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Conformal Symmetry Breaking Operators for Differential Forms on Spheres

  • Toshiyuki Kobayashi
  • Toshihisa Kubo
  • Michael Pevzner

Part of the Lecture Notes in Mathematics book series (LNM, volume 2170)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 1-12
  3. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 13-30
  4. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 31-39
  5. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 41-49
  6. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 51-65
  7. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 67-85
  8. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 87-91
  9. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 93-109
  10. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 111-119
  11. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 121-129
  12. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 131-139
  13. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 141-153
  14. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 155-172
  15. Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
    Pages 173-184
  16. Back Matter
    Pages 185-192

About this book

Introduction

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 (XY) = (SnSn-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.

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

Authors and affiliations

  • Toshiyuki Kobayashi
    • 1
  • Toshihisa Kubo
    • 2
  • Michael Pevzner
    • 3
  1. 1.Kavli IPMU and Graduate School of Mathematical SciencesThe University of Tokyo, 3-8-1 KomabaMeguroJapan
  2. 2.Ryukoku UniversityKyotoJapan
  3. 3.Mathematics Laboratory, FR 3399 CNRSUniversity of Reims-Champagne-ArdenneReimsFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-10-2657-7
  • Copyright Information Springer Nature Singapore Pte Ltd. 2016
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-10-2656-0
  • Online ISBN 978-981-10-2657-7
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site