Symmetry Breaking for Representations of Rank One Orthogonal Groups II (Lecture Notes in Mathematics, 2234)

This work provides the first classification theory of matrix-valued symmetry breaking operators from principal series representations of a reductive group to those of its subgroup.The study of __symmetry breaking operators__ (intertwining operators for restriction) is an important and very active research area in modern representation theory, which also interacts with various fields in mathematics and theoretical physics ranging from number theory to differential geometry and quantum mechanics.The first author initiated a program of the general study of symmetry breaking operators. The present book pursues the program by introducing new ideas and techniques, giving a systematic and detailed treatment in the case of orthogonal groups of real rank one, which will serve as models for further research in other settings.In connection to automorphic forms, this work includes a proof for a multiplicity conjecture by Gross and Prasad for tempered principal series representations in the case (__SO__(__n__ + 1, 1), __SO__(__n__, 1)). The authors propose a further multiplicity conjecture for nontempered representations.Viewed from differential geometry, this seminal work accomplishes the classification of all conformally covariant operators transforming differential forms on a Riemanniann manifold __X__ to those on a submanifold in the model space (__X__, __Y__) = (__S____n__, __Sn__-1). Functional equations and explicit formulæ of these operators are also established.This book offers a self-contained and inspiring introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups. This feature will be helpful for active scientists and accessible to graduate students and young researchers in representation theory, automorphic forms, differential geometry, and theoretical physics. Preface......Page 6 Contents......Page 7 1 Introduction......Page 16 Notations......Page 27 2.1.1 Subgroups of G=O(n+1,1) and G'=O(n,1)......Page 28 2.1.2 Isotropic Cone......Page 31 2.1.4 The Center ZG(g) and the Harish-Chandra Isomorphism......Page 32 2.2.1 Notation for Irreducible Representations of O(N)......Page 35 2.2.2 Branching Laws for O(N) SO(N)......Page 37 2.3 Principal Series Representations Iδ(V,λ) of the Orthogonal Group G=O(n+1,1)......Page 38 2.3.1 C∞-induced Representations Iδ(V,λ)......Page 39 2.3.2 Tensoring with Characters χ of G......Page 40 2.3.3 K-structure of the Principal Series Representation Iδ(V,λ)......Page 41 2.4.1 ZG(g)-infinitesimal Character of Iδ(i,λ)......Page 42 2.4.3 Basic K-types of Iδ(i,λ)......Page 43 2.4.4 Reducibility of Iδ(i,λ)......Page 44 2.4.5 Irreducible Subquotients of Iδ(i,i)......Page 45 3.1 Generalities......Page 48 3.2.1 Symmetry Breaking Operators When [V:W] =0......Page 49 3.2.2 Differential Symmetry Breaking Operators When [V:W] =0......Page 50 3.2.3 Sporadic Symmetry Breaking Operators When [V:W]=0......Page 51 3.2.4 Existence Condition for Regular Symmetry Breaking Operators......Page 52 3.2.5 Integral Operators, Analytic Continuation, and Normalization Factors......Page 53 3.3 Classification Scheme of Symmetry Breaking Operators: General Case......Page 54 3.4 Summary: Vanishing of Regular Symmetry Breaking Operators Aλ,ν,V,W......Page 57 3.5 The Classification of Symmetry Breaking Operators for Differential Forms......Page 58 3.5.1 Vanishing Condition for the Regular Symmetry Breaking Operators Aλ,ν,γi,j......Page 59 3.5.2 Differential Symmetry Breaking Operators......Page 60 3.5.3 Formula of the