Homogeneous bounded domain

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 $n$ 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