Symmetry breaking operators for strongly spherical reductive pairs

Abstract

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 ${\rm dim\,Hom}_H(π|_H,τ)<\infty$ for all smooth admissible representations $π$ of $G$ and $τ$ of $H$. In this paper we explicitly construct for all strongly spherical pairs $(G,H)$ intertwining operators in ${\rm Hom}_H(π|_H,τ)$ for $π$ and $τ$ 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 ${\rm Hom}_H(π|_H,τ)$. 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.