On the symplectic forms of Groechenig's Higgs moduli over an elliptic curve

Gorsky, Nekrasov, and Rubtsov introduced the moduli space of marked Higgs bundles over an elliptic curve $E$ and identified it with the Hilbert scheme of points on the cotangent bundle. Later, Groechenig constructed four analogous isomorphisms on marked rational curves, of affine Dynkin types $\wt D_4$, $\wt E_6$, $\wt E_7$, and $\wt E_8$, for the $Γ$-Hilbert schemes of $T^*E$ for cyclic groups $Γ$ with $|Γ|\in\{2,3,4,6\}$. We prove that the isomorphisms in all five cases are holomorphic symplectomorphisms.