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A [[Homogeneous complex manifold|homogeneous complex manifold]] that is isomorphic to a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476301.png" />. An example of a homogeneous bounded domain is the "complex unit ball"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476302.png" /></td> </tr></table>
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which is acted upon transitively by the pseudo-unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476303.png" />, represented by the projective transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476304.png" />.
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A [[Homogeneous complex manifold|homogeneous complex manifold]] that is isomorphic to a bounded domain in  $  \mathbf C  ^ {n} $.  
 +
An example of a homogeneous bounded domain is the "complex unit ball"  
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476305.png" /> is any bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476306.png" />, then the Hermitian differential form
+
$$
 +
\{ {z \in \mathbf C  ^ {n} } : {
 +
| z _ {1} |  ^ {2} + \dots + | z _ {n} |  ^ {2} < 1 } \}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476307.png" /></td> </tr></table>
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which is acted upon transitively by the pseudo-unitary group  $  \mathop{\rm SU} _ {n,1} $,
 +
represented by the projective transformations of the space  $  \mathbf C  ^ {n} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476308.png" /> is the [[Bergman kernel function|Bergman kernel function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h0476309.png" />, defines a [[Kähler metric|Kähler metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763010.png" />, called the Bergman metric, which is invariant under all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763011.png" /> (see [[#References|[1]]], [[#References|[2]]]). The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763012.png" /> of all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763013.png" /> is a real Lie group containing no non-trivial connected complex subgroups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763014.png" /> is homogeneous, then the Bergman metric is complete (cf. [[Complete metric space|Complete metric space]]).
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If $  D $
 +
is any bounded domain in  $  \mathbf C  ^ {n} $,  
 +
then the Hermitian differential form
  
Within the homogeneous bounded domains one can distinguish the symmetric domains. A bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763015.png" /> is called symmetric if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763016.png" /> there exists an involutory automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763017.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763018.png" /> as an isolated fixed point. Every symmetric domain is homogeneous and is a [[Hermitian symmetric space|Hermitian symmetric space]] with respect to the Bergman metric. A classification of symmetric spaces has been obtained [[#References|[3]]]. There are 4 series of irreducible symmetric domains, related to the classical simple Lie groups, and two special domains of dimensions 16 and 27. The classical symmetric domains include, in particular, the complex ball and the Siegel upper half-plane (see [[Siegel domain|Siegel domain]]). Every [[Symmetric domain|symmetric domain]] is isomorphic to a direct product of irreducible symmetric domains [[#References|[1]]].
+
$$
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d  ^  \prime  d  ^ {\prime\prime}
 +
\mathop{\rm ln}  K ( z , \overline{z}\; ) = \
 +
\sum _ { ij }
  
Every homogeneous bounded domain of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763019.png" /> is symmetric [[#References|[3]]]. Beginning with dimension 4, there are also non-symmetric homogeneous bounded domains (see [[#References|[4]]]). Moreover, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763020.png" /> there is a continuum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047630/h04763021.png" />-dimensional homogeneous bounded domains, of which only finitely many are symmetric. Every homogeneous bounded domain is homeomorphic to a cell and analytically isomorphic to an affinely-homogeneous Siegel domain uniquely determined up to an affine isomorphism. The classification of homogeneous bounded domains has been carried out by algebraic methods [[#References|[5]]].
+
\frac{\partial  ^ {2}  \mathop{\rm ln}  K ( z , \overline{z}\; ) }{\partial  z _ {i} \partial  \overline{z}\; _ {j} }
 +
\
 +
d z _ {i}  d \overline{z}\; _ {j} ,
 +
$$
 +
 
 +
where  $  K $
 +
is the [[Bergman kernel function|Bergman kernel function]] of  $  D $,
 +
defines a [[Kähler metric|Kähler metric]] on  $  D $,
 +
called the Bergman metric, which is invariant under all automorphisms of  $  D $(
 +
see [[#References|[1]]], [[#References|[2]]]). The group  $  G ( D) $
 +
of all automorphisms of  $  D $
 +
is a real Lie group containing no non-trivial connected complex subgroups. If  $  D $
 +
is homogeneous, then the Bergman metric is complete (cf. [[Complete metric space|Complete metric space]]).
 +
 
 +
Within the homogeneous bounded domains one can distinguish the symmetric domains. A bounded domain $  D $
 +
is called symmetric if for any point  $  z \in D $
 +
there exists an involutory automorphism of  $  D $
 +
having  $  z $
 +
as an isolated fixed point. Every symmetric domain is homogeneous and is a [[Hermitian symmetric space|Hermitian symmetric space]] with respect to the Bergman metric. A classification of symmetric spaces has been obtained [[#References|[3]]]. There are 4 series of irreducible symmetric domains, related to the classical simple Lie groups, and two special domains of dimensions 16 and 27. The classical symmetric domains include, in particular, the complex ball and the Siegel upper half-plane (see [[Siegel domain|Siegel domain]]). Every [[Symmetric domain|symmetric domain]] is isomorphic to a direct product of irreducible symmetric domains [[#References|[1]]].
 +
 
 +
Every homogeneous bounded domain of dimension  $  \leq  3 $
 +
is symmetric [[#References|[3]]]. Beginning with dimension 4, there are also non-symmetric homogeneous bounded domains (see [[#References|[4]]]). Moreover, for $  n \geq  7 $
 +
there is a continuum of $  n $-
 +
dimensional homogeneous bounded domains, of which only finitely many are symmetric. Every homogeneous bounded domain is homeomorphic to a cell and analytically isomorphic to an affinely-homogeneous Siegel domain uniquely determined up to an affine isomorphism. The classification of homogeneous bounded domains has been carried out by algebraic methods [[#References|[5]]].
  
