Embeddings of Certain Exceptional Shimura Varieties into Siegel Modular Varieties

We define a class of local Shimura varieties that contains some local Shimura varieties for exceptional groups, and for this class, we construct a functor from $(G,\mu )$-displays to $p$-divisible groups. As an application, we prove that for this class, the local Shimura variety is representable and perfectoid at the infinite level. Considering the global counterpart of this class, we embed certain exceptional Shimura varieties into Siegel modular varieties. In particular, we prove that they are perfectoid at the infinite level.