TITLE: A class of unipotent group actions
QUESTION [6 upvotes]: Consider algebraic actions of unipotent groups $G$ on affine spaces $X=\mathbb{C}^n$. I am looking for a condition that would guarantee that the quotient $X/G$ exists and is also an affine space. For instance:
Suppose $X$ is itself isomorphic to a unipotent group and $G$ acts via compositions of left translations with group automorphisms. Suppose the action is such that every point of $X$ has trivial stabilizer. Is it true that $X/G$ is isomorphic to affine space?
UPDATE: It turns out, by replacing $X/G$ with $G\backslash X\rtimes A/A$ where $A$ is the image of $G$ in the group of automorphisms of $X$, the question is reduced to the corresponding question for double cosets, which is again a special case of the original question. So here is an equivalent question:
Suppose a unipotent group $G$ contains unipotent subgroups $G_1, G_2$ such that $G_1\cap x G_2 x^{-1} = \{e\}$ for all $x\in G$.  Is the double coset space $G_1 \backslash G / G_2$ isomorphic to affine space?

REPLY [2 votes]: The answer is "no". In this paper: Winkelmann, J. On free holomorphic ℂ-actions on ℂn and homogeneous stein manifolds. Math. Ann. 286, 593–612 (1990) a free affine linear action of $G_a\times G_a$ on $\mathbb{C}^6$ is given in such a way that the quotient is not an affine variety. So for $X=G_a^6$, $G=G_a^2$, we obtain a counter-example.
By the way, in op. cit. this action is reduced to a triangular algebraic action of $G_a$ on $\mathbb{C}^5$. There is also an example there of a free triangular algebraic action of $G_a$ on $\mathbb{C}^4$ with non-Hausdorff quotient. It is not hard to check that the class of triangular algebraic actions coincides with the class of actions of $G_1$ on $G/G_2$ where $G_1, G_2$ range over unipotent subgroups of unipotent groups $G$.