Equivariant realizations of Hermitian symmetric space of noncompact type

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