JournalsjemsVol. 7 , No. 1DOI 10.4171/jems/23

Coxeter group actions on the complement of hyperplanes and special involutions

  • Giovanni Felder

    ETH Zürich, Switzerland
  • Alexander P. Veselov

    Loughborough University, UK
Coxeter group actions on the complement of hyperplanes and special involutions cover

Abstract

We consider both standard and twisted action of a (real) Coxeter group GG on the complement MG\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 GG and give explicit formulae which describe both actions on the total cohomology H(MG,C)H^*(\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 SnS_n, the Weyl groups of type D2m+1D_{2m+1}, E6E_6 and dihedral groups I2(2k+1).I_2 (2k+1). We discuss also the relations with the cohomology of Brieskorn's braid groups.