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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
Front Cover for Symmetry Breaking for Representations of Rank One Orthogonal Groups
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
Front Cover for Symmetry Breaking for Representations of Rank One Orthogonal Groups
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  • Front Cover for Symmetry Breaking for Representations of Rank One Orthogonal Groups
  • Back Cover for Symmetry Breaking for Representations of Rank One Orthogonal Groups
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.

  • Table of Contents
     
     
    • 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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