Fourier transform on the locus of cyclic spectral curves in the Hitchin base

TL;DR

This paper computes the Fourier transform of certain sheaf summands related to the Hitchin map for SL_n, focusing on cyclic spectral curves, and provides support estimates for other summands, advancing understanding of Hitchin fibration properties.

Contribution

It introduces explicit Fourier transform calculations for sheaves on the Hitchin base restricted to cyclic spectral curves, a novel analysis in the context of Hitchin fibrations.

Findings

Fourier transforms of specific summands are explicitly computed.

Support estimates are provided for the Fourier transforms of remaining summands.

Results enhance understanding of the structure of Hitchin fibrations for SL_n.

Abstract

We compute the Fourier transform of some of the summands of the push-forward of the constant sheaf under the Hitchin map for $\mathrm{SL}_n$ restricted to the locus of cyclic spectral curves inside the Hitchin base (for $\mathrm{SL}_2$ all spectral curves are cyclic) and give an estimate on the support of the Fourier transforms of the other summands.