# Integral Geometry and Representation Theory

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Generalized Functions, Volume 5: Integral Geometry and Representation Theory is devoted to the theory of representations, focusing on the group of two-dimensional complex matrices of determinant one. This book emphasizes that the theory of representations is a good example of the use of algebraic and geometric methods in functional analysis, in which transformations are performed not on the points of a space, but on the functions defined on it. The topics discussed include Radon transform on a real affine space, integral transforms in the complex domain, and representations of the group of complex unimodular matrices in two dimensions. The properties of the Fourier transform on G, integral geometry in a space of constant curvature, harmonic analysis on spaces homogeneous with respect to the Lorentz Group, and invariance under translation and dilation are also described. This volume is suitable for mathematicians, specialists, and students learning integral geometry and representation theory.

## Table of Contents

Translator's Note

Foreword

Chapter I Radon Transform of Test Functions and Generalized Functions on a Real Affine Space

1. The Radon Transform on a Real Affine Space

1.1. Definition of the Radon Transform

1.2. Relation Between Radon and Fourier Transforms

1.3. Elementary Properties of the Radon Transform

1.4. The Inverse Radon Transforms

1.5. Analog of Plancherel's Theorem for the Radon Transform

1.6. Analog of the Paley-Wiener Theorem for the Radon Transform

1.7. Asymptotic Behavior of Fourier Transforms of Characteristic Functions of Regions

2. The Radon Transform of Generalized Functions

2.1. Definition of the Radon Transform for Generalized Functions

2.2. Radon Transform of Generalized Functions Concentrated on Points and Line Segments

2.3. Radon Transform of (x1)+λ δ(x2,...,xn)

2.3a. Radon Transform of (x1)+λ δ(x2,...,xn) for Nonnegative Integer k

2.4. Integral of a Function over a Given Region in Terms of Integrals over Hyperplanes

2.5. Radon Transform of the Characteristic Function of One Sheet of a Cone

Appendix to Section 2.5

2.6. Radon Transform of the Characteristic Function of One Sheet of a Two-Sheeted Hyperboloid

2.7. Radon Transform of Homogeneous Functions

2.8. Radon Transform of the Characteristic Function of an Octant

2.9. The Generalized Hypergeometric Function

3. Radon Transforms of Some Particular Generalized Functions

3.1. Radon Transforms of the Generalized Functions (P + i0)λ, (P — i0)λ, and P+λ for Nondegenerate Quadratic Forms P

Appendix to Section 3.1

3.2. Radon Transforms of {P + c + i0)+λ, (P + c — i0)λ, and (P + c)+λ for Nondegenerate Quadratic Forms

3.3. Radon Transforms of the Characteristic Functions of Hyperboloids and Cones

3.4. Radon Transform of a Delta Function Concentrated on a Quadratic Surface

4. Summary of Radon Transform Formulas

Chapter II Integral Transforms In the Complex Domain

1. Line Complexes in a Space of Three Complex Dimensions and Related Integral Transforms

1.1. Plücker Coordinates of a Line

1.2. Line Complexes

1.3. A Special Class of Complexes

1.4. The Problem of Integral Geometry for a Line Complex

1.5. The Inversion Formula. Proof of the Theorem of Section 1.4

1.6. Examples of Complexes

1.7. Note on Translation Operators

2. Integral Geometry on a Quadratic Surface in a Space of Four Complex Dimensions

2.1. Statement of the Problem

2.2. Line Generators of Quadratic Surfaces

2.3. Integrals of ƒ(z) over Quadratic Surfaces and Along Complex Lines

2.4. Expression for ƒ(z) on a Quadratic Surface in Terms of Its Integrals Along Line Generators

2.5. Derivation of the Inversion Formula

2.6. Another Derivation of the Inversion Formula

2.7. Rapidly Decreasing Functions on Quadratic Surfaces. The Paley-Wiener Theorem

3. The Radon Transform in the Complex Domain

3.1. Definition of the Radon Transform

3.2. Representation of ƒ(z) in Terms of Its Radon Transform

3.3. Analog of Plancherel's Theorem for the Radon Transform

3.4. Analog of the Paley-Wiener Theorem for the Radon Transform

3.5. Radon Transform of Generalized Functions

3.6. Examples

3.7. The Generalized Hypergeometric Function in the Complex Domain

Chapter III Representations of the Group of Complex Unimodular Matrices in Two Dimensions

