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Delve into the second part of a lecture exploring the relationship between the weight filtration W and the perverse filtration P on the cohomology of the moduli space of stable twisted Higgs bundles. Examine the action of the Lie algebra H_2 of polynomial Hamiltonian vector fields on the plane, and its connection to the cohomology of the moduli space. Investigate how this action is generated by cup products with tautological classes and Hecke operators acting via correspondences. Discover how both the P and W filtrations coincide with the filtration associated with the sl_2 subalgebra of H_2. Learn about the motivations behind this research, including computations of Khovanov-Rozansky homology of links, and understand the implications for the conjecture by De Cataldo, Hausel, and Migliorini regarding the equality of these filtrations in arbitrary rank.
