Remarks on asymptotic isometric embeddings of conic transforms for torus actions

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.