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For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associatedto the Hitchin map on the cohomology of the moduli space of twisted G-Higgsbundles on a Riemann surface C agrees with the weight filtration on thecohomology of the twisted G character variety of C, when the cohomologies areidentified via non-Abelian Hodge theory. The proof is accomplished by means ofa study of the topology of the Hitchin map over the locus of integral spectralcurves.

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1