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Building on our previous joint work with A. Schmitt [7] we explain arecursive algorithm to determine the cohomology of moduli spaces of Higgsbundles on any given curve (in the coprime situation). As an application of themethod we compute the y-genus of the space of PGL_n-Higgs bundles for any rankn, confirming a conjecture of T. Hausel.

The y-genus of the moduli space of PGL_n-Higgs bundles on a curve (for degree coprime to n)