  Consider a Hodge manifold and assume that a torus acts on it in a Hamiltonian
and holomorphic manner and that this action linearizes on a given quantizing
line bundle. Inside the dual of the line bundle one can define the circle
bundle, which is a strictly pseudoconvex CR manifold. Then, there is an
associated unitary representation on the Hardy space of the circle bundle.
Under suitable assumptions on the moment map, we consider certain loci in unit
circle bundle, naturally associated to a ray through an irreducible weight.
Their quotients are called conic transforms. We introduce maps which are
asymptotic embeddings of conic transforms making use of the corresponding
equivariant Szeg\H{o} projector.