Dimension of HomG' (Iδ(i,λ)|G',J(j,ν))......Page 63 3.5.4 Classification of Symmetry Breaking Operators Iδ(i,λ) →J(j,ν)......Page 64 3.6 Consequences of Main Theorems in Sections 3.3 and 3.5......Page 65 3.6.2 Complementary Series Representations......Page 66 3.6.3 Singular Complementary Series Representations......Page 67 3.7.1 Generalities: The Action of Character Group of G G' on {HomG'(|G', π)} in the General Case......Page 69 3.7.2 Actions of the Character Group of the Component Group on {HomG' (Iδ(i,λ)|G', J(j,ν))}......Page 70 3.7.3 Actions of Characters of the Component Group on HomG'(i,δ|G', πj,)......Page 72 4.1 Main Theorems......Page 73 4.2 Graphic Description of the Multiplicity for Irreducible Representations with Infinitesimal Character ρ......Page 74 5.1.1 Distribution Kernels of Symmetry Breaking Operators......Page 76 5.1.2 Invariant Bilinear Forms on Admissible Smooth Representations and Symmetry Breaking Operators......Page 78 5.2.1 Bruhat and Iwasawa Decompositions for G=O(n+1,1)......Page 81 5.2.2 Distribution Kernels for Symmetry Breaking Operators......Page 83 5.2.3 Distribution Sections for Dualizing Bundle Vλ,δ over G/P......Page 84 5.2.4 Pair of Distribution Kernels for Symmetry Breaking Operators......Page 87 5.3 Distribution Kernels near Infinity......Page 89 5.4 Vanishing Condition of Differential Symmetry Breaking Operators: Proof of Theorem 3.12 (1)......Page 91 5.5 Upper Estimate of the Multiplicities......Page 92 5.6 Proof of Theorem 3.10: Analytic Continuation of Symmetry Breaking Operators Aλ,ν,V, W......Page 94 5.6.2 Preliminary Results in the Scalar-Valued Case......Page 95 5.6.4 Step 2: Reduction to the Scalar-Valued Case......Page 98 5.6.5 Step 3: Proof of Holomorphic Continuation......Page 101 5.7 Existence Condition for Regular Symmetry Breaking Operators: Proof of Theorem 3.9......Page 102 5.9 Generic Multiplicity-one Theorem: Proof of Theorem 3.3......Page 104 5.10 Lower Estimate of the Multiplicities......Page 105 5.11.1 Expansion of Aλ,ν,γV, W Along ν=Constant......Page 106 5.11.2 Renormalized Regular Symmetry Breaking Operator Ã̃λ,ν,γV,W......Page 107 6 Differential Symmetry Breaking Operators......Page 110 6.1 Differential Operators Between Two Manifolds......Page 111 6.2 Duality for Differential Symmetry Breaking Operators......Page 112 6.3 Parabolic Subgroup Compatible with a Reductive Subgroup......Page 113 6.4 Character Identity for Branching in the Parabolic BGG Category......Page 115 6.5 Branching Laws for Generalized Verma Modules......Page 116 6.7 Existence of Differential Symmetry Breaking Operators: Extension to Special Parameters......Page 118 6.8 Proof of Theorem 3.13 (2-b)......Page 120 7.1 Some Notation on Index Sets......Page 123 7.1.2 Signatures for Index Sets......Page 124 7.2 Minor Determinant for ψ:RN - {0}→O(N)......Page 125 7.3 Minor Summation Formulæ......Page 127 8.1 Basic K-types in the Compact Picture......Page 131 8.2.1 Explicit K-finite Vectors in the N-picture......Page 133 8.2.2 Basic K-types in the N-picture......Page 135 8.3.1 Knapp–Stein Intertwining Operator......Page 137 8.3.2 K-spectrum of the Knapp–Stein Intertwining Operator......Page 138 8.3.3 Vanishing of the Knapp–Stein Operator......Page 139 8.3.4 Integration Formula for the (K,K)-spectrum......Page 140 8.4 Renormalization of the Knapp–Stein Intertwining Operator......Page 143 8.5 Kernel of the Knapp–Stein Operator......Page 144 9 Regular Symmetry Breaking Operators Aλ,ν,δEi,j from Iδ(i,λ) to JE(j,ν)......Page 146 9.1.1 Existence Condition for Regular Symmetry Breaking Operators......Page 