Conformally invariant trilinear forms on the sphere

Abstract

To each complex number $λ$ is associated a representation $π_λ$ of the conformal group $SO_0(1,n)$ on $\mathcal C^\infty(S^{n-1})$ (spherical principal series). For three values $λ_1,λ_2,λ_3$, 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 $π_{λ_1}\otimes π_{λ_2}\otimes π_{λ_3}$. The trilinear form, first defined for $(λ_1, λ_2,λ_3)$ 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.