On the $\mathrm{\Sigma }$-invariants of Artin groups satisfying the $K(\pi ,1)$-conjecture

We consider $\mathrm{\Sigma }$-invariants of Artin groups that satisfy the $K(\pi ,1)$-conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC-type, giving a sufficient condition for a character $\chi :A_{\mathrm{\Gamma }}\to \mathbb{R}$ to belong to ${\mathrm{\Sigma }}^n(A_{\mathrm{\Gamma }},\mathbb{Z})$. We also prove some partial converses. As applications, we prove that the ${\mathrm{\Sigma }}^1$-conjecture holds true when there is a prime $p$ that divides $l(e)\mathrm{/}2$ for any edge with even label $l(e)>2$, we generalize to Artin groups the homological version of Bestvina-Brady theorem and we compute the $\mathrm{\Sigma }$-invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the $F_n$ and $F{P}_n$ properties of their derived subgroup.