Hilbert schemes of elliptic surfaces: group actions and derived categories

Let $X\to C$ be an elliptic surface with integral fibers and a section. The Hilbert scheme $X^{[n]}$ fibers over $C^{[n]}$. We construct a commutative group scheme over the entire base $C^{[n]}$ that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of $X^{[n]}$. We show that the action is $\delta$-regular in the sense of Ng\^o. Using the derived McKay correspondence, we construct an exact autoequivalence of $D^b\operatorname{Coh}(X^{[n]})$ whose kernel is a maximal Cohen-Macaulay sheaf on the fiber product. We show that this Fourier-Mukai transform intertwines with our group action, i.e. theorem of the square holds. We also discuss the case without a section using the theory of Tate-Shafarevich twists.