Complex K-theory of moduli spaces of Higgs bundles

We establish an isomorphism of complex $K$-theory of the moduli space $\check{\mathcal{M}}$ of $``SL_n"$-Higgs bundles of degree $d$ and rank $n$ (in the sense of Hausel--Thaddeus) and twisted complex $K$-theory of the orbifold $\hat{\mathcal{M}}$ of $PGL_n$-Higgs bundles of degree $e$, where $(n,d)=(n,e)=1$. Along the way we prove the vanishing of torsion for $H^*(\check{\mathcal{M}})$ and certain twisted complex $K$-theory groups of $\hat{\mathcal{M}}$. We also extend Arinkin's autoduality of compactified Jacobian to a derived equivalence between $SL_n$ and $PGL_n$-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.