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

TL;DR

This paper develops uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products of semisimple groups and vector spaces, improving existing bounds and aiding in calculating Kazhdan constants.

Contribution

It introduces sharper bounds for matrix coefficients of unitary representations on semidirect products, extending previous results and providing a new method for Kazhdan constant computation.

Findings

Bounds are sharper than those for $G$ alone in some cases.

Method for calculating Kazhdan constants for pairs involving semidirect products.

Applicable to representations without non-trivial $V$-fixed vectors.

Abstract

Let $k$ be a local field of characteristic 0, and let $G$ be a connected semisimple almost $k$-algebraic group. Suppose rank$_kG\geq 1$ 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 $G\ltimes_\rho V$ 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 $(G\ltimes_\rho V,V)$.