  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.
