Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants

Let k be a local field of characteristic not 2, and let G be the group of k-rational points of a connected reductive linear algebraic group defined over k with a simple derived group of k-rank at least 2. We construct new uniform pointwise bounds for the matrix coefficients of all infinite-dimensional irreducible unitary representations of G. These bounds turn out to be optimal for SL n (k), n 3, and Sp 2n (k), n 2. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of semisimple G.