Radial rapid decay does not imply rapid decay
Annales de l'Institut Fourier, Online first, 32 p.

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group Γ:=SL 2 (A), where A:=𝔽 q [X,X -1 ] is the ring of Laurent polynomials with coefficients in 𝔽 q , endowed with the length function coming from a natural action of Γ on a product of two trees, and show that it has the radial rapid decay (RRD) property and doesn’t have the rapid decay (RD) property. We show that the criterion also applies to all irreducible lattices (uniform or not) in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. When the rank is greater or equal to two and the lattice is non-uniform, the lattice has RRD but not RD. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn’t inherited by open subgroups.

Nous établissons un nouveau critère dynamique entraînant la propriété de décroissance rapide radiale. Nous explicitons le cas particulier du groupe Γ:=SL 2 (A), où A:=𝔽 q [X,X -1 ] est l’anneau des polynômes de Laurent à coefficients dans le corps fini 𝔽 q , muni d’une fonction longueur provenant d’une action naturelle de Γ sur le produit de deux arbres. Nous prouvons que pour cette fonction longueur, ce groupe vérifie la propriété de décroissance rapide radiale (RRD), mais ne vérifie pas la propriété de décroissance rapide (RD). Nous prouvons aussi que notre critère s’applique à tout réseau irréductible (uniforme ou non), de tout groupe de Lie semi-simple à centre fini, muni d’une certaine fonction longueur définie à l’aide d’une métrique de Finsler. Lorsque le rang réel est supérieur ou égal à deux et que le réseau n’est pas uniforme, le réseau vérifie la propriété RRD, mais pas la propriété RD. Ces exemples répondent à une question de Chatterji et montrent que, contrairement à la propriété RD, la propriété RRD n’est pas héréditaire par passage à un sous-groupe ouvert.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3552
Classification: 42A85,  22F10
Keywords: RD property, Koopman representation, semi-simple group, lattice, Harish-Chandra function, convolution operator norm, length function.
Boyer, Adrien 1; Pinochet Lobos, Antoine 2; Pittet, Christophe 2, 3

1 Université Paris 7 (France)
2 I2M, CNRS UMR7373, Université d’Aix-Marseille (France)
3 Section de Mathématiques, Faculté des Sciences, Université de Genève (Switzerland)
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Boyer, Adrien; Pinochet Lobos, Antoine; Pittet, Christophe. Radial rapid decay does not imply rapid decay. Annales de l'Institut Fourier, Online first, 32 p.

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