Symmetric domain
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>
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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 ![]() |
[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 |
Symmetric domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_domain&oldid=16315