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# memo_has_moved_text();Symmetry breaking for representations of rank one orthogonal groups

Toshiyuki Kobayashi and Birgit Speh

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 238, Number 1126
ISBNs: 978-1-4704-1922-6 (print); 978-1-4704-2615-6 (online)
DOI: http://dx.doi.org/10.1090/memo/1126
Published electronically: May 12, 2015
Keywords:Branching law, reductive Lie group, symmetry breaking, Lorentz group, conformal geometry, Verma module

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Chapters

• Chapter 1. Introduction
• Chapter 2. Symmetry breaking for the spherical principal series representations
• Chapter 3. Symmetry breaking operators
• Chapter 4. More about principal series representations
• Chapter 5. Double coset decomposition $P’ \backslash G/P$
• Chapter 6. Differential equations satisfied by the distribution kernels of symmetry breaking operators
• Chapter 7. $K$-finite vectors and regular symmetry breaking operators $\protect \widetilde {\mathbb {A}} _{\lambda , \nu }$
• Chapter 8. Meromorphic continuation of regular symmetry breaking operators ${K}_{{\lambda },{\nu }}^{\mathbb {A}}$
• Chapter 9. Singular symmetry breaking operator $\protect \widetilde {\mathbb {B}}_{\lambda ,\nu }$
• Chapter 10. Differential symmetry breaking operators
• Chapter 11. Classification of symmetry breaking operators
• Chapter 12. Residue formulae and functional identities
• Chapter 13. Image of symmetry breaking operators
• Chapter 14. Application to analysis on anti-de Sitter space
• Chapter 15. Application to branching laws of complementary series
• Chapter 16. Appendix

### Abstract

We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of $G=O(n+1,1)$ and $G'=O(n,1)$. We 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. We 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.