Hermitian symmetric space

A connected complex manifold $M$ with a Hermitian structure in which each point $p\in M$ is an isolated fixed point of some holomorphic involutory isometry $s_p$ of $M$. The component of the identity of the group $G$ of holomorphic isometries of $M$ acts transitively on $M$. Let $K$ be the isotropy subgroup of $G$ relative to some point $0\in M$. Then $M$ is said to be of compact or non-compact type in accordance with the type of the globally symmetric Riemannian space $G/K$. Every Hermitian symmetric space $M$ is a direct product $M=M_0\times M_-\times M_+$, where all the factors are simply-connected Hermitian symmetric spaces, $M_0=\mathbf C^n$ and $M_-$ and $M_+$ are spaces of compact and non-compact type, respectively. Any Hermitian symmetric space of compact or non-compact type is simply connected and is a direct product of irreducible Hermitian symmetric spaces.

A non-compact irreducible Hermitian symmetric space is of the form $G/K$, where $G$ is a connected non-compact simple Lie group with trivial centre and $K$ is a maximal compact subgroup of $G$ having non-discrete centre. Compact irreducible Hermitian symmetric spaces are of the form $G/K$, where $G$ is a connected compact simple Lie group with trivial centre and $K$ is a maximal connected proper subgroup of $G$ having non-discrete centre.

A Hermitian symmetric space of non-compact type arises in the following way in the theory of functions of several complex variables. Let $\mathbf C^n$ be the $n$-dimensional complex vector space. A bounded domain is defined as a connected bounded open subset of $\mathbf C^n$. A bounded domain $D$ is said to to be symmetric if every point $p\in D$ is an isolated fixed point of some involutory holomorphic diffeomorphism of $D$ onto itself. The following theorem holds: a) every bounded symmetric domain $D$ equipped with the Bergman metric (see Bergman kernel function; Homogeneous bounded domain) is a Hermitian symmetric space of non-compact type, in particular, a bounded symmetric domain is necessarily simply connected; and b) let $M$ be a Hermitian space of non-compact type, then there is a bounded symmetric domain $D$ and a holomorphic diffeomorphism of $M$ onto $D$.

For references see Symmetric space.