1 Introduction
		1.1 Orthosymplectic Spinors
		1.2 Berezin Formulas
		1.3 Aims of This Chapter: Orthosymplectic Spinors
		1.4 Aims of This Chapter: Gauss–Berezin Integral Operators
		1.5 Aims of This Chapter: Possible Applications
		1.6 Structure of This Chapter
	2 A Survey of Orthogonal Spinors: Berezin Operators and Lagrangian Linear Relations
		A. Grassmann Algebras and Berezin Operators
		2.1 Grassmann Variables and Grassmann Algebra
		2.2 Derivatives
		2.3 Exponentials
		2.4 Berezin Integral
		2.5 Integrals with Respect to Odd Gaussian Measure'
		2.6 Integral Operators
		2.7 Berezin Operators in the Narrow Sense
		2.8 Product Formula
		2.9 Pfaffians and Odd Gaussian Integrals
		2.10 The Definition of Berezin Operators
		2.11 The Space of Berezin Operators
		2.12 Examples of Berezin Operators
		2.13 Another Definition of Berezin Operators
		2.14 The Category of Berezin Operators: Groups of Automorphisms
		B. Linear Relations and the Category GD
		2.15 Linear Relations
		2.16 Product of Linear Relations
		2.17 Imitation of Some Standard Definitions of Matrix Theory
		2.18 Lagrangian Grassmannian and Orthogonal Groups
		2.19 Imitation of Orthogonal Groups: Category GD
		C. Explicit Correspondence
		2.20 Coordinates on the Lagrangian Grassmannian
		2.21 Atlas on the Lagrangian Grassmannian
		2.22 Atlas on the Lagrangian Grassmannian: Elementary Reflections
		2.23 Components of Lagrangian Grassmannian
		2.24 Coordinates on the Set of Morphisms of GD
		2.25 Creation-Annihilation Operators
		2.26 A Construction of the Correspondence
		2.27 Explicit Correspondence: Another Description
	3 A Survey of the Oscillator Representation: The Category of Gaussian Integral Operators
		3.1 Fock Space
		3.2  Operators
		3.3  Gaussian Operators
		3.4 Product Formula
		3.5 Complexification of a Linear Space with Bilinear Form
		3.6  The Category Sp
		3.7 Construction of Gaussian Operators from Linear Relations
		3.8 Construction of Linear Relations from Gaussian Operators
		3.9 Details: An Analogue of the Schwartz Space
		3.10 Details: The Olshanski Semigroup Sp(2n,R )
	4 Gauss–Berezin Integrals
		4.1  Phantom Algebra
		4.2  A Technical Comment
		4.3  Berezinian
		4.4 Functions
		4.5 Integral
		4.6  Exponential
		4.7  Gauss–Berezin Integrals: A Special Case
		4.8 Evaluation of the Gauss–Berezin Integral
		4.9  Grassmann Gaussian Integral
		4.10  More General Gauss–Berezin Integrals
		4.11  The First Way to Evaluate Super-Gaussian Integral
		4.12  The Second Way to Evaluate Gauss–Berezin Integrals
	5 Gauss–Berezin Integral Operators
		5.1  Fock–Berezin Spaces
		5.2  Another Form of the Gauss–Berezin Integral
		5.3  Integral Operators
		5.4  Linear and Antilinear Operators
		5.5  Gauss–Berezin Vectors in the Narrow Sense
		5.6  Gauss–Berezin Operators in the Narrow Sense
		5.7 General Gauss–Berezin Operators
		5.8  Operators π↓(B) and Boundedness of Gauss–Berezin Operators
		5.9 Products of Gauss–Berezin Operators
		5.10 General Gauss–Berezin Vectors
	6 Supergroups OSp(2p|2q)
		6.1  Modules Ap|q
		6.2  Matrices
		6.3  Super-Transposition
		6.4  The Supergroups GL(p|q;A)
		6.5  The Supergroup OSp(2p|2q;A)
		6.6  The Super-Olshanski Semigroup OSp(2p|2q;A)
	7 Super-Grassmannians
		7.1  Super-Grassmannians
		7.2  Intersections of Subspaces
		7.3  Atlas on the Super-Grassmannian
		7.4  Lagrangian Super-Grassmannians
		7.5  Coordinates on Lagrangian Super-Grassmannian
		7.6  Atlas on the Lagrangian Super-Grassmannian
		7.7  Elementary Reflections
	8 Superlinear Relations
		8.1  Superlinear Relations
		8.2  Transversality Conditions
		8.3  Transversality for Superlinear Relations
		8.4  Lagrangian Superlinear Relations
		8.5  Components of Lagrangian Super-Grassmannian
		8.6  Contractive Lagrangian Linear Relations
