Size of matrix coefficients characterizes anisotropic principal series representations of the free group

Abstract

Let Γ be a free nonabelian group and let Ω be its boundary. Let πh be one of the unitary representations of Γ introduced in More (Duke Math J 82:381–436, 1996). By its definition πh acts on L2(Ω, dνh) for a certain measure νh and satisfies certain genericity conditions. Those conditions guarantee that πh is not equivalent to a principal isotropic/anisotropic series representation of Figa-Talamanca–Picardello and Figa-Talamanca–Steger (J Funct Anal 47:281–304, 1982/Mem Am Math Soc 531:1–68). In this paper we show the converse: if the genericity conditions are not satisfied then, up to a twist by a unitary character, πh belongs to the isotropic/anisotropic series.