Symmetry Breaking Operators for Strongly Spherical Reductive Pairs

A real reductive pair (G, H) is called strongly spherical if the homogeneous space (G x H)/ diag(H) is real spherical. This geometric condition is equivalent to the representation-theoretic property that dimHomH(pi|H, iota) < infinity for all smooth admissible representations pi of G and iota of H. In this paper we explicitly construct for all strongly spherical pairs (G, H) intertwining operators in HomH(pi|(H), iota) for pi and iota 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 Hom(H)(pi|H, iota). 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.