Mirror of orbifold singularities in the Hitchin fibration: The case (SLn,PGLn)

Abstract

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