  The structure of subspaces of a Hilbert space that are invariant under
unitary representations of a discrete group is related to a notion of Hilbert
modules endowed with inner products taking values in spaces of unbounded
operators. A theory of reproducing systems in such modular structures is
developed, providing a general framework that includes fundamental results of
shift-invariant spaces. In particular, general characterizations of Riesz and
frame sequences associated to group representations are provided, extending
previous results for abelian groups and for cyclic subspaces of unitary
representations of noncommutative discrete groups.
