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Symmetry breaking for representations of rank one orthogonal groups

About this Title

Toshiyuki Kobayashi, Kavli IPMU (WPI), Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914 Japan and Birgit Speh, Department of Mathematics, Cornell University, Ithaca, New York 14853-4201

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)
Published electronically: May 12, 2015
Keywords: Branching law, reductive Lie group, symmetry breaking, Lorentz group, conformal geometry, Verma module, complementary series.
MSC: Primary 22E46; Secondary 33C45, 53C35

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Table of Contents


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


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.

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