Bounded bicircular domains in \({\mathbb{C}}^ n\)
It was proved by \textit{W. Kaup} and \textit{H. Upmeier} [Proc. Am. Math. Soc. 58, 129-133 (1976; Zbl 0337.32012)] that if D is the open unit ball in a complex Banach space E then there exists a \({\mathbb{C}}\)-linear subspace \(U\subset E\) such that the intersection \(A:=U\cap D\) equals the orbit G(0), where \(G=Aut(D)\) denotes the automorphism group of D. One can thus consider the following question: Given a bounded symmetric domain \(A\subset U\), where U is a linear subspace of \({\mathbb{C}}^ n\), which bounded circular domains \(D\subset {\mathbb{C}}^ n\) are extensions of A in the sense that \(A=C\cap U=Aut(D)(0)?\) The situation for extensions D of A which are Reinhardt domains was classified by \textit{T. Sunada} [Math. Ann. 235, 111-128 (1978; Zbl 0357.32001)]. The purpose of the present paper is to determine all extensions of a bounded symmetric domain \(A\subset U\) by bicircular domains \(D\subset E:={\mathbb{C}}^ n\). Here bicircular means that there is a linear subspace V with \(E=U\oplus V\) such that D is invariant under the bicircular group of transformations \((u,v)\mapsto (e^{is}u,e^{it}v)\) with s,t\(\in {\mathbb{R}}\). (Examples of such are provided by Reinhardt domains with respect to some direct decomposition of \({\mathbb{C}}^ n\) parallel to the coordinate axes.) In the case of bounded Reinhardt domains T. Sunada studied the Lie algebra \({\mathfrak g}\) of Aut(D) and showed that one has an analogue of the Cartan decomposition \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\). In this paper it is first proved that the Lie algebra of Aut(D) is reductive. Then the notion of hermitian representations of \(J^*\)-triples is introduced to describe the isomorphy classes of bicircular domains. A natural one-to-one correspondence between hermitian representations of \(J^*\)-triples and unitary representations of complex reductive Lie algebras which are the complexifications of the derivation algebras of the corresponding \(J^*\)-triples is constructed. For positive \(J^*\)-triples these correspond to the Lie algebras of the 0-orbits and are called ``of orbit type''. The final point is to show that the positive representations of Lie algebras of orbit type completely characterize the bounded bicircular domains.
scientific article; zbMATH DE number 4177505
scientific article