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

Abstract

Let $G=SO_0(1,n)$ be the conformal group acting on the $(n-1)$ dimensional sphere $S$, and let $(π_λ)_{λ\in \mathbb C}$ be the spherical principal series. For generic values of $\boldsymbol λ=(λ_1,λ_2,λ_3)$ 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 $π_{λ_1}\otimes π_{λ_2}\otimes π_{λ_3}$. 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 $\boldsymbol λ$. The family of generic invariant trilinear forms depend meromorphically on the parameter $\boldsymbol λ$ and the new forms are shown to be residues of this family.