Coxeter group actions on the complement of hyperplanes and special involutions

Details Coxeter group actions on the complement of hyperplanes and special involutions Coxeter group actions on the complement of hyperplanes and special involutions

2003-11-12

arxiv.org/abs/math/0311190v2

J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 101--116

Mathematics

Representation Theory

11 pages

Scientific paper

We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G 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(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no anti-invariants. We discuss also the relations with the cohomology of generalised braid groups.

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