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

Let k be a local field of characteristic 0, and let G be a connected semisimple almost k  -algebraic group. Suppose rankkG⩾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⋉ρVG⋉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⋉ρV,V)(G⋉ρV,V).