The Howe-Moore property for real and p-adic groups

TL;DR

This paper explores a relative version of the Howe-Moore Property for real and p-adic groups, linking it to ergodic actions and structural group properties, with implications for vanishing integrals.

Contribution

It introduces a characterization of the relative Howe-Moore Property for linear and p-adic Lie groups, connecting group structure and representation theory.

Findings

Characterization of the relative Howe-Moore Property via ergodic actions

Structural results on groups with all proper closed characteristic subgroups compact

Results on vanishing at infinity of oscillatory integrals

Abstract

We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or p-adic Lie groups, the pairs with the relative Howe-Moore Property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.