Abstract. For the group SLn, we study the geometry of singular Hitchin fibres over the elliptic locus of the Hitchin base. This gives interesting information on the Fourier-Mukai transform of a skyscraper sheaf supported at an orbifold singularity of the Hitchin moduli space for PGLn. The main results prove a classical version of the conjectures of Frenkel-Witten concerning the mirror of orbifold singularities.