The moduli spaces of semistable $SL_n$ and $PGL_n$ Higgs
bundles form dual abelian fibrations over (generic locus of) an affine
space called the Hitchin base. Motivated by the Hausel-Thaddeus mirror
symmetry picture, we construct an isomorphism between the complex
K-theory of the two moduli spaces on the spectra level. As part of our
proof, we show both K-theory groups are torsion-free. This is joint
work in progress with Michael Groechenig.