We consider both standard and twisted action of a (real) Coxeter group on the complement to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in and give explicit formulae which describe both actions on the total cohomology in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group , the Weyl groups of type , and dihedral groups We discuss also the relations with the cohomology of Brieskorn's braid groups.
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Giovanni Felder, Alexander P. Veselov, Coxeter group actions on the complement of hyperplanes and special involutions. J. Eur. Math. Soc. 7 (2005), no. 1, pp. 101–116DOI 10.4171/JEMS/23