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

TL;DR

This paper investigates the geometry of singular Hitchin fibers for SL_n and PGL_n over elliptic curves, revealing orbifold singularities and their relation to Fourier-Mukai transforms and fractional Hecke eigenproperties.

Contribution

It establishes a precise correspondence between orbifold singularities in PGL_n moduli and reducible fibers in SL_n, and proves a conjecture relating Fourier-Mukai transforms to fractional Hecke eigenproperties.

Findings

Orbifold singularities in PGL_n moduli occur exactly when SL_n fibers are reducible.

Fourier-Mukai transform of skyscraper sheaves at orbifold singularities satisfies fractional Hecke eigenproperty.

The results confirm a conjecture by Frenkel and Witten on the geometric Langlands correspondence.

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.