Singular conformally invariant trilinear forms, I Multiplicity one results

TL;DR

This paper investigates a family of conformally invariant trilinear forms on the sphere, characterizing their zero set and establishing that their multiplicity is generally one outside a specific zero set, with some exceptions.

Contribution

It provides a detailed description of the zero set and proves the multiplicity one property for conformally invariant trilinear forms, extending understanding of their structure.

Findings

Zero set of the family is explicitly described.

Multiplicity of invariant forms is one outside the zero set.

Exceptions occur for a denumerable subset.

Abstract

A normalized holomorphic family (depending on $\boldsymbol \lambda \in \mathbb C^3$) of conformally invariant trilinear forms on the sphere is studied. Its zero set $Z$ is described. For $\boldsymbol \lambda\notin Z$, the multiplicity of the space of conformally invariant trilinear forms is shown to be 1, except perhaps for a denumerable subset.