1. The Group of Complex Unimodular Matrices in Two Dimensions and Some of Its Realizations

1.1. Connection with the Proper Lorentz Group

1.2. Connection with Lobachevskian and Other Motions

2. Representations of the Lorentz Group Acting on Homogeneous Functions of Two Complex Variables

2.1. Representations of Groups

2.2. The Dx Spaces of Homogeneous Functions

2.3. Two Useful Realizations of the Dx

2.4. Representation of G on Dx

2.5. The Tx(g) Operators in Other Realizations of Dx

2.6. The Dual Representations

3. Summary of Basic Results concerning Representations on Dx

3.1. Irreducibility of Representations on the Dx and the Role of Integer Points

3.2. Equivalence of Representations on the Dx and the Role of Integer Points

3.3. The Problem of Equivalence at Integer Points

3.4. Unitary Representations

4. Invariant Bilinear Functionals

4.1. Statement of the Problem and the Basic Results

4.2. Necessary Condition for Invariance under Parallel Translation and Dilation

4.3. Conditions for Invariance under Inversion

4.4. Sufficiency of Conditions for the Existence of Invariant Bilinear Functionals (Nonsingular Case)

4.5. Conditions for the Existence of Invariant Bilinear Functionals (Singular Case)

4.6. Degeneracy of Invariant Bilinear Functionals

4.7. Conditionally Invariant Bilinear Functionals

5. Equivalence of Representations of G

5.1. Intertwining Operators

5.2. Equivalence of Two Representations

5.3. Partially Equivalent Representations

6. Unitary Representations of G

6.1. Invariant Hermitian Functionals on Dx

6.2. Positive Definite Invariant Hermitian Functionals

6.3. Invariant Hermitian Functionals for Noninteger ρ, | ρ | ≥ 1

6.4. Invariant Hermitian Functionals in the Special Case of Integer n1 = n2

6.5. Unitary Representations of G by Operators on Hubert Space

6.6. Subspace Irreducibility of the Unitary Representations

Chapter IV Harmonic Analysis on the Group of Complex Unimodular Matrices in Two Dimensions

1. Definition of the Fourier Transform on a Group. Statement of the Problems and Summary of the Results

1.1. Fourier Transform on the Line

1.2. Functions on G

1.3. Fourier Transform on G

1.4. Domain of Definition of F(x)

1.5. Summary of the Results of Chapter IV

Appendix. Functions on G

2. Properties of the Fourier Transform on G

2.1. Simplest Properties

2.2. Fourier Transform as Integral Operator

2.3. Geometric Interpretation of K(z1 , z2 ; x). The Functions ȹ(z1 , z2 ; λ) and Φ(u, v; u', v')

2.4. Properties of K(z1 , z2 ; x)

2.5. Continuity of K(z1 , z2 ; x)

2.6. Asymptotic Behavior of K(z1 , z2 ; x)

2.7. Trace of the Fourier Transform

3. Inverse Fourier Transform and Plancherel's Theorem for G

3.1. Statement of the Problem

3.2. Expression for ȹ(z1 , z2 ; λ) in Terms of K(z1 , z2 ; x)

3.3. Expression for ƒ(g) in Terms of ȹ(z1 , z2 ; x)

3.4. Expression for ƒ(g) in Terms of Its Fourier Transform F(x)

3.5. Analog of Plancherel's Theorem for G

3.6. Symmetry Properties of F(x)

3.7. Fourier Integral and the Decomposition of the Regular Representation of the Lorentz Group into Irreducible Representations

