Singular conformally invariant trilinear forms, II The higher multiplicity cases

Abstract

Let $S$ be the sphere of dimension $n-1, n\geq 4$. Let $(\pi_{\lambda})_{\lambda\in \mathbb C}$ be the scalar principle series of representations of the conformal group $SO_0(1,n)$, realized on $\mathcal C^\infty(S)$. For $\boldsymbol \lambda = (\lambda_1,\lambda_2,\lambda_3) \in \mathbb C^3$, let $Tri(\boldsymbol \lambda)$ be the space of continuous trilinear forms on $\mathcal C^\infty(S) \times \mathcal C^\infty(S) \times \mathcal C^\infty(S)$ which are invariant under $\pi_{\lambda_1} \otimes \pi_{\lambda_2} \otimes \pi_{\lambda_3} $. For each value of $\boldsymbol \lambda$, the dimension of $Tri(\boldsymbol \lambda)$ is computed and a basis of $Tri(\boldsymbol \lambda)$ is described.