We investigate properties of some spherical functions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson–Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (also called Haagerup’s inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. We also prove that the family of boundary representations studied in this paper, which can be regarded as a one parameter deformation of the boundary unitary representation, are slow growth representations acting on a Hilbert space admitting a proper 1-cocycle.

中文翻译：

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双曲群上的一些球函数

我们使用具有 Patterson-Sullivan 度量类的 Gromov 边界上的边界表示来研究在双曲群上定义的一些球面函数的性质。我们证明了球面函数的急剧衰减估计以及与边界表示相关的谱不等式。这种关于边界的观点让我们可以看到所谓的房地产研发（也叫Haagerup 不等式) 作为双曲群上球函数更一般行为的一个特例。我们还证明了本文研究的边界表示族，可以看作是边界酉表示的单参数变形，是作用在希尔伯特空间上的慢增长表示，允许一个适当的1-cocycle。