We associate a space of unitary representations (which we call quantizations)
with a volume-preserving action of a Lie group on R^d. These representations
are expressed in terms of certain Fourier integral operators deforming the
pullback of functions by diffeomorphisms. They are determined by a system of
amplitudes, which we show to be controlled by a Maurer-Cartan equation in an
appropriate differential graded algebra. We apply our quantization scheme to
the Heisenberg group by quantizing its adjoint action.