TITLE: Partial (or complete) flag varieties as GIT quotients of affine spaces
QUESTION [11 upvotes]: I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces.  That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would like to find examples of an affine variety space $V/\mathbb{C}$, a reductive group G acting on V, and a linearization $L$ such that the GIT quotient $V//G$ is equal to the flag variety $Fl(d_1,d_2,\dots,d_k)$.
Of course $\mathbb{P}^{n-1}$ is an easy example for $Fl(1,n)$, where $V = \mathbb{C}^{n}$ and $G = \mathbb{C}^{\times}$. And the Grassmannian $Fl(k,n) = Gr(k, {n})$ can be constructed as a GIT quotient of the vector space $M_{k,{n}}$ of $k\times n$ matrices $Gr(k, {n}) = M_{k,{n}}//GL(k)$, where $GL(k)$ acts as matrix multiplication on the left.  
Are there constructions that work for more general choices of $d_i$? Does anyone know of any other examples or have recommendations of where I might look for them?

REPLY [9 votes]: Short answer: Every spherical variety is a Mori Dream Space.
Longer answer: Every projective variety is a "GIT quotient", but the most obvious construction is probably unenlightening.  Let $X$ be a projective variety embedded in $\mathbb{P}^n$.  Let $V$ be the affine cone over $X$ with the natural action of $G=\mathbb{G}_m$.  Then $V$ is a $G$-invariant, Zariski closed subset of the affine space $\mathbb{A}^{n+1}$ that has the "standard" linear representation of $G$ (by scaling), and the GIT quotient $V//G$ is $X$.
However, the construction above depends on the choice of a projective embedding, or at least of an ample invertible sheaf, so it is not canonical.  There is a class of projective varieties that are canonically GIT quotients, namely the Mori Dream Spaces first studied by Hu and Keel.  Every flag variety, and indeed every projective variety homogeneous under a linear algebraic group, is a Mori Dream Space.  In fact, there is a class of varieties that contains both projective homogeneous varieties and toric varieties (another large class of Mori Dream Spaces), namely "spherical varieties".  Every spherical variety is a Mori Dream Space.
Edit.  The OP clarified that he wants $V$ to be an affine space, not an affine variety.  For most Mori Dream Spaces, $V$ is not an affine space (for toric varieties, $V$ is an affine space, as follows from the theory of the Cox ring).