147 9.1.2 Construction of Aλ,ν,i,j for j {i-1,i}......Page 148 9.2.1 Residue Formula of the Regular Symmetry Breaking Operator Aλ,ν,i,j......Page 149 9.2.2 Zeros of Aλ,ν,i,j......Page 150 9.3 (K,K')-spectrum for Symmetry Breaking Operators......Page 151 9.3.1 Generalities: (K,K')-spectrum of Symmetry Breaking Operators......Page 152 9.4 Explicit Formula of (K,K')-spectrum on Basic K-types for Regular Symmetry Breaking Operators Aλ,ν,i,j......Page 153 9.5 Proof of Vanishing Results on (K,K')-spectrum......Page 154 9.6 Proof of Theorem 9.8 on (K,K')-spectrum for the Normalized Symmetry Breaking Operator Aλ,v i,j: Iδ(i,λ) →Jδ(j,ν)......Page 155 9.6.1 Integral Expression of (K,K')-spectrum......Page 156 9.6.2 Integral Formula of the (K,K')-spectrum......Page 160 9.7 Proof of Theorem 9.8 on the (K,K')-spectrum for Aλ,ν,-i,j: Iδ(i,λ) →J-δ(j,ν)......Page 163 9.8 Matrix-Valued Functional Equations......Page 166 9.8.2 Proof of Functional Equations......Page 167 9.9.1 Functional Equations for the Renormalized Operator Ã̃λ,i,+i,i......Page 170 9.9.2 Functional Equations at Middle Degree for n Even......Page 171 9.9.4 Functional Equations at Middle Degree for n Odd......Page 173 9.10 Restriction Map Iδ(i,λ) →Jδ(i,λ)......Page 174 9.11 Image of the Differential Symmetry Breaking Operator Cλ,νi,j......Page 175 9.11.1 Surjectivity Condition of Cλ,νi,j......Page 176 9.11.2 Functional Equation for Cλ,νi,j......Page 177 9.11.3 The Case When Tν,n-1-νj=0......Page 179 9.11.4 Proof of Theorems 9.33 and 9.34......Page 180 10.1 Proof of the Vanishing Result (Theorem 4.1)......Page 182 10.2.1 Generators of Symmetry Breaking Operators Between Principal Series Representations Having the Trivial Infinitesimal Character ρ......Page 184 10.2.3 Multiplicity-one Property: Proof of Theorem 4.2......Page 186 10.2.4 First Construction i,δ →πi,δ (1 ≤i ≤n)......Page 188 10.2.5 Second Construction i,δ →πi,δ (0 ≤i ≤n-1)......Page 189 10.2.6 Third Construction i,δ →πi,δ......Page 190 10.3 Splitting of Iδ(m,m) and Its Symmetry Breaking for (G,G')=(O(2m+1,1),O(2m,1))......Page 191 10.3.1 HomG'(Iδ(m,m)|G', J(m,m)) with δ=+......Page 192 10.3.2 HomG'(Iδ(m,m)|G',J(m,m)) with δ=-......Page 194 10.4 Splitting of J(m,m) and Symmetry Breaking Operators for (G,G')=(O(2m+2,1),O(2m+1,1))......Page 195 10.4.1 HomG'(Iδ(m+1,m)|G', Jδ(m,m)) for n=2m+1......Page 196 10.5 Symmetry Breaking Operators from i,δ to πi-1,δ......Page 199 11.1 Vogan Packets of Tempered Induced Representations......Page 203 11.3 Embedding the Group G=SO(n-2p,2p+1) into the Group G=SO(n-2p+1,2p+1)......Page 205 11.4 The Gross–Prasad Conjecture I: Tempered Principal Series Representations......Page 206 11.5 The Gross–Prasad Conjecture II: Tempered Representations with Trivial Infinitesimal Character ρ......Page 210 11.5.1 The Gross–Prasad Conjecture II: Symmetry Breaking from m,(-1)m+1 to the Discrete Series Representation πm......Page 211 11.5.2 The Gross–Prasad Conjecture II: Symmetry Breaking from the Discrete Series Representation πm to m-1,(-1)m......Page 212 12.1.1 Periods......Page 216 12.1.2 Distinguished Representations......Page 218 12.2 Proofs of Theorems 12.4 and 12.5......Page 219 12.3 Bilinear Forms on (g,K)-cohomologies via Symmetry Breaking: General Theory for Nonvanishing......Page 221 12.3.1 Pull-back of (g,K)-cohomologies via Symmetry Breaking......Page 222 12.3.2 Nonvanishing of Pull-back of (g, K)-cohomologies of Aq via Symmetry Breaking......Page 223 12.4.1 Nonvanishing Theorem for O(n+1,1) O(n,1)......Page 225 12.4.2 Special Cycles......Page 228 13.1.1 Definition of Hasse