Stein quotients of connected complex Lie groups (Q1068996)

Let G be a connected complex Lie group and H a closed, connected complex Lie subgroup of G. The author investigates the conditions for the quotient complex manifold \(X=G/H\) to be a Stein manifold, in terms of various algebraic and geometric properties of the groups G and H. If G acts effectively on X, we know that H has a semi-direct product decomposition \(H=L\cdot V\), where L is a maximal reductive subgroup of H and V is a simply connected normal solvable subgroup of H. The author first asserts that if \(X=G/H\) is a Stein manifold, then \(V\cap M=\{e\}\) for every maximal reductive subgroup M of G. The author conjectures that the above condition is also sufficient for X to be Stein. In fact, the author proves, among other things, that this is true for solvable G.