Fourier-Mukai transform for fine compactified Prym varieties

Consider a finite covering $\beta : C \to X$ of a smooth projective curve $X$ by a reduced, projective, planar curve $C$. Associated to two general polarizations on $C$, $q$ and $q'$, one can construct the corresponding compactified Prym varieties $\overline{\mathrm{P}}_\beta(q)$ and $\overline{\mathrm{P}}_\beta(q')$. Consider $\Gamma$ to be the group of line bundles whose torsion coincides with the order of $\beta$. In this article we construct a Fourier-Mukai transform between the derived categories of $\overline{\mathrm{P}}_\beta(q)$ and the $\Gamma$-equivariant derived category of $\overline{\mathrm{P}}_\beta(q')$. Hence, we obtain a derived equivalence between the $\mathrm{SL}(n,\mathbb{C})$-Hitchin fibre and its associated $\mathrm{PGL}(n,\mathbb{C})$-Hitchin fibre for a dense class of singular spectral curves. Our work then provides the extension of the Fourier-Mukai transform constructed by Arinkin and Melo-Rapagnetta-Viviani, which corresponds to autoduality of $\mathrm{GL}(n,\mathbb{C})$-Hitchin fibres in this class of singular spectral curves.