Conformal Symmetry Breaking Operators for Differential Forms on Spheres

Description

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. Front Matter....Pages i-ix Introduction....Pages 1-12 Symmetry Breaking Operators and Principal Series Representations of G = O(n+1, 1)....Pages 13-30 F-method for Matrix-Valued Differential Operators....Pages 31-39 Matrix-Valued F-method for O(n+1, 1)....Pages 41-49 Application of Finite-Dimensional Representation Theory....Pages 51-65 F-system for Symmetry Breaking Operators ( j = i–1, i case)....Pages 67-85 F-system for Symmetry Breaking Operators ( j = i–2, i+1 case)....Pages 87-91 Basic Operators in Differential Geometry and Conformal Covariance....Pages 93-109 Identities of Scalar-Valued Differential Operators \( \mathfrak{D}_\mathrm{l}^\mu \) ....Pages 111-119 Construction of Differential Symmetry Breaking Operators....Pages 121-129 Solutions to Problems A and B for (Sn, Sn–1)....Pages 131-139 Intertwining Operators....Pages 141-153 Matrix-Valued Factorization Identities....Pages 155-172 Appendix: Gegenbauer Polynomials....Pages 173-184 Back Matter....Pages 185-192