In 2003, T. Hausel and M. Thaddeus proved that the Hitchin systems on the moduli spaces of $\operatorname{SL}(n,\mathbb{C})$- and $\operatorname{PGL}(n,\mathbb{C})$-Higgs bundles on a curve, verify the requirements to be considered SYZ-mirror partners, in the mirror symmetry setting proposed by Strominger-Yau-Zaslow (SYZ). These were the first non-trivial known examples of SYZ-mirror partners of dimension greater than $2$.

According to the expectations coming from physicists, the generalized Hodge numbers of these moduli spaces should thus agree — this is the so-called topological mirror symmetry. Hausel and Thaddeus proved that this is the case for $n=2,3$ and gave strong indications that the same holds for any $n$ prime (and degree coprime to $n$). In joint work in progress with P. Gothen, we perform a similar study but for parabolic Higgs bundles. We will roughly explain this setting, our study and some questions which naturally arise from it.