JournalsjemsVol. 7, No. 1pp. 101–116

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
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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.

Cite this article

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. 101–116

DOI 10.4171/JEMS/23