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

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

2011-02-09

arxiv.org/abs/1102.1861v1

Mathematics

Representation Theory

Scientific paper

Let $G=SO_0(1,n)$ 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 $\boldsymbol \lambda =(\lambda_1,\lambda_2,\lambda_3)$ 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 $SO_0(1,n)$, we construct new invariant trilinear forms corresponding to singular values of $\boldsymbol \lambda$. The family of generic invariant trilinear forms depend meromorphically on the parameter $\boldsymbol \lambda$ and the new forms are shown to be residues of this family.

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