# Symmetry breaking operators for real reductive groups of rank one

## TL;DR

This paper classifies and constructs symmetry breaking operators between spherical principal series representations of rank one real reductive groups, generalizing previous work to a broader class of groups and explicitly describing the distribution kernels.

## Contribution

It provides an explicit classification and construction of symmetry breaking operators for a wide class of rank one real reductive groups, extending prior results to new group pairs.

## Key findings

- Explicit classification of symmetry breaking operators.
- Construction of operators via meromorphic distribution families.
- Identification of sporadic operators beyond the main family.

## Abstract

For a pair of real reductive groups $G'\subset G$ we consider the space ${\rm Hom}_{G'}(\pi|_{G'},\tau)$ of intertwining operators between spherical principal series representations $\pi$ of $G$ and $\tau$ of $G'$, also called \emph{symmetry breaking operators}. Restricting to those pairs $(G,G')$ where ${\rm dim\,Hom}_{G'}(\pi|_{G'},\tau)<\infty$ and $G$ and $G'$ are of real rank one, we classify all symmetry breaking operators explicitly in terms of their distribution kernels. This generalizes previous work by Kobayashi--Speh for $(G,G')=({\rm O}(1,n+1),{\rm O}(1,n))$ to the reductive pairs $$ (G,G') = ({\rm U}(1,n+1;\mathbb{F}),{\rm U}(1,m+1;\mathbb{F})\times F) \qquad \mbox{with $\mathbb{F}=\mathbb{C},\mathbb{H},\mathbb{O}$ and $F<{\rm U}(n-m;\mathbb{F})$.} $$ In most cases, all symmetry breaking operators can be constructed using one meromorphic family of distributions whose poles and residues we describe in detail. In addition to this family, there may occur some sporadic symmetry breaking operators which we determine explicitly.