Holomorphic Imbeddings of Symmetric Domains into a Siegel Space

Abstract

A symmetric domain is a bounded domain in C N which becomes a symmetric Riemannian space with respect to its Bergman metric. For a symmetric domain 𝒟, we denote by G the connected component of the group of all analytic automorphisms of 𝒟 (or, what is the same, that of the group of all isometries of 𝒟 onto itself) with its natural topology and by K the isotropy subgroup of G at an (arbitrary) point z 0 ∈ 𝒟. Then, as is well-known ([1], [2]), G is a (connected) semi-simple Lie group of non-compact type with center reduced to the identity, K is a maximal compact subgroup of G and 𝒟 is identified with the coset-space G/K. A symmetric domain 𝒟 is decomposed uniquely into the direct product 𝒟 1 × ⋯ × 𝒟 S of irreducible symmetric domains 𝒟 i (i. e. the domains which cannot be decomposed any further) corresponding to the direct decomposition G 1 × ⋯ × G s of G into simple components. Irreducible symmetric domains have been classified completely by E. Cartas [1] into four main series (I), (II), (III) and (IV) of classical domains (see no 3 below) and two exceptional domains corresponding to the simple Lie groups of type (E 6) and (E 7).