On the P=W conjecture for ${\mathrm{SL}}_n$

Abstract

Let $p$ be a prime number. We prove that the $P=W$ conjecture for ${\mathrm{SL}}_p$ is equivalent to the $P=W$ conjecture for ${\mathrm{GL}}_p$. As a consequence, we verify the $P=W$ conjecture for genus 2 and ${\mathrm{SL}}_p$. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the ${\mathrm{SL}}_p$-Hitchin moduli space and the ${\mathrm{SL}}_p$-twisted character variety, relying on Gröchenig-Wyss-Ziegler's recent proof of the topological mirror conjecture by Hausel-Thaddeus. Finally we discuss obstructions of studying the cohomology of the ${\mathrm{SL}}_n$-Hitchin moduli space via compact hyper-Kähler manifolds.

18 Pages

References in corpus (1)

• On the E-polynomials of a family of Character Varieties