Sequence and StandardSequence......Page 232 13.1.2 Existence of Hasse Sequence......Page 234 13.1.3 Langlands Parameter of the Representations in the Hasse Sequence......Page 235 13.2.1 Conjecture: Version 1......Page 241 13.2.2 Conjecture: Version 2......Page 242 13.3 Supporting Evidence......Page 244 13.3.2 Evidence E.2......Page 245 13.3.3 Evidence E.3......Page 246 13.3.4 Evidence E.4......Page 248 14.1 Finite-dimensional Representations of O(N-1,1)......Page 259 14.2 Singular Parameters for V O(n): S(V) and SY(V)......Page 264 14.2.1 Infinitesimal Character r(V,λ) of Iδ(V,λ)......Page 265 14.2.2 Singular Integral Parameter: S(V) and SY(V)......Page 267 14.3 Irreducibility Condition of Iδ(V,λ)......Page 268 14.4.2 Subrepresentations of Iδ(V,n2) for V of Type Y......Page 270 14.5 Definition of the Height i(V,λ)......Page 271 14.6.1 K-type Formula of Iδ(V, λ)......Page 275 14.6.2 K-types of Subquotients Iδ(V,λ) and Iδ(V,λ)......Page 276 14.7 (δ, V, λ) (δ,V, λ) and (δ,V, λ)......Page 278 14.8.1 Characterizations of the Irreducible Subquotients δ(V,λ)......Page 281 14.9.1 Cohomological Parabolic Induction Aq(λ)=RqS(Cλ+ρ(u))......Page 283 14.9.2 θ-stable Parabolic Subalgebra qi for G=O(n+1,1)......Page 285 14.9.3 Irreducible Representations , δ and (Aqi),......Page 289 14.9.4 Irreducible Representations with Nonzero (g,K)-cohomologies......Page 290 14.9.5 Description of Subquotients in Iδ(V,λ)......Page 291 14.9.6 Proof of Theorem 14.46......Page 292 14.10 Hasse Sequence in Terms of θ-stable Parameters......Page 293 14.11 Singular Integral Case......Page 294 15.1 Restriction of Representations of G=O(n+1,1) to G= SO(n+1,1)......Page 297 15.2 Restriction of Principal Series Representation of G=O(n+1,1) to G=SO(n+1,1)......Page 299 15.2.1 Restriction Iδ(V,λ)|G When Iδ(V,λ) Is Irreducible......Page 300 15.2.2 Restriction Iδ(V,λ)|G When V Is of Type Y......Page 301 15.3 Proof of Theorem 14.15: Irreducibility Criterion of Iδ(V,λ)......Page 303 15.4 Socle Filtration of Iδ(V,λ): Proof of Proposition 14.19......Page 304 15.5 Restriction of ,δ to SO(n+1,1)......Page 306 15.6 Symmetry Breaking for Tempered Principal Series Representations......Page 308 15.7 Symmetry Breaking from Iδ(i,λ) to J(j,ν)......Page 310 15.8 Symmetry Breaking Between Irreducible Representations of G and G' with Trivial Infinitesimal Character ρ......Page 314 16.1 Some Features of Translation Functors for Reductive Groups that Are Not of Harish-Chandra Class......Page 315 16.2.1 Primary Decomposition of Admissible Smooth Representations......Page 316 16.2.2 Translation Functor ψμμ+τ for G=O(n+1,1)......Page 317 16.2.3 The Translation Functor and the Restriction G downarrow......Page 318 16.3.1 Main Results: Translation of Principal Series Representations......Page 319 16.3.3 Basic Lemmas for the Translation Functor......Page 321 16.4 Definition of an Irreducible Finite-dimensional Representation F(V,λ) of G=O(n+1,1)......Page 323 16.4.1 Definition of σ(i)(λ) and widehatσ(i)......Page 324 16.4.2 Definition of a Finite-dimensional Representation F(V,λ) of G......Page 325 16.4.3 Reformulation of Theorems 16.6 and 16.8......Page 327 16.4.5 Proof of Theorems 16.22 and 16.23......Page 328 16.5.1 Irreducible Summands for O(n+2) downarrow O(n) O(2) and for Tensor Product Representations......Page 330 16.5.2 Irreducible Summand for the Restriction G downarrow M A and for Tensor Product Representations......Page 333 16.6 Proof of Theorem 16.6......Page 335 16.6.2 Case: (V,λ) RedI I......Page 336 16.7 Proof of Theorem 16.8......Page 338 References......Page 340 List of