This work provides the first classification theory of matrix-valued
symmetry breaking operators from principal series representations of a
reductive group to those of its subgroup.The study of symmetry breaking
operators (intertwining operators for restriction) is an important and
very active research area in modern representation theory, which also
interacts with various fields in mathematics and theoretical physics
ranging from number theory to differential geometry and quantum
mechanics.The first author initiated a program of the general study of
symmetry breaking operators. The present book pursues the program by
introducing new ideas and techniques, giving a systematic and detailed
treatment in the case of orthogonal groups of real rank one, which will
serve as models for further research in other settings.In connection to
automorphic forms, this work includes a proof for a multiplicity
conjecture by Gross and Prasad for tempered principal series
representations in the case (SO(n + 1, 1), SO(n, 1)). The
authors propose a further multiplicity conjecture for nontempered
representations.Viewed from differential geometry, this seminal work
accomplishes the classification of all conformally covariant operators
transforming differential forms on a Riemanniann manifold X to those
on a submanifold in the model space (X, Y) = (Sn,
Sn-1). Functional equations and explicit formulæ
of these operators are also established.This book offers a
self-contained and inspiring 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 representation
theory, automorphic forms, differential geometry, and theoretical
physics.
