  In the wake of a preceding article \cite{RogUnt06} introducing the
Schr\"odinger-Virasoro group, we study its affine action on a space of
$(1+1)$-dimensional Schr\"odinger operators with time- and space-dependent
potential $V$ periodic in time. We focus on the subspace corresponding to
potentials that are at most quadratic in the space coordinate, which is in some
sense the natural quantization of the space of Hill (Sturm-Liouville) operators
on the one-dimensional torus. The orbits in this subspace have finite
codimension, and their classification by studying the stabilizers can be
obtained by extending Kirillov's results on the orbits of the space of Hill
operators under the Virasoro group. We then explain the connection to the
theory of Ermakov-Lewis invariants for time-dependent harmonic oscillators.
These exact adiabatic invariants behave covariantly under the action of the
Schr\"odinger-Virasoro group, which allows a natural classification of the
orbits in terms of a monodromy operator on $L^2(\R)$ which is closely related
to the monodromy matrix for the corresponding Hill operator.
