The Howe-Moore theorem implies that any ergodic action of a connected
noncompact, finite-center, simple Lie group is mixing. Thus, for
example, ergodicity inherits to all noncompact closed subgroups, a
very useful fact which immediately yields ergodicity of many
geometrically-motivated actions. The proofs I know of Howe-Moore go
via unitary representation theory. By thinking of a Hilbert space as
an infinite-dimensional Riemannian manifold and by adjusting
techniques originally used in studying Lorentzian (and Riemannian)
dynamics, we can obtain a version of Howe-Moore that is valid for all
connected Lie groups, though only useful for nonAbelian
groups. Specifically, for any connected Lie group $G$, for any
faithful irreducible unitary representation of $G$, we have: any
matrix coefficient tends to zero, as $Ad(g)$ leaves compact subsets of
$GL({\frak g})$, where ${\frak g}$ is the Lie algebra of $G