# Symmetry Breaking for Representations of Rank One Orthogonal Groups

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*Toshiyuki Kobayashi; Birgit Speh*

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.

#### Table of Contents

# Table of Contents

## Symmetry Breaking for Representations of Rank One Orthogonal Groups

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Chapter 2. Symmetry breaking for the spherical principal series representations 1118 free
- Chapter 3. Symmetry breaking operators 2330
- 3.1. Restriction of representations and symmetry breaking operators 2330
- 3.2. Distribution kernels of symmetry breaking operators 2330
- 3.3. Differential intertwining operators 2633
- 3.4. Smooth representations and intertwining operators 2835
- 3.5. Symmetry breaking operators for principal series representations 2936
- 3.6. Meromorphic continuation of symmetry breaking operators 3037

- Chapter 4. More about principal series representations 3340
- Chapter 5. Double coset decomposition ๐โ\๐บ/๐ 4148
- Chapter 6. Differential equations satisfied by the distribution kernels of symmetry breaking operators 4552
- Chapter 7. ๐พ-finite vectors and regular symmetry breaking operators ฬ๐ธ_{๐,๐} 4956
- Chapter 8. Meromorphic continuation of regular symmetry breaking operators \ka{๐}๐ 5764
- Chapter 9. Singular symmetry breaking operator \B_{๐,๐} 6370
- Chapter 10. Differential symmetry breaking operators 6976
- Chapter 11. Classification of symmetry breaking operators 7582
- Chapter 12. Residue formulae and functional identities 8390
- Chapter 13. Image of symmetry breaking operators 8996
- Chapter 14. Application to analysis on anti-de Sitter space 93100
- Chapter 15. Application to branching laws of complementary series 101108
- Chapter 16. Appendix 105112
- References 109116
- List of Symbols 111118
- Back Cover Back Cover1124