Symmetry Breaking for Representations of Rank One Orthogonal Groups.

Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Symmetry breaking for the spherical principal series representations -- 2.1. Notation and review of previous results -- 2.2. Finite-dimensional subquotients of disconnected groups -- 2.3. Symmetry breaking operators and spherical principal series representations -- 2.4. Multiplicities for composition factors -- Chapter 3. Symmetry breaking operators -- 3.1. Restriction of representations and symmetry breaking operators -- 3.2. Distribution kernels of symmetry breaking operators -- 3.3. Differential intertwining operators -- 3.4. Smooth representations and intertwining operators -- 3.5. Symmetry breaking operators for principal series representations -- 3.6. Meromorphic continuation of symmetry breaking operators -- Chapter 4. More about principal series representations -- 4.1. Models of principal series representations -- 4.2. Explicit -finite functions in the non-compact model -- 4.3. Normalized Knapp-Stein intertwining operator -- Chapter 5. Double coset decomposition '\ / -- Chapter 6. Differential equations satisfied by the distribution kernels of symmetry breaking operators -- 6.1. A system of differential equations for symmetry breaking operators -- 6.2. The solutions ℴ (ℝⁿ∖{0} -- , ) -- Chapter 7. -finite vectors and regular symmetry breaking operators ̃ _{ , } -- 7.1. Distribution kernel \ka{ } and its normalization -- 7.2. Preliminary results -- 7.3. Proof of Proposition 7.3 -- Chapter 8. Meromorphic continuation of regular symmetry breaking operators \ka{ } -- 8.1. Recurrence relations of the distribution kernels \ka{ } -- 8.2. Functional equations -- 8.3. 8.3 Support of \KA{ } -- 8.4. Renormalization \AAt_{ , } for ∈-ℕ -- Chapter 9. Singular symmetry breaking operator \B_{ , } -- 9.1. Singular symmetry breaking operator \B_{ , }.

9.2. -finite vectors and singular operators \tB{ } -- 9.3. Proof of Theorem 9.1 -- 9.4. Support of the distribution kernel of \B_{ , } -- 9.5. Renormalization \BB_{ , } for \nulambda∈ _{ } with odd -- Chapter 10. Differential symmetry breaking operators -- 10.1. Power of the Laplacian -- 10.2. Juhl's family of differential operators -- 10.3. The kernel of the differential symmetry breaking operator \C_{ , } -- Chapter 11. Classification of symmetry breaking operators -- 11.1. Classification of symmetry breaking operators -- 11.2. Strategy of the proof of Theorem 11.1 -- 11.3. Lower bounds of the multiplicities -- 11.4. Extension of solutions from ℝⁿ∖{0} to ℝⁿ -- 11.5. Regular symmetry breaking operators -- 11.6. Singular symmetry breaking operators -- Chapter 12. Residue formulae and functional identities -- 12.1. Residues of symmetry breaking operators -- 12.2. Functional equations satisfied by singular symmetry breaking operators -- Chapter 13. Image of symmetry breaking operators -- 13.1. Finite-dimensional image for ∈-ℕ -- 13.2. Image for ∈ +ℕ -- 13.3. Spherical vectors and symmetry breaking operators -- Chapter 14. Application to analysis on anti-de Sitter space -- 14.1. Harmonic analysis on Lorentzian hyperbolic spaces -- 14.2. Application of symmetry breaking operators to anti-de Sitter spaces -- 14.3. Analysis on vector bundles over anti-de Sitter spaces -- Chapter 15. Application to branching laws of complementary series -- 15.1. Discrete spectrum in complementary series -- 15.2. ²-model of complementary series representations -- Chapter 16. Appendix -- 16.1. Gegenbauer polynomials -- 16.2. -Bessel function and its renormalization -- 16.3. Zuckerman derived functor modules _{ }( ) -- Acknowledgments -- References -- List of Symbols -- Back Cover.

The authors give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). They construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explicitly. Symmetry breaking operators at exceptional discrete parameters are thoroughly studied. The authors obtain closed formulae for the functional equations which the composition of the symmetry breaking operators with the Knapp--Stein intertwining operators of G and G' satisfy, and use them to determine the symmetry breaking operators between irreducible composition factors of the spherical principal series representations of G and G'. Some applications are included.

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