Coxeter group actions on the complement of hyperplanes and special involutions

Abstract

We consider both standard and twisted action of a (real) Coxeter group $G$ on the complement ${\mathcal{M}}_G$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^{\ast }({\mathcal{M}}_G,\mathbb{C})$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_n$, the Weyl groups of type $D_{2m+1}$, $E_6$ and dihedral groups $I_2(2k+1).$ We discuss also the relations with the cohomology of Brieskorn's braid groups.