Fourier–Mukai transform for fine compactified Prym varieties

Consider a finite flat morphism [Formula: see text] between a reduced, projective, locally planar curve [Formula: see text] and a smooth projective curve [Formula: see text]. Associated to two generic polarizations [Formula: see text] and [Formula: see text] on [Formula: see text], one can construct the corresponding compactified Prym varieties [Formula: see text] and [Formula: see text]. With [Formula: see text] the rank of [Formula: see text], the finite group [Formula: see text] of [Formula: see text]-torsion line bundles acts on [Formula: see text] by tensorization. In this paper, we construct a Fourier–Mukai transform between the derived category of [Formula: see text] and the [Formula: see text]-equivariant derived category of [Formula: see text], providing a derived equivalence between the [Formula: see text]-Hitchin fiber and the dual [Formula: see text]-Hitchin fiber for a dense class of singular spectral curves. This is an extension of Fourier–Mukai transforms on compactified Jacobian varieties constructed by Arinkin and Melo–Rapagnetta–Viviani, which correspond to autoduality of [Formula: see text]-Hitchin fibers.