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> 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