The P = W conjecture for fixed-determinant (SL_n) Higgs moduli in composite rank

Theorem 1.1: $P_k=W_{2k}=W_{2k+1}$ on all of $H^*(X^B,\mathbb Q)$ for the smooth coprime fixed-determinant, trace-free moduli in every composite rank $n\ge4$, genus $g\ge2$, including the variant (non-$\Gamma$-invariant) cohomology; with the earlier prime-rank theorems this gives the fixed-determinant conjecture in every coprime rank. A corollary answers Maulik-Shen's Question 5.5 on Betti weights of endoscopic images. Non-coprime (singular) cases and other groups are not treated.

Theorem 1.1: $P_k=W_{2k}=W_{2k+1}$ on all of $H^*(X^B,\mathbb Q)$ for the smooth coprime fixed-determinant, trace-free moduli in every composite rank $n\ge4$, genus $g\ge2$, including the variant (non-$\Gamma$-invariant) cohomology; with the earlier prime-rank theorems this gives the fixed-determinant conjecture in every coprime rank. A corollary answers Maulik-Shen's Question 5.5 on Betti weights of endoscopic images. Non-coprime (singular) cases and other groups are not treated.