Representation Theory and Automorphic Functions

Title page
Preface
Chapter 1: HOMOGENEOUS SPACES WlTH A DISCRETE STABILlTY GROUP
1 Generalities
1 Homogeneous Spaces and Their Stability Subgroups
2 The Connection Between the Homogeneous Spaces X = Γ\G and Riemann Surfaces
3 The Fundamental Domain of a Discrete Group Γ
4 Discrete Groups with a Compact Fundamental Domain
5 The Structure of a Fundamental Domain in the Lobachevskii Plane
2 Representations of a Group G Induced by a Discrete Subgroup
1 Definition of Induced Representations
2 The Operators T_φ
3 The Discreteness of the Spectrum of the Induced Representation in the Case of a Compact Space X = Γ\G
4 The Trace Formula
5 Another Form of the Trace Formula
3 Irreducible Unitary Representations of the Group of Real Unimodular Matrices of Order 2
1 The Principal Series of Irreducible Unitary Representations
2 The Supplementary Series of Representations
3 The Discrete Series of Representations
4 Another Realization of the Representations of the Principal and Supplementary Series
5 The Laplace Operator Δ The Spaces Ω_s
4 The Duality Theorem
1 Automorphic Forms
2 Statement of the Duality Theorem,47 3 The Laplace Operator
4 Proof of the Duality Theorem for Representations of the Continuous Series
5 Proof of the Duality Theorem for Representations of the Discrete Series
6 The General Duality Theorem
5 The Trace Formula for the Group G of Real U nimodular Matrices of Order 2
1 Statement of the Problem
2 The Function h
3 Contribution of the Hyperbolic Elements to the Trace Formula
4 Contribution of the Elliptic Elements
5 Contribution of the Elements e and -e to the Trace Formula
6 The Final Traee Formula
7 Formulae for the Multiplicities of the Representations of the Discrete Series
8 Complete Splitting of the Trace Formula
9 Construction of the Functions φ_n^+(g) and φ_n^-(g)
10 The Asymptotic Formula
11 The Trace Formula for the Case When -e Does Not Belong to Γ,84
Appendix I to 5 A Theorem on Continuous Deformations of a Discrete Subgroup
Appendix II to 5 The Trace Formula for the Group of Complex Unimodular Matrices of Order 2
1 Irreducible Unitary Representations of G
2 The Trace Formula for G
3 The Asyrnptotic Formula
6 Investigation of the Spectrum of a Representation Generated by a Noncompact Space X = Γ\G (Separation of the Discrete Part of the Spectrum)
1 Horospheres in a Homogeneous Space
2 Statement of the Main Theorern
3 Cylindrical Sets
4 Reduction of the Main Theorem
5 Proof that the Trace etc is Finite
Appendix to Chapter 1 Arithmetic Subgroups of the Group G of Real Unimodular Matrices of Order 2
1 Definition of an Arithmetic Subgroup
2 The Modular Group
3 Some Subgroups of the Modular Group
4 Quaternion groups
Chapter 2: REPRESENTATIONS OF THE GROUP OF UNIMODULAR MATRICES OF ORDER 2 WlTH ELEMENTS FROM A LOCALLY COMPACT TOPOLOGICAL FIELD
1 Structure of Locally Compact Fields
1 Classification of Locally Compact Fields
2 The Norm in K
3 Structure of Disconnected Fields
4 Additive and Multiplicative Characters of K
5 The Structure of the Subgroup A The Functions exp x and ln x
6 Quadratic Extensions of a Disconnected Field
7 The Multiplicative Characters sign_τ x
8 Circles in K(√τ)
9 Cartesian and Polar Coordinates in K(√τ)
10 Invariant Measures on K and in its Quadratic Extension K(√τ)
11 Additive and Multiplicative Characters on the "Plane" K√τ
2 Test and Generalized Functions on a Locally Compact Disconnected Field K
1 The Space of Test Functions
2 Generalized Functions Conccntrated at a Point
3 Homogeneous Generalized Functions
4 The Fourier Transform of Test Functions
5 The Fourier Transform of Generalized Homogeneous Functions The Gamma-Function and Beta-Function
6 Additional Information on the Gamma-Function
7 The Integral etc
8 Functions Resembling Analytic Functions in the Upper and the Lower Half-Plane
9 The Mellin Transform
10 The Relation Between the Gamma-Function Connected with the Ground Field K and the Gamma-Function Connected with the Quadratic Extension K(√τ) of K
3 Irreducible Representations of the Group of Matrices of Order 2 with Elements from a Locally Compact Field (the Continuous Series)
1 The Continuous Series of U nitary Representations of G
2 Another Realization of the Representations of the Continuous Series
3 Equivalence of Representations of the Continuous Series
4 The Irreducibility of the Representations of the Continuous Series
5 The Decomposition of the Representations etc into Irreducible Representations
6 The Quasiregular Representation of G and its Decomposition into Irreducible Representations
7 The Supplementary Series of Irreducible Unitary Representations of G
8 The Singular Representation of G
9 Representations in the Spaces D_π
10 Spherical Functions, 174 11 The Operator of the Horospherical Automorphism
4 The Discrete Series of Irreducible U nitary Representations of G
1 Description of the Representations of the Discrete Series
