Coxeter group actions on the complement of hyperplanes and special involutions

Abstract

We consider both standard and twisted actions of a (real) Coxeter group G on the complement MG 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*(MG, ℂ) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group Sn, the Weyl groups of type D2m+1, E6, and dihedral groups 1 2(2k + 1). We also discuss the relations with the cohomology of Brieskorn's braid groups.