Radial rapid decay does not imply rapid decay. (La décroissance rapide radiale n’implique pas la décroissance rapide.) (English. French summary) Zbl 1527.42003
Summary: We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group \(\Gamma:=\mathrm{SL}_2(A)\), where \(A:=\mathbb{F}_q[X,X^{-1}]\) is the ring of Laurent polynomials with coefficients in \(\mathbb{F}_q\), endowed with the length function coming from a natural action of \(\Gamma\) 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 I. Chatterji [Contemp. Math. 691, 53–72 (2017; Zbl 1404.20032)] and moreover show that, unlike the RD property, the RRD property isn’t inherited by open subgroups.
MSC:
| 42A85 | Convolution, factorization for one variable harmonic analysis |
| 22F10 | Measurable group actions |
| 37A99 | Ergodic theory |
Keywords:
RD property; Koopman representation; semi-simple group; lattice; Harish-Chandra function; convolution operator norm; length functionCitations:
Zbl 1404.20032References:
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