Mirror of Orbifold Singularities in the Hitchin Fibration: the case $({\text{SL}}_n,{\text{PGL}}_n)$

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.