Partial (or complete) flag varieties as GIT quotients of affine spaces

Question

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?

Answer 1

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].

Answer 2

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).