Normalization in integral models of Shimura varieties of Hodge type

Let $(G,X)$ be a Shimura datum of Hodge type, and $\mathscr{S}_K(G,X)$ its integral model with hyperspecial (resp. parahoric, assuming the group is unramified) level structure. We prove that $\mathscr{S}_K(G,X)$ admits a closed embedding, which is compatible with moduli interpretations, into the integral model $\mathscr{S}_{K'}(\mathrm{GSp},S^{\pm})$ for a Siegel modular variety. In particular, the normalization step in the construction of $\mathscr{S}_K(G,X)$ is redundant. In particular, our results apply to the earlier integral models constructed by Rapoport, Kottwitz etc. (resp. Rapoport-Zink etc.), as those models agree with the Hodge type integral models for appropriately chosen Shimura data. Moreover, combined with a result of Lan's on the boundary components of toroidal compactifications of integral models, our result also implies that there exist closed embeddings of toroidal compactifications of integral models of Hodge type into toroidal compactifications of Siegel integral models, for suitable choices of cone decompositions.