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Conformal Symmetry Breaking Operators for Differential Forms on Spheres (Paperback, 1st ed. 2016)
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Conformal Symmetry Breaking Operators for Differential Forms on Spheres (Paperback, 1st ed. 2016)
Series: Lecture Notes in Mathematics, 2170
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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 formulae 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.
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