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We study the geometry of singular $\text{SL}_n$-Hitchin fibres over theelliptic 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. Ourmain theorem shows that the Fourier-Mukai transform of a skyscraper sheafsupported at an orbifold singularity in $M^{ell}(\text{PGL}_n)$ satisfies aversion of the fractional Hecke eigenproperty, as conjectured by Frenkel andWitten.

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