Symmetry breaking operators for strongly spherical reductive pairs

A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(G\times H)\mathrm{/}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(H)$ is real spherical. This geometric condition is equivalent to the representation theoretic property that ${\rm dim,Hom}_H(\pi|_H,\tau)<\infty$forallsmoothadmissiblerepresentations$\pi$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 ${\mathrm{H}\mathrm{o}\mathrm{m}}_H(\pi {\mathrm{\mid }}_H,\tau )$. 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.