Coxeter group actions on the complement of hyperplanes and special involutions

We consider both standard and twisted action of a (real) Coxeter group G onthe complement M_G to the complexified reflection hyperplanes by combining thereflections with complex conjugation. We introduce a natural geometric class ofspecial involutions in G and give explicit formulae which describe both actionson the total cohomology H(M_G,C) in terms of these involutions. As a corollarywe prove that the corresponding twisted representation is regular only for thesymmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groupsI_2 (2k+1) and that the standard action has no anti-invariants. We discuss alsothe relations with the cohomology of generalised braid groups.