  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.
