  A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous
space $(G\times H)/{\rm diag}(H)$ is real spherical. This geometric condition
is equivalent to the representation theoretic property that ${\rm
dim\,Hom}_H(\pi|_H,\tau)<\infty$ for all smooth admissible representations
$\pi$ of $G$ and $\tau$ of $H$. In this paper we explicitly construct for all
strongly spherical pairs $(G,H)$ intertwining operators in ${\rm
Hom}_H(\pi|_H,\tau)$ for $\pi$ and $\tau$ spherical principal series
representations of $G$ and $H$. These so-called symmetry breaking operators
depend holomorphically on the induction parameters and we further show that
they generically span the space ${\rm Hom}_H(\pi|_H,\tau)$. In the special case
of multiplicity one pairs we extend our construction to vector-valued principal
series representations and obtain generic formulas for the multiplicities
between arbitrary principal series.
  As an application, we prove an early version of the Gross-Prasad conjecture
for complex orthogonal groups, and also provide lower bounds for the dimension
of the space of Shintani functions.

Representation Theory;Number Theory