Uniform pointwise bounds for Matrix coefficients of unitary representations on semidirect products

2012/06/22 by Zhenqi Jenny Wang, Wang, Zhenqi Jenny
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1206.5045

Keywords: Matrix coefficient, unitary representation, Fourier transform, projection-valued measure, Mackey machine, Kazhdan constant

arxiv created 2012/06/22 · arxiv updated 2012/06/25

Abstract

Let k be a local field of characteristic 0, and let G be a connected semisimple almost k-algebraic group. Suppose rank_kG≥ 1 and ρ is an excellent representation of G on a finite dimensional k-vector space V. We construct uniform pointwise bounds for the K-finite matrix coefficients restricted on G of all unitary representations of the semi-direct product G\ltimes_ρV without non-trivial V-fixed vectors. These bounds turn out to be sharper than the bounds obtained from G itself for some cases. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of the pair (G\ltimes_ρV,V).

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