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| − | A connected [[Complex manifold|complex manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471101.png" /> with a [[Hermitian structure|Hermitian structure]] in which each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471102.png" /> is an isolated fixed point of some holomorphic involutory isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471104.png" />. The component of the identity of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471105.png" /> of holomorphic isometries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471106.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471107.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471108.png" /> be the isotropy subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h0471109.png" /> relative to some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711010.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711011.png" /> is said to be of compact or non-compact type in accordance with the type of the [[Globally symmetric Riemannian space|globally symmetric Riemannian space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711012.png" />. Every Hermitian symmetric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711013.png" /> is a direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711014.png" />, where all the factors are simply-connected Hermitian symmetric spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711017.png" /> 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. | + | {{TEX|done}} |
| | + | A connected [[Complex manifold|complex manifold]] $M$ with a [[Hermitian structure|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|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. |
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| − | A non-compact irreducible Hermitian symmetric space is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711019.png" /> is a connected non-compact simple Lie group with trivial centre and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711020.png" /> is a maximal compact subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711021.png" /> having non-discrete centre. Compact irreducible Hermitian symmetric spaces are of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711023.png" /> is a connected compact simple Lie group with trivial centre and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711024.png" /> is a maximal connected proper subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711025.png" /> having non-discrete centre. | + | 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. |
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| − | A Hermitian symmetric space of non-compact type arises in the following way in the theory of functions of several complex variables. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711026.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711027.png" />-dimensional complex vector space. A bounded domain is defined as a connected bounded open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711028.png" />. A bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711029.png" /> is said to to be symmetric if every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711031.png" /> is an isolated fixed point of some involutory holomorphic diffeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711032.png" /> onto itself. The following theorem holds: a) every bounded symmetric domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711033.png" /> equipped with the Bergman metric (see [[Bergman kernel function|Bergman kernel function]]; [[Homogeneous bounded domain|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711034.png" /> be a Hermitian space of non-compact type, then there is a bounded symmetric domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711035.png" /> and a holomorphic diffeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711036.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047110/h04711037.png" />. | + | 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|Bergman kernel function]]; [[Homogeneous bounded domain|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 $D$. |
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| | For references see [[Symmetric space|Symmetric space]]. | | For references see [[Symmetric space|Symmetric space]]. |
Revision as of 12:45, 27 September 2014
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.
How to Cite This Entry:
Hermitian symmetric space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_symmetric_space&oldid=16806
This article was adapted from an original article by A.S. Fedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article