  We study the geometry of singular $\text{SL}_n$-Hitchin fibres over the
elliptic locus. We show that orbifold singularities appear in the
$\text{PGL}_n$-moduli space $M^{ell}(\text{PGL}_n)$ exactly when the
$\text{SL}_n$ side $M^{ell}(\text{SL}_n)$ has a reducible Hitchin fibre. Our
main theorem shows that the Fourier-Mukai transform of a skyscraper sheaf
supported at an orbifold singularity in $M^{ell}(\text{PGL}_n)$ satisfies a
version of the fractional Hecke eigenproperty, as conjectured by Frenkel and
Witten.
