P=W conjecture for the Hitchin system

The $P=W$ conjecture identifies the perverse filtration from the Hitchin map with the weight filtration on the corresponding character variety under non-abelian Hodge theory. The broad conjecture is not fully settled across all reductive groups and moduli-space variants.

Known results

• $\operatorname{GL}_2$, $\operatorname{SL}_2$, and $\operatorname{PGL}_2$: de Cataldo–Hausel–Migliorini.
• Arbitrary rank in genus $2$: de Cataldo–Maulik–Shen for $\operatorname{GL}_n$; related prime-rank $\operatorname{SL}_p$ cases were also known.
• The intersection-cohomology formulation remained conjectural for general complex reductive groups in the 2022 literature.

Full $\operatorname{GL}_n$ proof (date unavailable)

A paper proves $P=W$ for $\operatorname{GL}_n$ in every rank and every curve of genus $g\ge 2$, using tautological-class perversity, vanishing cycles, global Springer theory, parabolic support, and Mellit’s curious hard Lefschetz theorem. A later source reports independent proofs, including work of Hausel–Mellit–Minets–Schiffmann.

Current status (as of August 2026): $P=W$ is settled for smooth $\operatorname{GL}_n$ moduli spaces in all ranks and genera $g\ge 2$, with related type-$A$ consequences, but remains open in general reductive-group, singular, intersection-cohomology, stack, and parabolic formulations.