Injectivity of the complexified action map and saturation equality for groups satisfying the Lie condition

Determine whether, for a homogeneous Siegel domain X ⊂ C^d, with B a maximal simply connected split solvable subgroup of Aut_O(X) acting simply transitively, a closed subgroup H < B that fulfills the Lie condition and S = exp(s), the holomorphic map Φ: H^C × (S · x0) → Z defined by Φ(h, s · x0) = h s · x0, where Z := H^C · M_H = H^C S · x0 and M_H = μ_H^{-1}(0), is injective; additionally, determine whether X is contained in Z (equivalently, whether Z = H^C · X).

Background

In the paper, the author studies Hamiltonian actions on homogeneous bounded domains, realized as homogeneous Siegel domains. Let X be a homogeneous Siegel domain in C^d and B a maximal simply connected split solvable subgroup of Aut_O(X) acting simply transitively. For a closed subgroup H < B satisfying the Lie condition, with Lie subalgebra s and S = exp(s), the momentum map μ_H has zero fiber M_H = μ_H^{-1}(0). Theorem 3.6 shows M_H = H S * x0 and that the restriction of the quotient map identifies M_H/H biholomorphically and isometrically with the complex submanifold S * x0, implying the symplectic reduction is Stein.

The remark introduces the complexified saturation Z := H^C * M_H = H^C S * x0 and the holomorphic map Φ: H^C × S * x0 → Z, Φ(h, s * x0) = h s * x0. It is established that Φ is a surjective holomorphic immersion. The authors explicitly raise whether Φ is injective and whether X is contained in Z, i.e., whether Z equals the H^C-saturation of X. They note that in a related work [Ku] they prove injectivity for subgroups of the linear automorphism group of the Lorentz tube, leaving the general case unresolved.