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.

Classification : 43A80, 22E25
Keywords: Gelfand pair, Schwartz space, nilpotent Lie group
Mot clés : paire de Gelfand, classe de Schwartz, groupe de Lie nilpotent

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