Grassmannian inside a hyperkahler manifold I am currently looking at stratified Mukai flop. Roughly speaking, this is a construction that, starting with a grassmannian $G$ inside a hyperkahler $X$ produces a birational manifold $X^*$ (with a dual grassmannian $G^*$ inside). The easiest example is the standard Mukai flop, namely the case in which the grassmannian $G$ is a projective space $\mathbb{P}^n$. 
I was looking for a concrete example of this situation.
In the case $G=\mathbb{P}^n$, I can construct such an example starting from a rational curve $C\cong\mathbb{P}^1$ inside a $K3$ surface $S$. The Hilbert scheme of points then provides $C^{[n]}\cong\mathbb{P}^n\subset S^{[n]}$. 
What about a generic grassmannian $G=G(k,n)$ with $k\neq1,n-1$? 
Thank you very much for any comments, remarks and answers.
 A: This is just the general case of the example from my comment above.  Let $S\subset \mathbb{P}^g_{\mathbb{C}}$ be a K3 surface of degree $2g-2$ that contains no curve that spans a $\mathbb{P}^r$ with $r<g-1$.  Any such curve would be contained in (many) hyperplane sections of $S$, hence would have degree $< 2g-2$.  So if $\text{Pic}(S)$ is generated by $[\mathcal{O}_{\mathbb{P}^g}(1)|_S]$, then $S$ contains no such curve.
Now let $G$ be the Grassmannian of dimension $2g-2$ parameterizing linear subspace $\Lambda\subset \mathbb{P}^g$ of dimension $g-2$.  There is a morphism $G \to \text{Hilb}^{2g-2}_{S/\mathbb{C}}$ by $[\Lambda] \mapsto [\Lambda \cap S]$.  Since $\text{Hilb}^{2g-2}_{S/\mathbb{C}}$ has dimension $4g-4$, $G$ is a maximal isotropic subvariety of $\text{Hilb}^{2g-2}_{S/\mathbb{C}}$.  So it does not factor through any projective spaces embedded in $\text{Hilb}^{2g-2}_{S/\mathbb{C}}$.  
Edit. I originally had an incorrect comment here speculating on a modular interpretation of the Mukai flop.
