Langlands duality for the Hitchin fibration

Let $C$ be the underlying smooth projective curve, let $r$ and $d$ be the rank and degree, let $M=\mathcal M(d,r,\omega_C)$ be the moduli space of $S$-equivalence classes of Higgs bundles of rank $r$ and degree $d$, and let $H:M\to\mathcal A$ be the Hitchin fibration. A Langlands duality conjecture for the Hitchin fibration. There exists a universal Poincaré sheaf $\overline P$ on $M\times_{\mathcal A}M$ such that the Fourier–Mukai transform with kernel $\overline P$,

$$\Phi^{\overline P}:D^b(M)\to D^b(M),$$

is an auto-equivalence of the bounded derived category $D^b(M)$ of coherent sheaves on $M$.

This is the autoduality property expected as a classical limit of the geometric Langlands correspondence. Donagi and Pantev proved the statement over the open subset of $\mathcal A$ parametrizing smooth spectral curves; the assertion over all of $\mathcal A$ remains conjectural.