Fourier transform on the locus of cyclic spectral curves in the Hitchin base

Abstract: We compute the Fourier transform of some of the summands of the push-forward of the constant sheaf under the Hitchin map for $\mathrm{SL}_n$ restricted to the locus of cyclic spectral curves inside the Hitchin base (for $\mathrm{SL}_2$ all spectral curves are cyclic) and give an estimate on the support of the Fourier transforms of the other summands.