Endoscopic decompositions and the Hausel–Thaddeus conjecture

Abstract We construct natural operators connecting the cohomology of moduli spaces stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover topological mirror symmetry conjecture Hausel Thaddeus concerning $\mathrm {SL}_n$ - {PGL}_n$ -Higgs bundles. This provides a complete description space in terms tautological classes, gives new proof Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss Ziegler via p -adic integration. Our method is to relate decomposition theorem for Hitchin fibration, using vanishing cycle functors, twisted whose supports are simpler.