On the P = W conjecture for SL_n

Let p be a prime number. We prove that the P= W conjecture for SL _p is equivalent to the P= W conjecture for GL _p. As a consequence, we verify the P= W conjecture for genus 2 and SL _p. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the SL _p-Hitchin moduli space and the 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 SL _n-Hitchin moduli space via compact hyper-Kähler manifolds.