By $H_2$ we denote the Lie algebra of polynomial hamiltonian vector fields on the plane. We consider the moduli space of stable twisted Higgs bundles on an algebraic curve of a given coprime rank and degree. De Cataldo, Hausel and Migliorini proved in the case of rank 2 and conjectured in arbitrary rank that two natural filtrations on the cohomology of the moduli space coincide. One is the weight filtration W coming from the Betti realization, and the other one is the perverse filtration P induced by the Hitchin map. Motivated by computations of the Khovanov-Rozansky homology of links by Gorsky, Hogancamp and myself, we look for an action of $H_2$ on the cohomology of the moduli space. We find it in the algebra generated by two kinds of natural operations: on the one hand, we have the operations of cup product by tautological classes, and on the other hand, we have the Hecke operators acting via certain correspondences. We then show that both P and W coincide with the filtration canonically associated to the $\mathfrak{sl}_2$ subalgebra of $H_2$.
Based on joint work with Hausel, Minets and Schiffmann.