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

Abstract

Let $G=S{O}_0(1,n)$ 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 $\mathbf{\lambda }=({\lambda }_1,{\lambda }_2,{\lambda }_3)$ 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 $S{O}_0(1,n)$, we construct new invariant trilinear forms corresponding to singular values of $\mathbf{\lambda }$. The family of generic invariant trilinear forms depend meromorphically on the parameter $\mathbf{\lambda }$ and the new forms are shown to be residues of this family.

Cited by in corpus (3)

• Conformally Equivariant Quantization - a Complete Classification
• Singular conformally invariant trilinear forms and generalized Rankin Cohen operators
• Tensor products of complementary series of rank one Lie groups