Equivariant realizations of Hermitian symmetric space of noncompact type

2020/12/09 by Hashinaga, Takahiro, Kajigaya, Toru
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.05039

Abstract

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 T_oM 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 T_oM 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.

Related

• Equivariant embeddings of Hermitian symmetric spaces
• Equivariant Embeddings of Kälerian Symmetric Spaces
• Embeddings from noncompact symmetric spaces to their compact duals
• Submanifold geometry in symmetric spaces of noncompact type
• On symplectic geometry of tangent bundles of Hermitian symmetric spaces