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Symmetry breaking for representations of rank one orthogonal groups II

Author: Toshiyuki Kobayashi; Birgit Speh
Publisher: Singapore : Springer, 2018.
Series: Lecture notes in mathematics (Springer-Verlag), 2234.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
"This work provides the first classification theory of matrix-valued symmetry breaking operators from principal series representations of a reductive group to those of its subgroup. The study of symmetry breaking operators (intertwining operators for restriction) is an important and very active research area in modern representation theory, which also interacts with various fields in mathematics and theoretical  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Kobayashi, Toshiyuki, 1962-
Symmetry breaking for representations of rank one orthogonal groups II.
Singapore : Springer, [2018]
(DLC) 2015027247
(OCoLC)1052875671
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Toshiyuki Kobayashi; Birgit Speh
ISBN: 9789811329012 981132901X
OCLC Number: 1081173560
Description: 1 online resource (xv, 344 pages) : illustrations (some color).
Contents: Review of principal series representations --
Symmetry breaking operators for principal series representations : general theory --
Symmetry breaking for irreducible representations with infinitesimal character [rho] --
Regular symmetry breaking operators --
Differential symmetry breaking operators --
Minor summation formulae related to exterior tensor [wedge]i(Cn) --
The Knapp-Stein intertwining operators revisted : renormalization and the K-spectrum --
Regular symmetry breaking operators Ã[i,j/lambda, v,&epsilon] from I[delta](i,[lambda]) to J[epsilon](j, v) --
Symmetry breaking operators for irreducible representations with innitesimal character [rho] : proof of theorems 4.1 and 4.2 --
Application I : some conjectures by B. Gross and D. Prasad : restrictions of tempered representations of SO(n+1, 1) to SO(n, 1) --
Application II : periods, distinguished representations and (g, K)-cohomologies --
A conjecture : symmetry breaking for irreducible representations with regular integral infinitesimal character --
Appendix I. Irreducible representations of G=O(n+1, 1), [theta]-stable parameters, and cohomological induction --
Appendix II. Restriction to Ḡ=SO(n+1, 1) --
Appendix III. A translation functor for G=O(n+1, 1).
Series Title: Lecture notes in mathematics (Springer-Verlag), 2234.
Responsibility: Toshiyuki Kobayashi, Birgit Speh.

Abstract:

"This work provides the first classification theory of matrix-valued symmetry breaking operators from principal series representations of a reductive group to those of its subgroup. The study of symmetry breaking operators (intertwining operators for restriction) is an important and very active research area in modern representation theory, which also interacts with various fields in mathematics and theoretical physics ranging from number theory to differential geometry and quantum mechanics. The first author initiated a program of the general study of symmetry breaking operators. The present book pursues the program by introducing new ideas and techniques, giving a systematic and detailed treatment in the case of orthogonal groups of real rank one, which will serve as models for further research in other settings. In connection to automorphic forms, this work includes a proof for a multiplicity conjecture by Gross and Prasad for tempered principal series representations in the case (SO(n + 1, 1), SO(n, 1)). The authors propose a further multiplicity conjecture for nontempered representations. Viewed from differential geometry, this seminal work accomplishes the classification of all conformally covariant operators transforming differential forms on a Riemanniann manifold X to those on a submanifold in the model space (X, Y) = (Sn, Sn-1). Functional equations and explicit formulae of these operators are also established. This book offers a self-contained and inspiring introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups. This feature will be helpful for active scientists and accessible to graduate students and young researchers in representation theory, automorphic forms, differential geometry, and theoretical physics"--Print version, page 4 of cover.

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