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 SLn(k), n ≥ 3, and Sp2n(k), n ≥ 2. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of semisimple G.