Overview
- Introduces a new method to construct and classify matrix-valued symmetry breaking operators in representation theory
- Includes hot topics of automorphic forms and conformal geometry as applications of branching rules in representation theory
- Provides the complete classification of all conformally equivariant operators on differential forms on the model space (Sn, Sn-1})
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2234)
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Table of contents (16 chapters)
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Front Matter
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Back Matter
Keywords
- Symmetry breaking operator
- branching law
- Gross-Prasad conjecture
- automorphic form
- conformal geometry
- reductive group
- restriction of representation
- period
- orthogonal group
- tempered representation
- Verma module
- intertwining operator
- F-method
- Lorentz group
- (g,K) cohomology
- Lie group
- unitary representation
- Gegenbauer polynomial
- Juhl operator
- differential form
About this book
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 formulæ 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.
Authors and Affiliations
Bibliographic Information
Book Title: Symmetry Breaking for Representations of Rank One Orthogonal Groups II
Authors: Toshiyuki Kobayashi, Birgit Speh
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-981-13-2901-2
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2018
Softcover ISBN: 978-981-13-2900-5Published: 28 December 2018
eBook ISBN: 978-981-13-2901-2Published: 27 December 2018
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XV, 344
Number of Illustrations: 4 b/w illustrations, 11 illustrations in colour
Topics: Mathematical Physics, Topological Groups, Lie Groups, Number Theory, Differential Geometry, Partial Differential Equations, Global Analysis and Analysis on Manifolds