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

Abstract

Let $k$ be a local field of characteristic 0, and let $G$ be a connected semisimple almost $k$-algebraic group. Suppose rank${}_{k}G\ge 1$ and $\rho$ 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 $G{⋉}_{\rho }V$ 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 $(G{⋉}_{\rho }V,V)$.

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