Equivariant realizations of Hermitian symmetric space of noncompact type

TL;DR

This paper develops a method to construct $K$-equivariant embeddings of Hermitian symmetric spaces of noncompact type into their tangent spaces, providing new insights into their holomorphic and symplectic structures and their submanifolds.

Contribution

It introduces a novel approach using polarity of the $K$-action to realize embeddings, including the Harish-Chandra realization and symplectomorphisms, and characterizes these embeddings via polarity.

Findings

Reconstructed the Harish-Chandra realization as a $K$-equivariant embedding.

Characterized holomorphic and symplectic embeddings through polarity of the $K$-action.

Identified totally geodesic submanifolds as linear or bounded domains in the tangent space.

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 $M$.