Conformally invariant trilinear forms on the sphere

Clerc, Jean-Louis; Ørsted, Bent

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1807-1838.

À chaque nombre complexe $\lambda$ est associée une représentation ${\pi }_{\lambda }$ du groupe conforme $S{O}_0(1,n)$ sur ${𝒞}^{\infty }\left(S^{n-1}\right)$ (série principale sphérique). Pour chaque triplet $({\lambda }_1,{\lambda }_2,{\lambda }_3)$, nous construisons une forme trilinéaire sur ${𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)$ qui est invariante par ${\pi }_{{\lambda }_1}\otimes {\pi }_{{\lambda }_2}\otimes {\pi }_{{\lambda }_3}$. La forme trilinéaire, d’abord définie dans un ouvert de ${ℂ}^3$ est étendue méromorphiquement, avec des pôles simples en une famille explicite de plans de ${ℂ}^3$. Pour les valeurs génériques des paramètres, nous démontrons l’unicité d’une telle forme trilinéaire invariante.

To each complex number $\lambda$ is associated a representation ${\pi }_{\lambda }$ of the conformal group $S{O}_0(1,n)$ on ${𝒞}^{\infty }\left(S^{n-1}\right)$ (spherical principal series). For three values ${\lambda }_1,{\lambda }_2,{\lambda }_3$, we construct a trilinear form on ${𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)$, 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 ${ℂ}^3$ 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.

Classification : 22E45, 43A85
Keywords: Trilinear invariant forms, conformal group, meromorphic continuation
Mot clés : formes trilinéaires invariantes, groupe conforme, prolongement méromorphe

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