On the P = W conjecture for $$SL_n$$ S L n

Abstract

Abstract Let p be a prime number. We prove that the

$$P=W$$

P = W

conjecture for

$$\mathrm {SL}_p$$

SL p

is equivalent to the

$$P=W$$

P = W

conjecture for

$$\mathrm {GL}_p$$

GL p

. As a consequence, we verify the

$$P=W$$

P = W

conjecture for genus 2 and

$$\mathrm {SL}_p$$

SL p

. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the

$$\mathrm {SL}_p$$

SL p

-Hitchin moduli space and the

$$\mathrm {SL}_p$$

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

$$\mathrm {SL}_n$$

SL n

-Hitchin moduli space via compact hyper-Kähler manifolds.