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?



Partial (or complete) flag varieties as GIT quotients of affine spaces
If you're willing to quotient by a nonreductive group, then $M\_n//B$ will get you the $GL(n)$ flag manifold. (People are usually afraid to do so, worrying that the ring of invariants won't be Noetherian, but this one is.)


That flag manifold is also available reductively. Let $V\_0,V\_1\ldots,V\_n$ be a list of vector spaces with those dimensions, and let $Hom := \prod\_{i=1}^n Hom(V\_{i-1},V\_i)$. If we quotient this by $GL(V\_1)\times \cdots \times GL(V\_{n-1})$, it forgets the actual maps and only remembers the images inside $V\_n$, so the result is (or to be precise, can be chosen to be) the manifold of flags in $V\_n$. I forget whom this is due to, but it's pretty old.


You can get some of this to work for symplectic and orthogonal groups, using the $O(V) \times Sp(W)$ action on $V\otimes W$; the reference I know is [[Lerman-Montgomery-Sjamaar]](http://www.math.cornell.edu/~sjamaar/papers/lms.pdf "[Lerman-Montgomery-Sjamaar]").


