Overview
- Introduces a cutting-edge method for effective construction of symmetry breaking operators for branching rules in representation theory
- Includes hot topics of conformal geometry and global analysis as applications of representation theory
- Provides the complete classification of all conformally equivariant differential operators on forms on the model space (Sn, Sn-1)
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2170)
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Table of contents (14 chapters)
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Front Matter
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Back Matter
Keywords
- Symmetry breaking operators
- branching law
- F-method
- conformal geometry
- Verma module
- Lorentz group
- Lie group
- reductive group
- homogeneous space
- Riemannian geometry
- unitary representation
- Yamabe operator
- Paneitz operator
- conformal holography
- Fradkin-Tseytlin operator
- Gegenbauer polynomial
- hypergeometric function
- differential form
- Hodge operator
- hyperbolic space
About this book
This work is the first systematic study of all possible conformally covariant differential operators transforming differential forms on a Riemannian manifold X into those on a submanifold Y with focus on the model space (X, Y) = (Sn, Sn-1).
The authors give a complete classification of all such conformally covariant differential operators, and find their explicit formulæ in the flat coordinates in terms of basic operators in differential geometry and classical hypergeometric polynomials. Resulting families of operators are natural generalizations of the Rankin–Cohen brackets for modular forms and Juhl's operators from conformal holography. The matrix-valued factorization identities among all possible combinations of conformally covariant differential operators are also established.
The main machinery of the proof relies on the "F-method" recently introduced and developed by the authors. It is a general method to construct intertwining operators between C∞-induced representations or to find singular vectors of Verma modules in the context of branching rules, as solutions to differential equations on the Fourier transform side. The book gives a new extension of the F-method to the matrix-valued case in the general setting, which could be applied to other problems as well.
This book offers a self-contained 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 differential geometry, representation theory, and theoretical physics.
Authors and Affiliations
Bibliographic Information
Book Title: Conformal Symmetry Breaking Operators for Differential Forms on Spheres
Authors: Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-981-10-2657-7
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2016
Softcover ISBN: 978-981-10-2656-0Published: 12 October 2016
eBook ISBN: 978-981-10-2657-7Published: 11 October 2016
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: IX, 192
Topics: Differential Geometry, Topological Groups, Lie Groups, Mathematical Physics, Fourier Analysis, Partial Differential Equations