Natural vector bundle on flag variety coming from the variety of nilpotent matrices of fixed rank?

Question

Let $F$ be a field and $V$ be a $n$-dimensional $F$-vector space, then $\{A \in End(V) | A^2 =0, \operatorname{rank} A=k \}$ gives an algerbraic variety $\mathcal{N}_{n,k}$ over $F$. There is a natural map $f$ from $\mathcal{N}_{n,k}$ to a flag variety $Fl_{(k,n-k)}$ sending $A$ to $0 \subseteq Im A \subseteq Ker A \subseteq V$.

$f$ is surjective (over the algebraic closure of $F$), and the fiber of $f$ can be identified with $GL_k$, so does $f$ define a $GL_k$-torsor hence a vecotor bundle on $Fl_{(k,n-k)}$ ? What is this rank $k$ vector bundle ?

Edit: the comment says it's not a $GL_k$ torsor in general.

Can we generalize such construction to other linear algebraic groups (using the nilpotent cone)?

Example: if $k=1$ and $n=2$, then this gives $O(-2)$ on $\mathbb P^1$, the canonical bundle.

Answer 1

If $k=n/2$ exactly, then any rank $k$ matrix with $A^2=0$ has kernel = image. Then you can (non-canonically) identify the fiber with $GL_k$.

However, there is no natural action of $GL_k$ on the fiber. (At least no obvious one.) Instead, the fiber is the space of isomorphisms between two vector bundles, those being the $k$-dimensional subspace and the $k$-dimensional quotient space.

The case of $k=1$ is special because $GL_1$ is abelian, so the $\operatorname{Isom}(V_1, V_2)$ can be viewed as a single torsor, for instance as $\operatorname{Isom} (1, V_2 \otimes V_1^{-1})$ and so the associated vector bundle is $V_2 \otimes V_1^{-1}$.

But this doesn't work in general, we just have two vector bundles and the tensor product $V_2 \otimes V_1^{\vee}$.