Gelfand transforms of $SO\left(3\right)$-invariant Schwartz functions on the free group $N_{3,2}$

Fischer, Véronique; Ricci, Fulvio

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2143-2168.

Il est toujours possible d’injecter dans un espace euclidien le spectre d’une paire de Gelfand du type $(K⋉N,K)$, où $N$ est un groupe de Lie nilpotent. Nous démontrons que de manière générale, les fonctions de la classe de Schwartz sur le spectre sont les transformées des fonctions de la classe de Schwartz sur $N$ qui sont invariantes par $K$. Nous prouvons également l’inclusion inverse dans le cas où $N=N_{3,2}$ est le groupe de Lie nilpotent libre à trois générateurs et $K=SO\left(3\right)$. Ceci étend des résultats récents sur le groupe de Heisenberg.

The spectrum of a Gelfand pair $(K⋉N,K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz $K$-invariant functions on $N$. We also show the converse in the case of the Gelfand pair $(SO\left(3\right)⋉N_{3,2},SO\left(3\right))$, where $N_{3,2}$ is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.

[1] Astengo, F.; Di Blasio, B.; Ricci, F. Gelfand transforms of polyradial Schwartz functions on the Heisenberg group, J. Funct. Anal., Volume 251 (2007) no. 2, pp. 772-791 | DOI | MR | Zbl

[2] Astengo, F.; Di Blasio, B.; Ricci, F. Gelfand pairs on the Heisenberg group and Schwartz functions (2008) (to appear, http://arxiv.org/abs/0805.3809v1) | MR | Zbl

[3] Benson, C.; Jenkins, J.; Ratcliff, G. On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc., Volume 321 (1990) no. 1, pp. 85-116 | DOI | MR | Zbl

[4] Benson, C.; Jenkins, J.; Ratcliff, G. The spherical transform of a Schwartz function on the Heisenberg group, J. Funct. Anal., Volume 154 (1998) no. 2, pp. 379-423 | DOI | MR | Zbl

[5] Ferrari R., F. The topology of the spectrum for Gelfand pairs on Lie groups, Bull. Un. Mat. It., Volume 10 (2007), pp. 569-579 (http://arxiv.org/abs/0706.0708v1) | MR

[6] Folland, G. B. Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., Volume 13 (1975) no. 2, pp. 161-207 | DOI | MR | Zbl

[7] Folland, G. B.; Stein, E. M. Hardy spaces on homogeneous groups, Mathematical Notes, 28, Princeton University Press, Princeton, N.J., 1982 | MR | Zbl

[8] Geller, D. Fourier analysis on the Heisenberg group. I. Schwartz space, J. Funct. Anal., Volume 36 (1980) no. 2, pp. 205-254 | DOI | MR | Zbl

[9] Geller, D. Liouville’s theorem for homogeneous groups, Comm. Partial Differential Equations, Volume 8 (1983) no. 15, pp. 1665-1677 | MR | Zbl

[10] Goodman, R.; Wallach, N. R. Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, 68, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[11] Helffer, B.; Nourrigat, J. Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations, Volume 4 (1979) no. 8, pp. 899-958 | DOI | MR | Zbl

[12] Helgason, S. Groups and geometric analysis, Pure and Applied Mathematics, 113, Academic Press Inc., Orlando, FL, 1984 | MR | Zbl

[13] Helgason, S. The Radon transform, Progress in Mathematics, 5, Birkhäuser Boston Inc., Boston, MA, 1999 | MR | Zbl

[14] Hulanicki, A. A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math., Volume 78 (1984) no. 3, pp. 253-266 | MR | Zbl

[15] Mather, J. N. Differentiable invariants, Topology, Volume 16 (1977) no. 2, pp. 145-155 | DOI | MR | Zbl

[16] Schwarz, G. W. Smooth functions invariant under the action of a compact Lie group, Topology, Volume 14 (1975), pp. 63-68 | DOI | MR | Zbl

[17] Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993 | MR | Zbl