Symmetry Breaking Operators for Strongly Spherical Reductive Pairs

Abstract

A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(G\times H)/\mathrm{diag}(H)$ is real spherical. This geometric condition is equivalent to the representation-theoretic property that $\dim {\mathrm{Hom}}_H(\pi {|}_H,\tau )<\infty$ 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 ${\mathrm{Hom}}_H(\pi {|}_H,\tau )$ 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 ${\mathrm{Hom}}_H(\pi {|}_H,\tau )$. 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.