The non-abelian Hodge theorem gives a diffeomorphism between the moduli space of Higgs bundles on a smooth projective complex curve and the character variety of (twisted) representations of its fundamental group. The $P=W$ conjecture of de Cataldo, Hausel and Migliorini predicts that via the corresponding isomorphism on cohomology, the perverse filtration for the Hitchin fibration on the Higgs moduli space is identified with the weight filtration of the mixed Hodge structure on the character variety. We will discuss two (recent) proofs of the $P=W$ conjecture due to Maulik–Shen and Hausel–Mellit–Minets–Schiffmann. Since the cohomology of the Higgs moduli space is generated by tautological classes (Markman) and their weights on the character variety are known (Shende), the $P=W$ conjecture reduces to describing the interaction between the tautological classes and the perverse filtration. The proof of Maulik–Shen combines support theorems for meromorphic Hitchin fibrations (after Ngô and Chaudouard–Laumon), vanishing cycle techniques and Yun’s global Springer theory, which allows them to determine the strong perversity of tautological classes by pulling back to a parabolic Higgs moduli space. The proof of Hausel–Mellit–Minets–Schiffmann shows the $P$- and $W$-filtrations both agree with a third representation-theoretic filtration for an $\mathfrak{sl}_2$-triple in a Lie algebra of polynomial Hamiltonian vector fields, which acts on the cohomology via Hecke operators and cup products by tautological classes.

[After Maulik–Shen, Hausel–Mellit–Minets–Schiffmann]