# On a generalization of the Howe-Moore property

## TL;DR

This paper introduces a generalized Howe-Moore property relative to a set of subgroups and proves that semisimple groups possess this property concerning their factors, extending classical results on matrix coefficient decay.

## Contribution

It generalizes the Howe-Moore property to a relative setting involving subgroups and establishes this for semisimple groups and their factors.

## Key findings

- Semisimple groups have the Howe-Moore property relative to their factors.
- The property ensures matrix coefficients vanish at infinity under certain restrictions.
- The generalization broadens understanding of decay properties in representation theory.

## Abstract

We define a Howe-Moore property relative to a set of subgroups. Namely, a group $G$ has the Howe-Moore property relative to a set $\mathcal{F}$ of subgroups if for every unitary representation $\pi$ of $G$, whenever the restriction of $\pi$ to any element of $\mathcal{F}$ has no non-trivial invariant vectors, the matrix coefficients vanish at infinity. We prove that a semisimple group has the Howe-Moore property relatively to the family of its factors.