Complex $K$-theory of moduli spaces of Higgs bundles

We establish an isomorphism of complex K -theory of the moduli space \widecheck{\mathcal{M}} of “ \mathrm{SL}_{n} ”-Higgs bundles of degree d and rank n (in the sense of Hausel–Thaddeus) and twisted complex K -theory of the orbifold \widehat{\mathcal{M}} of \mathrm{PGL}_{n} -Higgs bundles of degree e , where (n,d)=(n,e)=1 . Along the way, we prove the vanishing of torsion for H^{*}(\widecheck{\mathcal{M}}) and certain twisted complex K -theory groups of \widehat{\mathcal{M}} . We also extend Arinkin’s autoduality of compactified Jacobian to a derived equivalence between \mathrm{SL}_{n} - and \mathrm{PGL}_{n} -Hitchin systems over the elliptic locus. In the appendix, we develop a formalism of G -sheaves of spectra, generalising equivariant homotopy theory to a relative setting.