4. Differential Operators on G

4.1. Tangent Space to G

4.2. Lie Operators

4.3. Relation Between Left and Right Derivative Operators

4.4. Commutation Relations for the Lie Operators

4.5. Laplacian Operators

4.6. Functions on G with Rapidly Decreasing Derivatives

4.7. Fourier Transforms of Lie Operators

5. The Paley-Wiener Theorem for the Fourier Transform on G

5.1. Integrals of ƒ(g) Along "Line Generators"

5.2. Behavior of Φ(u, ν; u', v') under Translation and Differentiation of ƒ(g)

5.3. Differentiability and Asymptotic Behavior of Φ(u, v; w', v')

5.4. Conditions on K(z1, z2 ; x)

5.5. Moments of ƒ(g) and Their Expression in Terms of the Kernel

5.6. The Paley-Wiener Theorem for the Fourier Transform on G

Chapter V Integral Geometry in a Space of Constant Curvature

1. Spaces of Constant Curvature

1.1. Spherical and Lobachevskian Spaces

1.2. Some Models of Lobachevskian Spaces

1.3. Imaginary Lobachevskian Spaces

1.3a. Isotropie Lines of an Imaginary Lobachevskian Space

1.4. Spheres and Horospheres in a Lobachevskian Space

1.5. Spheres and Horospheres in an Imaginary Lobachevskian Space

1.6. Invariant Integration in a Space of Constant Curvature

1.7. Integration over a Horosphere

1.8. Measures on the Absolute

2. Integral Transform Associated with Horospheres in a Lobachevskian Space

2.1. Integral Transform Associated with Horospheres

2.2. Inversion Formula for n = 3

2.3. Inversion Formula for Arbitrary Dimension

2.4. Functions Depending on the Distance from a Point to a Horosphere, and Their Averages

3. Integral Transform Associated with Horospheres in an Imaginary Lobachevskian Space

3.1. Statement of the Problem and Preliminary Remarks

3.2. Regularizing Integrals by Analytic Continuation in the Coordinates

3.3. Derivation of the Inversion Formula

3.4. Derivation of the Inversion Formula (Continued)

3.4a. Parallel Isotropie Lines

3.5. Calculation of Φ(x, a; μ)

Chapter VI Harmonic Analysis on Spaces Homogeneous with Respect to the Lorentz Group

1. Homogeneous Spaces and the Associated Representations of the Lorentz Group

1.1. Homogeneous Spaces

1.2. Representations of the Lorentz Group Associated with Homogeneous Spaces

1.3. The Relation between Representation Theory and Integral Geometry

1.4. Homogeneous Spaces and Associated Subgroups of Stability

1.5. Examples of Spaces Homogeneous with Respect to the Lorentz Group

1.6. Group-Theoretical Definition of Horospheres

1.7. Fourier Integral Expansions of Functions on Homogeneous Spaces

2. Representations of the Lorentz Group Associated with the Complex Affine Plane and with the Cone, and Their Irreducible Components

2.1. Unitary Representations of the Lorentz Group Associated with the Complex Affine Plane

2.2. Unitary Representation of the Lorentz Group Associated with the Cone

3. Decomposition of the Representation of the Lorentz Group Associated with Lobachevskian Space

3.1. Representation of the Lorentz Group Associated with Lobachevskian Space

3.2. Decomposition by the Horosphere Method

3.3. The Analog of Plancherel's Theorem for Lobachevskian Space

4. Decomposition of the Representation of the Lorentz Group Associated with Imaginary Lobachevskian Space

4.1. Representation of the Lorentz Group Associated with Imaginary Lobachevskian Space

4.2. Decomposition of the Representation Associated with Horospheres of the First Kind

4.3. Decomposition of the Representation Associated with Isotropie Lines

4.4. Decomposition of the Representation Associated with Imaginary Lobachevskian Space

4.5. The Analog of Plancherel's Theorem for Imaginary Lobachevskian Space

4.6. Integral Transform Associated with Planes in Lobachevskian Space

5. Integral Geometry and Harmonic Analysis on the Point Pairs on the Complex Projective Line

Chapter VII Representations of the Group of Real Unimodular Matrices in Two Dimensions

1. Representations of the Real Unimodular Matrices in Two Dimensions Acting on Homogeneous Functions of Two Real Variables

1.1. The Dx Spaces of Homogeneous Functions

1.2. Two Useful Realizations of Dx

1.3. Representation of G on Dx

1.4. The Tx(g) Operators in Other Realizations of Dx

1.5. The Dual Representations

2. Summary of the Basic Results concerning Representations on Dx

2.1. Irreducibility of Representations on Dx

2.2. Equivalence of Representations on Dx and the Role of Integer Points

2.3. The Problem of Equivalence at Integer Points

2.4. Unitary Representations

3. Invariant Bilinear Functionals

3.1. Invariance Under Translation and Dilation

3.2. Necessary and Sufficient Conditions for the Existence of an Invariant Bilinear Functional

3.3. Degenerate Invariant Bilinear Functionals for Analytic Representations

3.4. Conditionally Invariant Bilinear Functionals

4. Equivalence of Two Representations

4.1. Intertwining Operators

4.2. Equivalence of Two Representations

4.3. Partially Equivalent Representations

4.4. Other Models of F8+ and F8-

5. Unitary Representations of G

5.1. Existence of an Invariant Hermitian Functional

5.2. Positive Definite Invariant Hermitian Functionals (Nonanalytic Representations)

5.3. Invariant Hermitian Functionals for Analytic Representations

5.4. Invariant Positive Definite Hermitian Functionals on the Analytic Function Spaces F8+ and F8-

5.5. Unitary Representations of G by Operators on Hubert Space

5.6. Inequivalence of the Representations of the Discrete Series

5.7. Subspace Irreducibility of the Unitary Representations

Notes and References to the Literature

Bibliography

Index

## Product details

- No. of pages: 468
- Language: English
- Copyright: © Academic Press 1966
- Published: January 1, 1966
- Imprint: Academic Press
- eBook ISBN: 9781483262253

## About the Authors

### I. M. Gel'fand

### M. I. Graev

### N. Ya. Vilenkin

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