To each complex number is associated a representation of the conformal group on (spherical principal series). For three values , we construct a trilinear form on , which is invariant by . The trilinear form, first defined for in an open set of 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 est associée une représentation du groupe conforme sur (série principale sphérique). Pour chaque triplet , nous construisons une forme trilinéaire sur qui est invariante par . La forme trilinéaire, d’abord définie dans un ouvert de est étendue méromorphiquement, avec des pôles simples en une famille explicite de plans de . Pour les valeurs génériques des paramètres, nous démontrons l’unicité d’une telle forme trilinéaire invariante.
Keywords: Trilinear invariant forms, conformal group, meromorphic continuation
Mot clés : formes trilinéaires invariantes, groupe conforme, prolongement méromorphe
Clerc, Jean-Louis 1; Ørsted, Bent 2
@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}, pages = {1807--1838}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {5}, year = {2011}, doi = {10.5802/aif.2659}, mrnumber = {2961841}, zbl = {1252.22008}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2659/} }
TY - JOUR AU - Clerc, Jean-Louis AU - Ørsted, Bent TI - Conformally invariant trilinear forms on the sphere JO - Annales de l'Institut Fourier PY - 2011 SP - 1807 EP - 1838 VL - 61 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2659/ DO - 10.5802/aif.2659 LA - en ID - AIF_2011__61_5_1807_0 ER -
%0 Journal Article %A Clerc, Jean-Louis %A Ørsted, Bent %T Conformally invariant trilinear forms on the sphere %J Annales de l'Institut Fourier %D 2011 %P 1807-1838 %V 61 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2659/ %R 10.5802/aif.2659 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2659/
[1] Estimates of automorphic functions, Mosc. Math. J., Volume 4 (2004) no. 1, pp. 19-37 | MR | Zbl
[2] Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France, Volume 84 (1956), pp. 97-205 | Numdam | MR | Zbl
[3] Generalized Bernstein- Reznikov integrals (to be published in Mathematische Annalen, DOI 10.1007/ s0028-010-0516-4)
[4] Orbits of triples in the Shilov boundary of a bounded symmetric domain, Transform. Groups, Volume 11 (2006) no. 3, pp. 387-426 | DOI | MR | Zbl
[5] Invariant triple products, Int. J. Math. Math. Sci. (2006), pp. 22 (Art. ID 48274) | DOI | MR | Zbl
[6] Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977] (Properties and operations, Translated from the Russian by Eugene Saletan) | MR | Zbl
[7] The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256, Springer-Verlag, Berlin, 1983 (Distribution theory and Fourier analysis) | MR | Zbl
[8] On the transverse symbol of vectorial distributions and some applications to harmonic analysis, Indag. Math. (N.S.), Volume 7 (1996) no. 1, pp. 67-96 | DOI | MR | Zbl
[9] On spherical double cones, J. Algebra, Volume 166 (1994) no. 1, pp. 142-157 | DOI | MR | Zbl
[10] Trilinear forms of , Pacific J. Math., Volume 197 (2001) no. 1, pp. 119-144 | DOI | MR | Zbl
[11] Multiple flag varieties of finite type, Adv. Math., Volume 141 (1999) no. 1, pp. 97-118 | DOI | MR | Zbl
[12] Tensor products of unitary representations of the three-dimensional Lorentz group, Izv. Akad. Nauk SSSR Ser. Mat., Volume 43 (1979) no. 4, p. 860-891, 967 | MR | Zbl
[13] Trilinear Lorentz invariant forms, Comm. Math. Phys., Volume 29 (1973), pp. 189-217 | DOI | MR
[14] Polynômes de Bernstein-Sato à plusieurs variables, Séminaire sur les équations aux dérivées partielles 1986–1987, École Polytech., Palaiseau, 1987 (Exp. No. XIX, 6) | Numdam | MR | Zbl
[15] Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France, Volume 91 (1963), pp. 289-433 | Numdam | MR | Zbl
[16] The principal series for a reductive symmetric space. I. -fixed distribution vectors, Ann. Sci. École Norm. Sup. (4), Volume 21 (1988) no. 3, pp. 359-412 | Numdam | MR | Zbl
[17] Harmonic analysis on homogeneous spaces, Marcel Dekker Inc., New York, 1973 (Pure and Applied Mathematics, No. 19) | MR | Zbl
[18] Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York, 1972 (Die Grundlehren der mathematischen Wissenschaften, Band 188) | MR | Zbl
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