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.
