The $P=W$ conjecture for $\mathrm{GL}_n$

TL;DR

This paper proves the $P=W$ conjecture for the general linear group $ ext{GL}_n$ across all ranks and higher genus curves, using advanced geometric and algebraic techniques.

Contribution

It establishes the $P=W$ conjecture for $ ext{GL}_n$ for all ranks and genera, combining tautological class analysis with Mellit's Hard Lefschetz theorem.

Findings

Proves the $P=W$ conjecture for $ ext{GL}_n$ in all cases

Develops a strong perversity result for tautological classes

Applies vanishing cycles and global Springer theory techniques

Abstract

We prove the $P=W$ conjecture for $\mathrm{GL}_n$ for all ranks $n$ and curves of arbitrary genus $g\geq 2$. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the perversity statement, we apply the vanishing cycles constructions in our earlier work to global Springer theory in the sense of Yun, and prove a parabolic support theorem.