Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve

Let $C$ be a smooth projective curve of genus $2$. Following a method by O’ Grady, we construct a semismall desingularization ${\tilde{\mathcal{M}}}_{Dol}^G$ of the moduli space ${\mathcal{M}}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}),SL(2,\mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of ${\tilde{\mathcal{M}}}_{Dol}^G$ as a direct sum of the intersection cohomology of ${\mathcal{M}}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of ${\mathcal{M}}_{Dol}^G$ and prove that the mixed Hodge structure on it is actually pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.