# Symmetry breaking operators for strongly spherical reductive pairs

## TL;DR

This paper constructs explicit symmetry breaking operators for strongly spherical reductive pairs, providing new tools for understanding representation restrictions and applications to conjectures like Gross-Prasad.

## Contribution

It explicitly constructs intertwining operators for all strongly spherical pairs, extending to vector-valued cases and deriving formulas for multiplicities in principal series.

## Key findings

- Constructed symmetry breaking operators depending holomorphically on parameters.
- Showed these operators generically span the Hom space for spherical principal series.
- Applied results to prove cases of the Gross-Prasad conjecture and bounds for Shintani functions.

## 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 ${\rm dim\,Hom}_H(\pi|_H,\tau)<\infty$ 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.