Conformally invariant trilinear forms on the sphere

Clerc, Jean-Louis; Ørsted, Bent

doi:10.5802/aif.2659

Conformally invariant trilinear forms on the sphere
[Formes trilinéaires conformément invariantes sur la sphère]

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)

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

Résumé
Abstract

À chaque nombre complexe $λ$ est associée une représentation $π_{λ}$ du groupe conforme $S O_{0} (1, n)$ sur $𝒞^{\infty} (S^{n - 1})$ (série principale sphérique). Pour chaque triplet $(λ_{1}, λ_{2}, λ_{3})$ , nous construisons une forme trilinéaire sur $𝒞^{\infty} (S^{n - 1}) \times 𝒞^{\infty} (S^{n - 1}) \times 𝒞^{\infty} (S^{n - 1})$ qui est invariante par $π_{λ_{1}} \otimes π_{λ_{2}} \otimes π_{λ_{3}}$ . 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 $λ$ is associated a representation $π_{λ}$ of the conformal group $S O_{0} (1, n)$ on $𝒞^{\infty} (S^{n - 1})$ (spherical principal series). For three values $λ_{1}, λ_{2}, λ_{3}$ , we construct a trilinear form on $𝒞^{\infty} (S^{n - 1}) \times 𝒞^{\infty} (S^{n - 1}) \times 𝒞^{\infty} (S^{n - 1})$ , which is invariant by $π_{λ_{1}} \otimes π_{λ_{2}} \otimes π_{λ_{3}}$ . The trilinear form, first defined for $(λ_{1}, λ_{2}, λ_{3})$ 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.

MR Zbl

DOI : 10.5802/aif.2659

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

Affiliations des auteurs :

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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     title = {Conformally invariant trilinear forms on the sphere},
     journal = {Annales de l'Institut Fourier},
     pages = {1807--1838},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Clerc, Jean-Louis; Ørsted, Bent. Conformally invariant trilinear forms on the sphere. Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1807-1838. doi : 10.5802/aif.2659. https://aif.centre-mersenne.org/articles/10.5802/aif.2659/

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