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

摘要

Let k be a local field of characteristic 0, and let G be a connected semisimple almost k-algebraic group. Suppose rank_k G ≥ 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×_ρ 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 ×_ρ V, V).