Representation Theory and Automorphic Functions

Description

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