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Symmetry Breaking for Representations of Rank One Orthogonal Groups

Toshiyuki Kobayashi University of Tokyo, Japan
Birgit Speh Cornell University, Ithaca, NY
Available Formats:
Electronic ISBN: 978-1-4704-2615-6
Product Code: MEMO/238/1126.E
112 pp
List Price: $80.00 MAA Member Price:$72.00
AMS Member Price: $48.00 Click above image for expanded view Symmetry Breaking for Representations of Rank One Orthogonal Groups Toshiyuki Kobayashi University of Tokyo, Japan Birgit Speh Cornell University, Ithaca, NY Available Formats:  Electronic ISBN: 978-1-4704-2615-6 Product Code: MEMO/238/1126.E 112 pp  List Price:$80.00 MAA Member Price: $72.00 AMS Member Price:$48.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2382015
MSC: Primary 22; Secondary 33; 53;

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.

• Chapters
• 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
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Volume: 2382015
MSC: Primary 22; Secondary 33; 53;

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.

• Chapters
• 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
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