Every V-twisted Higgs bundle on a compact Riemann surface corresponds, via a Hecke transformation, to a pair of commuting line-bundle-twisted Higgs bundles, yielding a spectral correspondence for their moduli space.
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Remarks on Higgs bundles twisted by a vector bundle
Every V-twisted Higgs bundle on a compact Riemann surface corresponds, via a Hecke transformation, to a pair of commuting line-bundle-twisted Higgs bundles, yielding a spectral correspondence for their moduli space.