2 Continuous Dependence of the Operators T_π(g) on g
3 Proof of the Re1ation T_π(glgZ) = T_π(gl)T_π(gz)
4 Unitariness of the Operators T_π(g)
5 The π-Realization of the Representations of the Discrete Series
6 Another Realization of the Representations of the Discrete Series
7 Equivalence of Representations of the Discrete Series
8 Discrete Series for the Field of 2-adic Numbers
5 The Traces of Irreducible Representations of G
1 Statement of the Problem
2 The Traces of the Representations of the Continuous Series
3 Trace of the Singular Representation
4 Traces of the Representations of the Discrete Series
5 Traces of the Representations of the Discrete Series for the Field of Real Numbers
6 The Inversion Formula and the Plancherel Formula on G
1 Statement of the Problem
2 The Inversion Formula for a Disconnected Field
3 Computation of Certain Integrais
4 Computation of the Constant c in the Inversion Formula
5 The Inversion Formulae for Connected Fields
Appendix to Chapter 2
1 Some Facts from the Theory of Operator Rings in Hilbert Space
2 Connection Between the Unitary Representations of the Group G of all Nonsingular Matrices of Order 2 and the Subgroup of Matrices of the Form etc
3 Theorem on the Complete Continuity of the Operator T_φ
4 The Decomposition of an Irreducible Representation of G Relative to Representations of its Maximal Compact Subgroup The Theorem on the Existence of a Trace
5 Representations of the Unimodular Group
6 Classification of all Irreducible Representations of G and ~G
Chapter 3: REPRESENTATlONS OF ADELE GROUPS
1 Adeles and Ideles
1 The Group of Characters of the Additive Group of Rational Numbers
2 Definition of Adeles and Ideles
3 Another Construction of the Group of Adeles
4 The Isomorphisms Q --> A and Q* --> A*
5 The Group of Additive Characters of the Ring of Adeles A
6 The Characters of the Group A/Q
7 Invariant Measures in the Group of Adeles and the Group of Ideles
8 The Function |λ|
9 The Characters of the Group of Ideles A*
10 The Characters of the Group A*/Q*
Appendix to l On a Zeta-Function
2 Analysis on the Group of Adeles
1 Schwartz-Bruhat Functions
2 The Fourier Transform of Schwartz-Bruhat Functions
3 The Poisson Summation Formula
4 The Mellin Transform of Schwartz-Bruhat Functions The Tate Formula
5 The Space A^n
Appendix to 2 Tate Rings
3 The Groups of Adeles G_A and their Representations
1 Definition of the Group of Adeles G_A
2 Irreducible Unitary Representations of the Group of Adeles
3 Proof of a Theorem on Tensor Products
4 Criteria for the Existence of a Single Linearly Independent Invariant Vector
5 Second Theorem on Tensor Products
4 The Adele Group of the Group of Unimodular Matrices of Order 2
1 Statement of the Problem and Summary of the Results
2 The Structure of the Space X
3 Description of the Space Ω of all Compact Horospheres of X
4 Cylindrical Sets
5 The Horospherical Map
6 Investigation of the Kernel of the Horospherical Map (Discreteness of the Spectrum)
7 The Spaces A², Υ and E
8 The Operation of Multiplication in the Spaces A², Υ and E
9 Decomposition of the Representations Generated by Υ and Ω into Irreducible Representations
10 The Operator B (Definition)
11 Properties of the Operator B
12 Schwartz-Bruhat Functions in Ω
13 The Fourier Transform in L₂(Ω)
14 The Operator M
15 An Explicit Expression for M
16 The Family M of Functions on Ω
17 Decomposition of the Representation in H' into Irreducible Representations
18 Connection of the Operator of the Horospherical Automorphism B with Dirichlet L-Functions
Appendix I to 4
1 Lemma on the Completeness of the Family Φ_∞)
2 Lemma on Functions Defined on the Half-Line 0<=τ<∞ and Belonging to L₂
Appendix II to 4
1 On the Connection Between the Homogeneous Space GQ\G A and the Homogeneous Spaces of the Group Ga)'
2 The Generalized Peterson Conjecture
5 The Space of Horospheres
1 Reductive Algebraic Groups
2 The Space L₂(etc.)
3 The Operators B_s
4 Properties of the Operators B_s
5 Main Theorem on the Operators B_s
6 Reduction to Rank l
6 Representations Generated by the Homogeneous Space G_Q\G_A
1 The Homogeneous Space G_Q\G_A
2 Investigation of the Spectrum of the Representation for a Compact Space G_Q\G_A/K_A
3 The Space of Horospheres
4 The Horospherical Map and the Operator M
5 An Explicit Expression for the Operator M
6 The Structure of the Space H'
7 Discreteness of the Spectrum
1 Horospheres in the Space X = G_Q\G_A
2 Statement of the Main Theorem
3 Siegel Sets on G_A
4 Regular Siegel Sets
5 Regular Siegel Sets Connected with Π-Horospheres
6 Reduction of the Main Theorem
7 The p-Norm
8 Proof of the Main Theorem
9 Solvable Algebras and Groups Statement of the Fundamental Lemrna
10 Proof of the Fundamental Lemma
Appendix to 7 Functions on Regular Nilpotent Lie Groups
1 Regular Nilpotent Algebras
2 Regular Nilpotent Lie Groups
GUIDE TO THE LITERATURE
BIBLIOGRAPHY
INDEX OF NAMES
SUBJECT IXDEX