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.