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A [[Complex manifold|complex manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916501.png" /> isomorphic to a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916502.png" /> and such that, for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916503.png" />, there is an involutory holomorphic transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916504.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916505.png" /> as unique fixed point. A symmetric domain is a [[Hermitian symmetric space|Hermitian symmetric space]] of negative curvature with respect to the Bergman metric (cf. [[Bergman kernel function|Bergman kernel function]]). Its automorphism group is contained, as a complex manifold, in the group of motions and has the same connected component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916506.png" />, which is a non-compact real semi-simple Lie group without centre. The stationary subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916507.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916508.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s0916509.png" /> is a connected compact Lie group with one-dimensional centre. As a real manifold, a symmetric domain is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165010.png" />.
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Every symmetric domain is uniquely decomposable as a direct product of irreducible symmetric domains, and these are listed in the following table (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165011.png" /> denotes the space of complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165012.png" />-matrices).''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Cartan type</td> <td colname="2" style="background-color:white;" colspan="1">Type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165013.png" /></td> <td colname="3" style="background-color:white;" colspan="1">Type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165014.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165015.png" /></td> <td colname="5" style="background-color:white;" colspan="1">Model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165016.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">I</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165017.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165018.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165019.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165020.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">II</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165021.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165022.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165023.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165024.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">III</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165025.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165026.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165027.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165028.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">IV</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165029.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165030.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165031.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165032.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">V</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165033.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165034.png" /></td> <td colname="4" style="background-color:white;" colspan="1">16</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">VI</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165035.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165036.png" /></td> <td colname="4" style="background-color:white;" colspan="1">27</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> </tbody> </table>
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A [[Complex manifold|complex manifold]]  $  D $
 +
isomorphic to a bounded domain in  $  \mathbf C  ^ {n} $
 +
and such that, for every point  $  p \in D $,
 +
there is an involutory holomorphic transformation  $  \sigma _ {p} :  D \rightarrow D $
 +
with  $  p $
 +
as unique fixed point. A symmetric domain is a [[Hermitian symmetric space|Hermitian symmetric space]] of negative curvature with respect to the Bergman metric (cf. [[Bergman kernel function|Bergman kernel function]]). Its automorphism group is contained, as a complex manifold, in the group of motions and has the same connected component  $  G ( D) $,
 +
which is a non-compact real semi-simple Lie group without centre. The stationary subgroup  $  H ( D) $
 +
of  $  p \in D $
 +
in  $  G ( D) $
 +
is a connected compact Lie group with one-dimensional centre. As a real manifold, a symmetric domain is diffeomorphic to  $  \mathbf R ^ {2 n } $.
 +
 
 +
Every symmetric domain is uniquely decomposable as a direct product of irreducible symmetric domains, and these are listed in the following table (where  $  M _ {p,q} $
 +
denotes the space of complex  $  ( p \times q) $-
 +
matrices).<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Cartan type</td> <td colname="2" style="background-color:white;" colspan="1">Type of $  G( D) $
 +
</td> <td colname="3" style="background-color:white;" colspan="1">Type of  $  H( D)  ^  \prime  $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  \mathop{\rm dim}  D $
 +
</td> <td colname="5" style="background-color:white;" colspan="1">Model of  $  D $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">I</td> <td colname="2" style="background-color:white;" colspan="1"> $  A _ {p + q - 1 }  $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  \begin{array}{c}
 +
A _ {p - 1 }  + A _ {q - 1 }  \\
 +
( p \geq  q)
 +
\end{array}
 +
$
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  pq $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \{ {Z\in M _ {p,q} } : {Z  ^ {*} Z < E } \} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">II</td> <td colname="2" style="background-color:white;" colspan="1"> $  D _ {p} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  A _ {p - 1 }  $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  {
 +
\frac{p ( p - 1) }{2}
 +
} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \{ {Z \in M _ {p,p} } : {Z  ^ {t} =- Z, Z  ^ {*} Z < E } \} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">III</td> <td colname="2" style="background-color:white;" colspan="1"> $  C _ {p} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  A _ {p - 1 }  $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  {
 +
\frac{p ( p + 1) }{2}
 +
} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \{ {Z \in M _ {p,p} } : {Z  ^ {t} = Z, Z  ^ {*} Z < E } \} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">IV</td> <td colname="2" style="background-color:white;" colspan="1"> $  \begin{array}{c}
 +
D _ {p/2 + 1 }  \\
 +
B _ {( p + 1)/2 } 
 +
\end{array}
 +
$
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  \begin{array}{c}
 +
D _ {p/2 - 1 }  \\
 +
B _ {( p - 1)/2 } 
 +
\end{array}
 +
$
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  p $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \{ {z \in \mathbf C  ^ {p}  } : {  \sum | z _ {i} |  ^ {2} < {
 +
\frac{1}{2}
 +
} \left ( 1 + \left | \sum z _ {i}  ^ {2} \right |  ^ {2} \right ) < 1  } \} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">V</td> <td colname="2" style="background-color:white;" colspan="1"> $  E _ {6} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  D _ {5} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1">16</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">VI</td> <td colname="2" style="background-color:white;" colspan="1"> $  E _ {7} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  E _ {6} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1">27</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
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A symmetric domain of type III can be represented as the Siegel upper half-plane:
 
A symmetric domain of type III can be represented as the Siegel upper half-plane:
  
<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/s/s091/s091650/s09165037.png" /></td> </tr></table>
+
$$
 +
\{ Z \in M _ {p,p} : Z  ^ {t} = Z,  \mathop{\rm Im}  Z > 0 \} .
 +
$$
  
 
Its points parametrize principally polarized Abelian varieties. The other symmetric domains can also be represented as Siegel domains (cf. [[Siegel domain|Siegel domain]]) of the first or second kind (see [[#References|[2]]]).
 