		8.7  Positive Domain in the Lagrangian Super-Grassmannian
	9 Correspondence Between Lagrangian Superlinear Relations and Gauss–Berezin Operators
		9.1  Creation–Annihilation Operators
		9.2 Supercommutator
		9.3  Annihilators of Gaussian Vectors
		9.4  Gauss–Berezin Operators and Superlinear Relations
		9.5  Products of Gauss–Berezin Operators
	10 Final Remarks
		10.1  Extension of Notion of Gaussian Operators?
		10.2  The Infinite-Dimensional Orthosymplectic Supergroup
		10.3  The Super-Virasoro Algebras
		10.4 ``Unitary Representations'' of Supergroups
	References
Toward Gan-Gross-Prasad-Type Conjecture for Discrete Series Representations of Symmetric Spaces
	1 Introduction
	2 Notation and Generalities
	3 The Symmetric Spaces and Their Discrete Series Representations
	4 Restriction of Representations and Relative Branching Laws
		4.1 Branching
		4.2  Relative Branching for Compact Unitary Symmetric Spaces and Period Integrals
		4.3 Relative Branching for Discrete Series Representations
		4.4 Period Integrals
	5 The Main Results
		5.1 Relative Branching Laws and Interlacing Patterns
		5.2 Characters A(ϕ) and Relative Branching
	6 Relative Branching for +
		6.1 Background
		6.2 Period Integrals of Flensted-Jensen Functions
		6.3 Branching in Stages
	7 Relative Branching for -
		7.1 Symmetry Breaking by Geometric Restriction
	Appendix 1
	Appendix 2
	References
Pseudo-dual Pairs and Branching of Discrete Series
	1 Introduction
	2 Preliminaries and Some Notation
	3 Duality Theorem, Explicit Isomorphism
		3.1 Statement and Proof of the Duality Result
		3.2 Kernel of the Restriction Map
		3.3 The Map r0D Is Injective
		3.4 Previous Work on Duality Formula and Harish-Chandra Parameters
			3.4.1 Computing Harish-Chandra Parameters from Theorem 1
			3.4.2 Gross-Wallach Multiplicity Formula
			3.4.3 Duflo-Vargas Multiplicity Formula DV
			3.4.4 Harris-He-Olafsson Multiplicity Formula HHO
		3.5 Completion of the Proof of Theorem 1: The Map r0D Is Surjective
	4 Duality Theorem, Proof of Dimension Equality
		4.1 Dimension Equality Theorem: Statement
			4.1.1 Conclusion Proof of Theorem 4
			4.1.2 Existence of D
			4.1.3 Representations πλ so that resL(τ) Is Irreducible
			4.1.4 Analysis of U(h0)W, Lλ, Existence of D, Case τ|L Is Irreducible
			4.1.5 Analysis of U(h0)W, Lλ, Existence of D, for General (τ, W)
		4.2 Explicit Inverse Map to r0D
			4.2.1 Case (τ, W) Restricted to L Is Irreducible
			4.2.2 Value of b=d=c when resL(τ) Is Irreducible
			4.2.3 Analysis of r0D for Arbitrary (τ, W), (σ, Z)
			4.2.4 Eigenvalues of r0r0
	5 Examples
		5.1 Multiplicity-Free Representations
			5.1.1 Holomorphic Representations
			5.1.2 Quaternionic Real Forms, Quaternionic Representations
			5.1.3 More Examples of Multiplicity-Free Restriction
		5.2 Explicit Examples
			5.2.1 Quaternionic Representations for Sp(1,b)
			5.2.2 Explicit Example I
			5.2.3 Explicit Example II
			5.2.4 Explicit Example III
			5.2.5 Comments on Admissible Restriction of Quaternionic Representations
			5.2.6 Explicit Example IV
		5.3 Existence of Discrete Series Whose Lowest K-Type Restricted to K1() Is Irreducible
	6 Symmetry Breaking Operators and Normal Derivatives
		6.1 Converse to Proposition 5
		6.2 Comments on the Interplay Among Subspaces, Lλ, U(h0)W, H2(G,τ)K-fin, and Symmetry Breaking Operators
		6.3 A Functional Equation for Symmetry Breaking Operators
	7 Tables
	8 Partial List of Symbols and Definitions
	References
Integral Transformations of Hypergeometric Functions with Several Variables
	1 Introduction
	2 Integral Transformations
	3 Some Hypergeometric Functions
	4 A Connection Problem
	5 More Transformations
	6 Differential Equations
	7 KZ Equations
	8 Fuchsian Ordinary Differential Equations
	References