Mirror symmetry for parabolic Higgs bundles via $p$-adic integration

Applying the technique of $p$-adic integration, we prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli spaces of (strongly) parabolic Higgs bundles for the structure groups ${\text{SL}}_n$ and ${\text{PGL}}_n$, building on previous work of Groechenig-Wyss-Ziegler on the non-parabolic case. We also prove the $E$-polynomial of the smooth moduli space of parabolic ${\text{GL}}_n$-Higgs bundles is independent of the degree of the underlying vector bundles.