  For a pair of real reductive groups $G'\subset G$ we consider the space ${\rm
Hom}_{G'}(\pi|_{G'},\tau)$ of intertwining operators between spherical
principal series representations $\pi$ of $G$ and $\tau$ of $G'$, also called
\emph{symmetry breaking operators}. Restricting to those pairs $(G,G')$ where
${\rm dim\,Hom}_{G'}(\pi|_{G'},\tau)<\infty$ and $G$ and $G'$ are of real rank
one, we classify all symmetry breaking operators explicitly in terms of their
distribution kernels. This generalizes previous work by Kobayashi--Speh for
$(G,G')=({\rm O}(1,n+1),{\rm O}(1,n))$ to the reductive pairs $$ (G,G') = ({\rm
U}(1,n+1;\mathbb{F}),{\rm U}(1,m+1;\mathbb{F})\times F) \qquad \mbox{with
$\mathbb{F}=\mathbb{C},\mathbb{H},\mathbb{O}$ and $F<{\rm U}(n-m;\mathbb{F})$.}
$$ In most cases, all symmetry breaking operators can be constructed using one
meromorphic family of distributions whose poles and residues we describe in
detail. In addition to this family, there may occur some sporadic symmetry
breaking operators which we determine explicitly.

Representation Theory