Symmetry breaking for representations of rank one orthogonal groups
We give a complete classification of intertwining operators ( We obtain closed formulae for the functional equations which the composition of the symmetry breaking operators with the Knapp–Stein intertwining operators of
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| Main Authors | , |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Providence, Rhode Island
American Mathematical Society
2015
|
| Edition | 1 |
| Series | Memoirs of the American Mathematical Society |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9781470419226 147041922X |
| ISSN | 0065-9266 1947-6221 |
| DOI | 10.1090/memo/1126 |
Cover
Table of Contents:
- Introduction -- Symmetry breaking for the spherical principal series representations -- Symmetry breaking operators -- More about principal series representations -- Double coset decomposition <inline-formula content-type="math/mathml"> P ′ ∖ G / P P’ \backslash G/P </inline-formula> -- Differential equations satisfied by the distribution kernels of symmetry breaking operators -- <inline-formula content-type="math/mathml"> K K </inline-formula>-finite vectors and regular symmetry breaking operators <inline-formula content-type="math/mathml"> A ~ λ , ν \widetilde {\mathbb {A}} _{\lambda , \nu } </inline-formula> -- Meromorphic continuation of regular symmetry breaking operators <inline-formula content-type="math/mathml"> K λ , ν A {K}_{{\lambda },{\nu }}^{\mathbb {A}} </inline-formula> -- Singular symmetry breaking operator <inline-formula content-type="math/mathml"> B ~ λ<!-- λ --> , ν \widetilde {\mathbb {B}}_{\lambda ,\nu } </inline-formula> -- Differential symmetry breaking operators -- Classification of symmetry breaking operators -- Residue formulae and functional identities -- Image of symmetry breaking operators -- Application to analysis on anti-de Sitter space -- Application to branching laws of complementary series -- Appendix
- 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