Abstract
Let Γ be a free nonabelian group and let Ω be its boundary. Let π h be one of the unitary representations of Γ introduced in More (Duke Math J 82:381–436, 1996). By its definition π h acts on L 2(Ω, d ν h ) for a certain measure ν h and satisfies certain genericity conditions. Those conditions guarantee that π h is not equivalent to a principal isotropic/anisotropic series representation of Figà-Talamanca–Picardello and Figà-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, π h belongs to the isotropic/anisotropic series.
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Kuhn, M.G., Steger, T. Size of matrix coefficients characterizes anisotropic principal series representations of the free group. Monatsh Math 160, 151–166 (2010). https://doi.org/10.1007/s00605-009-0105-3
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DOI: https://doi.org/10.1007/s00605-009-0105-3