Conformally invariant trilinear forms on the sphere
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 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: 10.5802/aif.2659
Classification: 22E45,  43A85
Keywords: Trilinear invariant forms, conformal group, meromorphic continuation
Clerc, Jean-Louis 1; Ørsted, Bent 2

1 Université Henri Poincaré (Nancy 1) Institut Élie Cartan 54506 Vandoeuvre-lès-Nancy (France)
2 Matematisk Institut Byg. 430, Ny Munkegade 8000 Aarhus C (Denmark)
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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://archive.numdam.org/articles/10.5802/aif.2659/

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