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

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