Conformally invariant trilinear forms on the sphere

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.