Conformally invariant trilinear forms on the sphere
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, p. 1807-1838

To each complex number λ is associated a representation π λ of the conformal group SO 0 (1,n) on 𝒞 (S n-1 ) (spherical principal series). For three values λ 1 ,λ 2 ,λ 3 , we construct a trilinear form on 𝒞 (S n-1 )×𝒞 (S n-1 )×𝒞 (S n-1 ), which is invariant by π λ 1 π λ 2 π λ 3 . The trilinear form, first defined for (λ 1 ,λ 2 ,λ 3 ) in an open set of 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.

À chaque nombre complexe λ est associée une représentation π λ du groupe conforme SO 0 (1,n) sur 𝒞 (S n-1 ) (série principale sphérique). Pour chaque triplet (λ 1 ,λ 2 ,λ 3 ), nous construisons une forme trilinéaire sur 𝒞 (S n-1 )×𝒞 (S n-1 )×𝒞 (S n-1 ) qui est invariante par π λ 1 π λ 2 π λ 3 . La forme trilinéaire, d’abord définie dans un ouvert de 3 est étendue méromorphiquement, avec des pôles simples en une famille explicite de plans de 3 . Pour les valeurs génériques des paramètres, nous démontrons l’unicité d’une telle forme trilinéaire invariante.

DOI : https://doi.org/10.5802/aif.2659
Classification:  22E45,  43A85
Keywords: Trilinear invariant forms, conformal group, meromorphic continuation
@article{AIF_2011__61_5_1807_0,
     author = {Clerc, Jean-Louis and \O rsted, Bent},
     title = {Conformally invariant trilinear forms on the sphere},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     pages = {1807-1838},
     doi = {10.5802/aif.2659},
     mrnumber = {2961841},
     zbl = {1252.22008},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_5_1807_0}
}
Clerc, Jean-Louis; Ørsted, Bent. Conformally invariant trilinear forms on the sphere. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1807-1838. doi : 10.5802/aif.2659. http://www.numdam.org/item/AIF_2011__61_5_1807_0/

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