A unified proof of the Howe-Moore property

TL;DR

This paper offers a unified proof demonstrating that various classes of locally compact groups exhibit the Howe-Moore property, ensuring matrix coefficients vanish at infinity for certain unitary representations.

Contribution

It provides a comprehensive, unified proof for all known examples of groups with the Howe-Moore property, consolidating previous separate proofs into one framework.

Findings

All examined groups have the Howe-Moore property.

The proof applies to real Lie groups, algebraic groups over non-Archimedean fields, and certain automorphism groups of trees.

The result confirms the vanishing of matrix coefficients at infinity for these groups.

Abstract

We provide a unified proof of all known examples of locally compact groups that enjoy the Howe-Moore property, namely, the vanishing at infinity of all matrix coefficients of the group unitary representations that are without non-zero invariant vectors. These examples are: connected, non-compact, simple real Lie groups with finite center, isotropic simple algebraic groups over non Archimedean local fields and closed, topologically simple subgroups of Aut(T) that act 2-transitively on the boundary at infinity of T, where T is a bi-regular tree with valence > 2 at every vertex.