  We investigate the structure of subspaces of a Hilbert space that are
invariant under unitary representations of a discrete group. We work with
square integrable representations, and we show that they are those for which we
can construct an isometry intertwining the representation with the right
regular representation, that we call a Helson map. We then characterize
invariant subspaces using the Helson map, and provide general characterizations
of Riesz and frame sequences of orbits. These results extend to the nonabelian
setting several known results for abelian groups. They also extend to countable
families of generators previous results obtained for principal subspaces.
