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.
