Singular conformally invariant trilinear forms and covariant differential operators on the sphere

TL;DR

This paper constructs new conformally invariant trilinear forms on the sphere by using covariant differential operators, extending the family of known forms to singular parameter values and analyzing their residues.

Contribution

It introduces a method to generate new invariant trilinear forms at singular parameters via covariant differential operators, expanding the understanding of conformal invariance on the sphere.

Findings

New invariant trilinear forms at singular parameters

Residue formulas for these forms

Meromorphic dependence on parameters

Abstract

Let $G=SO_0(1,n)$ be the conformal group acting on the $(n-1)$ dimensional sphere $S$, and let $(\pi_\lambda)_{\lambda\in \mathbb C}$ be the spherical principal series. For generic values of $\boldsymbol \lambda =(\lambda_1,\lambda_2,\lambda_3)$ in $\mathbb C^3$, there exits a (essentially unique) trilinear form on $\mathcal C^\infty(S)\times \mathcal C^\infty(S)\times \mathcal C^\infty(S)$ which is invariant under $\pi_{\lambda_1}\otimes \pi_{\lambda_2}\otimes \pi_{\lambda_3}$. Using differential operators on the sphere $S$ which are covariant under the conformal group $SO_0(1,n)$, we construct new invariant trilinear forms corresponding to singular values of $\boldsymbol \lambda$. The family of generic invariant trilinear forms depend meromorphically on the parameter $\boldsymbol \lambda$ and the new forms are shown to be residues of this family.