Symbols......Page 344 Index......Page 347 "This work provides the first classification theory of matrix-valued symmetry breaking operators from principal series representations of a reductive group to those of its subgroup. The study of symmetry breaking operators (intertwining operators for restriction) is an important and very active research area in modern representation theory, which also interacts with various fields in mathematics and theoretical physics ranging from number theory to differential geometry and quantum mechanics. The first author initiated a program of the general study of symmetry breaking operators. The present book pursues the program by introducing new ideas and techniques, giving a systematic and detailed treatment in the case of orthogonal groups of real rank one, which will serve as models for further research in other settings. In connection to automorphic forms, this work includes a proof for a multiplicity conjecture by Gross and Prasad for tempered principal series representations in the case (SO(n + 1, 1), SO(n, 1)). The authors propose a further multiplicity conjecture for nontempered representations. Viewed from differential geometry, this seminal work accomplishes the classification of all conformally covariant operators transforming differential forms on a Riemanniann manifold X to those on a submanifold in the model space (X, Y) = (Sn, Sn-1). Functional equations and explicit formulae of these operators are also established. This book offers a self-contained and inspiring introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups. This feature will be helpful for active scientists and accessible to graduate students and young researchers in representation theory, automorphic forms, differential geometry, and theoretical physics"--Print version, page 4 of cover This work provides the first classification theory of matrix-valued symmetry breaking operators from principal series representations of a reductive group to those of its subgroup. The study of symmetry breaking operators (intertwining operators for restriction) is an important and very active research area in modern representation theory, which also interacts with various fields in mathematics and theoretical physics ranging from number theory to differential geometry and quantum mechanics. The first author initiated a program of the general study of symmetry breaking operators. The present book pursues the program by introducing new ideas and techniques, giving a systematic and detailed treatment in the case of orthogonal groups of real rank one, which will serve as models for further research in other settings. In connection to automorphic forms, this work includes a proof for a multiplicity conjecture by Gross and Prasad for tempered principal series representations in the case (SO(n + 1, 1), SO(n, 1)). The authors propose a further multiplicity conjecture for nontempered representations. Viewed from differential geometry, this seminal work accomplishes the classification of all conformally covariant operators transforming differential forms on a Riemanniann manifold X to those on a submanifold in the model space (X, Y) = (Sn, Sn-1). Functional equations and explicit formulæ of these operators are also established. This book offers a self-contained and inspiring introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups. This feature will be helpful for active scientists and accessible to graduate students and young researchers in representation theory, automorphic forms, differential geometry, and theoretical physics