Conformal Symmetry Breaking Operators for Differential Forms on Spheres

Authors: Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner



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)

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.

Table of contents (14 chapters)

Front Matter

Introduction

Symmetry Breaking Operators and Principal Series Representations of G = O(n+1, 1)

F-method for Matrix-Valued Differential Operators

Matrix-Valued F-method for O(n+1, 1)

Application of Finite-Dimensional Representation Theory

F-system for Symmetry Breaking Operators ( j = i–1, i case)

F-system for Symmetry Breaking Operators ( j = i–2, i+1 case)

Basic Operators in Differential Geometry and Conformal Covariance

Identities of Scalar-Valued Differential Operators DμlDlμ

Construction of Differential Symmetry Breaking Operators

Solutions to Problems A and B for (Sn, Sn–1)

Intertwining Operators

Matrix-Valued Factorization Identities

Appendix: Gegenbauer Polynomials

Back Matter



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