logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Jean-Louis Clerc; Bent Ørsted
Conformally invariant trilinear forms on the sphere
(Formes trilinéaires conformément invariantes sur la sphère)
Annales de l'institut Fourier, 61 no. 5 (2011), p. 1807-1838, doi: 10.5802/aif.2659
Article PDF | Reviews MR 2961841 | Zbl 1252.22008
Class. Math.: 22E45, 43A85
Keywords: Trilinear invariant forms, conformal group, meromorphic continuation

Résumé - Abstract

To each complex number $\lambda $ is associated a representation $\pi _\lambda $ of the conformal group $SO_0(1,n)$ on $\mathcal{C}^\infty (S^{n-1})$ (spherical principal series). For three values $\lambda _1,\lambda _2,\lambda _3$, we construct a trilinear form on $\mathcal{C}^\infty (S^{n-1})\times \mathcal{C}^\infty (S^{n-1})\times \mathcal{C}^\infty (S^{n-1})$, 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 $\mathbb{C}^3$ 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.

Bibliography

[1] Joseph Bernstein & Andre Reznikov, “Estimates of automorphic functions”, Mosc. Math. J. 4 (2004) no. 1, p. 19-37  MR 2074982 |  Zbl 1081.11037
[2] François Bruhat, “Sur les représentations induites des groupes de Lie”, Bull. Soc. Math. France 84 (1956), p. 97-205 Numdam |  MR 84713 |  Zbl 0074.10303
[3] Jean-Louis Clerc, T. Kobayashi, B. Ørsted & M. Pevzner, “Generalized Bernstein- Reznikov integrals”, to be published in Mathematische Annalen, DOI 10.1007/ s0028-010-0516-4
[4] Jean-Louis Clerc & Karl-Hermann Neeb, “Orbits of triples in the Shilov boundary of a bounded symmetric domain”, Transform. Groups 11 (2006) no. 3, p. 387-426 Article |  MR 2264460 |  Zbl 1112.32010
[5] Anton Deitmar, “Invariant triple products”, Int. J. Math. Math. Sci. (2006), Art. ID 48274 Article |  MR 2251763 |  Zbl 1140.22018
[6] I. M. Gelʼfand & G. E. Shilov, 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 166596 |  Zbl 0115.33101
[7] Lars Hörmander, 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 717035 |  Zbl 0521.35001
[8] Johan A. C. Kolk & V. S. Varadarajan, “On the transverse symbol of vectorial distributions and some applications to harmonic analysis”, Indag. Math. (N.S.) 7 (1996) no. 1, p. 67-96 Article |  MR 1621372 |  Zbl 0892.22010
[9] Peter Littelmann, “On spherical double cones”, J. Algebra 166 (1994) no. 1, p. 142-157 Article |  MR 1276821 |  Zbl 0823.20040
[10] Hung Yean Loke, “Trilinear forms of $\mathfrak{gl}_2$”, Pacific J. Math. 197 (2001) no. 1, p. 119-144 Article |  MR 1810211 |  Zbl 1049.22007
[11] Peter Magyar, Jerzy Weyman & Andrei Zelevinsky, “Multiple flag varieties of finite type”, Adv. Math. 141 (1999) no. 1, p. 97-118 Article |  MR 1667147 |  Zbl 0951.14034
[12] V. F. Molčanov, “Tensor products of unitary representations of the three-dimensional Lorentz group”, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) no. 4, p. 860-891, 967  MR 548507 |  Zbl 0448.22010
[13] A. I. Oksak, “Trilinear Lorentz invariant forms”, Comm. Math. Phys. 29 (1973), p. 189-217 Article |  MR 340478
[14] C. Sabbah, Polynômes de Bernstein-Sato à plusieurs variables, Séminaire sur les équations aux dérivées partielles 1986–1987, École Polytech., 1987 Cedram |  MR 920037 |  Zbl 0634.32003
[15] Reiji Takahashi, “Sur les représentations unitaires des groupes de Lorentz généralisés”, Bull. Soc. Math. France 91 (1963), p. 289-433 Numdam |  MR 179296 |  Zbl 0196.15501
[16] E. P. van den Ban, “The principal series for a reductive symmetric space. I. $H$-fixed distribution vectors”, Ann. Sci. École Norm. Sup. (4) 21 (1988) no. 3, p. 359-412 Numdam |  MR 974410 |  Zbl 0714.22009
[17] Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Marcel Dekker Inc., New York, 1973, Pure and Applied Mathematics, No. 19  MR 498996 |  Zbl 0265.22022
[18] Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York, 1972, Die Grundlehren der mathematischen Wissenschaften, Band 188  MR 498999 |  Zbl 0265.22020
top