Classification of homogeneous bounded domains of lower dimension

Abstract

The theory of classification of homogeneous bounded domains in the complex number space Cn has been developed mainly in the recent papers [10], [6], [3] and [7]. As a result, the classification is reduced to that of S-algebras due to Takeuchi [7] which correspond to irreducible Siegel domains of type I or type II (For the definition of irreducibility see § 1). On the other hand Pjateckii-Sapiro [5] found large classes of homogeneous Siegel domains obtained from classical self-dual cones. Even in lower-dimensional cases, however, there are still homogeneous Siegel domains which do not appear in his results.

Siegel domains over self-dual cones and their automorphisms

“…A figure © in the plane is called a connected 2-skeleton (of type II) if © is one of the following ® x or © 2 ; 6) This definition is slightly different from that of [5], but these are equivalent.…”

“…It is known in [5] that for given positive integers n, m 19 there exists a unique homogeneous Siegel domain which corresponds to © lβ Furthermore the explicit forms of these domains are found in [5], [10]. 4.2.…”

“…(2ϊ) onto R n+2 ; (4.15) /: a = a n + a 22 + a 12 e R{%) >-> '(on, α 22 , a\ 2 , , α&) e ^T O+2 , where a 12 = 2] Λfjβ* We define the map F: c mi+m * X C mi+m2 >-> C n+2 by putting F = '(F 1 , ...,F n+2 ), where Proof. First we will show that the map F defined by (4.16) is a C(n + 2)-hermitian form on C mi + C W2 and the Siegel domain D(C(n + 2), F) thus constructed is the one which corresponds to (iV,< , >,/) in the sense of [5]. By Theorem A in [13], the homogeneous Siegel domain which corresponds to the Γ-algebra (SI, *,/) is given by the following V(N)-hermitian form F = Σ^^^2F fcl on Tf(Sί); If mi=m 2 in © 2> then this construction is reduced to Pjateckii-Sapiro's [10].…”

“…Suppose that D(C(n + 2),F) corresponds to © lβ Then it was proved in [5], [10] that the C(n + 2)-hermitian form F on C mi is given by…”

A Characterization of Symmetric Cones by the Degrees of Basic Relative Invariants

“…The presentation follows our previous paper [18]. The oriented graph which concerns us is of the type (S 5 4 ) * in the notation of Kaneyuki and Tsuji [10]. Thus, is as in Figure 1, where d 41 = m ≥ 2, and we do not indicate dim V kj (k > j ) if they are equal to one for simplicity.…”