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

Abstract

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.

References in corpus (5)

• Foncteur de Picard d'un champ algébrique
• Very stable Higgs bundles, equivariant multiplicity and mirror symmetry
• Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system
• Fourier-Mukai transform for fine compactified Prym varieties
• Complex K-theory of moduli spaces of Higgs bundles