Equivariant realizations of Hermitian symmetric space of noncompact type

Let $$M=G/K$$ be a Hermitian symmetric space of noncompact type. We provide way constructing K-equivariant embeddings from M to its tangent $$T_oM$$ at the origin by using polarity K-action. As an application, we reconstruct holomorphic embedding so called Harish-Chandra realization and symplectomorphism constructed Di Scala-Loi Roos under appropriate identifications spaces. Moreover, characterize holomorphic/symplectic means Furthermore, show special class totally geodesic submanifolds in is realized as either linear subspaces or bounded domains embeddings. also construct open dense subset compact dual $$M^*$$ into M.