Involutions of rank 2 Higgs bundle moduli spaces

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We consider the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over the complex numbers, and study involutions defined by tensoring the vector bundle with an element α of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by ± 1. We describe the fixed points of these involutions in terms of the Prym variety of the covering of X defined by α, and give an interpretation in terms of the moduli space of representations of the fundamental group.