On Siegel domains of finite type (Q1099305)

Let \(D=D(V,F)\subset {\mathbb{C}}^ N \)be a Siegel domain associated with a convex cone V and a V-hermitian form F. Let G be the identity component of the holomorphic automorphism group of D. By a result of \textit{K. Nakajima} [J. Math. Kyoto Univ. 14, 533-548 (1974; Zbl 0298.32020)], D is G-equivariantly and holomorphically imbedded into a complex coset space M of the complexification of G. Now, D is said to be of finite type, if there are only finitely many G-orbits in M. Next, let H be the identity component of linear automorphism group of D. Then, there exists a natural homomorphism \(\rho\) of H into the linear automorphism group of the cone. The cone V is said to be of \(\rho\) (H)-finite type, if there exists only a finite number of \(\rho\) (H)-orbits in the ambiant vector space, in which V is imbedded as an open set. The author proves, among other things, that d is of finite type if and only if V is of \(\rho\) (H)-finite type. The author notices that the class of Siegel domains of finite type properly contains the class of quasi-symmetric Siegel domains.