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Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups

We study mirror symmetry on the singular locus of the Hitchin system at two levels. Firstly, by covering it by (supports of) $(BBB)$-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup $\mathrm{P}$, whose conjectural dual $(BAA)$-branes we describe. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of $\mathrm{P}$. Secondly, when $\mathrm{P}$ is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the $(BBB)$-brane, and study the variation of the dual under this choice. We give evidence of both families of branes being dual under mirror symmetry via an integral functor induced by Fourier-Mukai in the moduli stack of Higgs bundles.