Its points parametrize principally polarized Abelian varieties. The other symmetric domains can also be represented as Siegel domains (cf. [[Siegel domain|Siegel domain]]) of the first or second kind (see [[#References|[2]]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.L. Siegel,  "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Pyatetskii-Shapiro] Piatetski-Shapiro,  "Automorphic functions and the geometry of classical domains" , Gordon &amp; Breach  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Cartan,  "Domains bornés homogènes de l'espace de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165038.png" /> variables complexes"  ''Abh. Math. Sem. Univ. Hamburg'' , '''1'''  (1935)  pp. 116–162</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Drucker,  "Exceptional Lie algebras and the structure of hermitian symmetric spaces" , Amer. Math. Soc.  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.L. Siegel,  "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Pyatetskii-Shapiro] Piatetski-Shapiro,  "Automorphic functions and the geometry of classical domains" , Gordon &amp; Breach  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Cartan,  "Domains bornés homogènes de l'espace de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165038.png" /> variables complexes"  ''Abh. Math. Sem. Univ. Hamburg'' , '''1'''  (1935)  pp. 116–162</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Drucker,  "Exceptional Lie algebras and the structure of hermitian symmetric spaces" , Amer. Math. Soc.  (1978)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The stationary subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091650/s09165039.png" /> has one-dimensional centre if and only if the symmetric domain is irreducible.
+
The stationary subgroup $  H( D) $
 +
has one-dimensional centre if and only if the symmetric domain is irreducible.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)  pp. Chapt. X</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)  pp. Chapt. X</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


A complex manifold $ D $ isomorphic to a bounded domain in $ \mathbf C ^ {n} $ and such that, for every point $ p \in D $, there is an involutory holomorphic transformation $ \sigma _ {p} : D \rightarrow D $ with $ p $ as unique fixed point. A symmetric domain is a Hermitian symmetric space of negative curvature with respect to the Bergman metric (cf. Bergman kernel function). Its automorphism group is contained, as a complex manifold, in the group of motions and has the same connected component $ G ( D) $, which is a non-compact real semi-simple Lie group without centre. The stationary subgroup $ H ( D) $ of $ p \in D $ in $ G ( D) $ is a connected compact Lie group with one-dimensional centre. As a real manifold, a symmetric domain is diffeomorphic to $ \mathbf R ^ {2 n } $.

Every symmetric domain is uniquely decomposable as a direct product of irreducible symmetric domains, and these are listed in the following table (where $ M _ {p,q} $ denotes the space of complex $ ( p \times q) $-

matrices).

<tbody> </tbody>
Cartan type Type of $ G( D) $ Type of $ H( D) ^ \prime $ $ \mathop{\rm dim} D $ Model of $ D $
I $ A _ {p + q - 1 } $ $ \begin{array}{c} A _ {p - 1 } + A _ {q - 1 } \\ ( p \geq q) \end{array} $ $ pq $ $ \{ {Z\in M _ {p,q} } : {Z ^ {*} Z < E } \} $
II $ D _ {p} $ $ A _ {p - 1 } $ $ { \frac{p ( p - 1) }{2} } $ $ \{ {Z \in M _ {p,p} } : {Z ^ {t} =- Z, Z ^ {*} Z < E } \} $
III $ C _ {p} $ $ A _ {p - 1 } $ $ { \frac{p ( p + 1) }{2} } $ $ \{ {Z \in M _ {p,p} } : {Z ^ {t} = Z, Z ^ {*} Z < E } \} $
IV $ \begin{array}{c} D _ {p/2 + 1 } \\ B _ {( p + 1)/2 } \end{array} $ $ \begin{array}{c} D _ {p/2 - 1 } \\ B _ {( p - 1)/2 } \end{array} $ $ p $ $ \{ {z \in \mathbf C ^ {p} } : { \sum | z _ {i} | ^ {2} < { \frac{1}{2} } \left ( 1 + \left | \sum z _ {i} ^ {2} \right | ^ {2} \right ) < 1 } \} $
V $ E _ {6} $ $ D _ {5} $ 16
VI $ E _ {7} $ $ E _ {6} $ 27

A symmetric domain of type III can be represented as the Siegel upper half-plane:

$$ \{ Z \in M _ {p,p} : Z ^ {t} = Z, \mathop{\rm Im} Z > 0 \} . $$

Its points parametrize principally polarized Abelian varieties. The other symmetric domains can also be represented as Siegel domains (cf. Siegel domain) of the first or second kind (see [2]).

References

[1] C.L. Siegel, "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen (1955)
[2] I.I. [I.I. Pyatetskii-Shapiro] Piatetski-Shapiro, "Automorphic functions and the geometry of classical domains" , Gordon & Breach (1969) (Translated from Russian)
[3] E. Cartan, "Domains bornés homogènes de l'espace de variables complexes" Abh. Math. Sem. Univ. Hamburg , 1 (1935) pp. 116–162
[4] D. Drucker, "Exceptional Lie algebras and the structure of hermitian symmetric spaces" , Amer. Math. Soc. (1978)

Comments

The stationary subgroup $ H( D) $ has one-dimensional centre if and only if the symmetric domain is irreducible.

References

[a1] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) pp. Chapt. X
How to Cite This Entry:
Symmetric domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_domain&oldid=16315
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article