Conformally invariant trilinear forms on the sphere

TL;DR

This paper constructs and analyzes conformally invariant trilinear forms on the sphere, extending their definition meromorphically and establishing their uniqueness for generic parameters, contributing to the understanding of conformal symmetry representations.

Contribution

It introduces a new family of conformally invariant trilinear forms on the sphere and proves their meromorphic extension and uniqueness for generic parameters.

Findings

Constructed conformally invariant trilinear forms on the sphere.

Extended the forms meromorphically with explicit pole locations.

Proved uniqueness of the forms for generic parameter values.

Abstract

To each complex number $\lambda$ is associated a representation $\pi_\lambda$ of the conformal group $SO_0(1,n)$ on $\mathcal C^\infty(S^{n-1})$ (spherical principal series). For three values $\lambda_1,\lambda_2,\lambda_3$, we construct a trilinear form on $\mathcal C^\infty(S^{n-1})\times\mathcal C^\infty(S^{n-1})\times \mathcal C^\infty(S^{n-1})$, which is invariant by $\pi_{\lambda_1}\otimes \pi_{\lambda_2}\otimes \pi_{\lambda_3}$. The trilinear form, first defined for $(\lambda_1, \lambda_2,\lambda_3)$ in an open set of $\mathbb C^3$ is extended meromorphically, with simple poles located in an explicit family of hyperplanes. For generic values of the parameters, we prove uniqueness of trilinear invariant forms.