 
Related to homogeneous bounded domains there are multi-dimensional generalizations of Eulerian integrals (Siegel integrals of the first and second kinds), as well as of hypergeometric functions [[#References|[6]]].
 
Related to homogeneous bounded domains there are multi-dimensional generalizations of Eulerian integrals (Siegel integrals of the first and second kinds), as well as of hypergeometric functions [[#References|[6]]].

Revision as of 22:10, 5 June 2020


A homogeneous complex manifold that is isomorphic to a bounded domain in $ \mathbf C ^ {n} $. An example of a homogeneous bounded domain is the "complex unit ball"

$$ \{ {z \in \mathbf C ^ {n} } : { | z _ {1} | ^ {2} + \dots + | z _ {n} | ^ {2} < 1 } \} , $$

which is acted upon transitively by the pseudo-unitary group $ \mathop{\rm SU} _ {n,1} $, represented by the projective transformations of the space $ \mathbf C ^ {n} $.

If $ D $ is any bounded domain in $ \mathbf C ^ {n} $, then the Hermitian differential form

$$ d ^ \prime d ^ {\prime\prime} \mathop{\rm ln} K ( z , \overline{z}\; ) = \ \sum _ { ij } \frac{\partial ^ {2} \mathop{\rm ln} K ( z , \overline{z}\; ) }{\partial z _ {i} \partial \overline{z}\; _ {j} } \ d z _ {i} d \overline{z}\; _ {j} , $$

where $ K $ is the Bergman kernel function of $ D $, defines a Kähler metric on $ D $, called the Bergman metric, which is invariant under all automorphisms of $ D $( see [1], [2]). The group $ G ( D) $ of all automorphisms of $ D $ is a real Lie group containing no non-trivial connected complex subgroups. If $ D $ is homogeneous, then the Bergman metric is complete (cf. Complete metric space).

Within the homogeneous bounded domains one can distinguish the symmetric domains. A bounded domain $ D $ is called symmetric if for any point $ z \in D $ there exists an involutory automorphism of $ D $ having $ z $ as an isolated fixed point. Every symmetric domain is homogeneous and is a Hermitian symmetric space with respect to the Bergman metric. A classification of symmetric spaces has been obtained [3]. There are 4 series of irreducible symmetric domains, related to the classical simple Lie groups, and two special domains of dimensions 16 and 27. The classical symmetric domains include, in particular, the complex ball and the Siegel upper half-plane (see Siegel domain). Every symmetric domain is isomorphic to a direct product of irreducible symmetric domains [1].

Every homogeneous bounded domain of dimension $ \leq 3 $ is symmetric [3]. Beginning with dimension 4, there are also non-symmetric homogeneous bounded domains (see [4]). Moreover, for $ n \geq 7 $ there is a continuum of $ n $- dimensional homogeneous bounded domains, of which only finitely many are symmetric. Every homogeneous bounded domain is homeomorphic to a cell and analytically isomorphic to an affinely-homogeneous Siegel domain uniquely determined up to an affine isomorphism. The classification of homogeneous bounded domains has been carried out by algebraic methods [5].

Related to homogeneous bounded domains there are multi-dimensional generalizations of Eulerian integrals (Siegel integrals of the first and second kinds), as well as of hypergeometric functions [6].

References

[1] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[2] B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)
[3] E. Cartan, "Domains bornés homogènes de l'espace de variables complexes" Abh. Math. Sem. Univ. Hamburg , 11 (1935) pp. 116–162
[4] I.I. [I.I. Pyatetskii-Shapiro] Piatetski-Shapiro, "Automorphic functions and the geometry of classical domains" , Gordon & Breach (1969) (Translated from Russian)
[5] E.B. Vinberg, S.G. Gindikin, I.I. Pyatetskii-Shapiro, "On the classification and canonical realization of complex bounded homogeneous domains" Trans. Moscow Math. Soc. , 12 (1963) pp. 404–437 Trudy Moskov. Mat. Obshch. , 12 (1963) pp. 359–388
[6] S.G. Gindikin, "Analysis in homogeneous domains" Russian Math. Surveys , 19 : 4 (1964) pp. 3–92 Uspekhi Mat. Nauk , 19 : 4 (1964) pp. 3–92
How to Cite This Entry:
Homogeneous bounded domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_bounded_domain&oldid=47250
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article