P=W via H_2

Explore a comprehensive lecture on the moduli space of stable twisted Higgs bundles and the relationship between natural filtrations on its cohomology. Delve into the work of De Cataldo, Hausel, and Migliorini, who proved the coincidence of two filtrations in the rank 2 case and conjectured it for arbitrary rank. Examine the weight filtration W from the Betti realization and the perverse filtration P induced by the Hitchin map. Investigate the motivation behind seeking an action of the Lie algebra H_2 (polynomial Hamiltonian vector fields on the plane) on the cohomology of the moduli space, inspired by computations of Khovanov-Rozansky homology of links. Discover how this action is found in the algebra generated by cup products with tautological classes and Hecke operators acting via correspondences. Learn how both P and W coincide with the filtration associated with the sl_2 subalgebra of H_2, providing insight into the structure and properties of these mathematical objects.