Coxeter group actions on the complement of hyperplanes and special involutions

• Autores: Alexander P. Veselov, G. Felder
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 7, Nº 1, 2005, págs. 101-116
• Idioma: inglés
• DOI: 10.4171/jems/23
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• Resumen

  • We consider both standard and twisted action of a (real) Coxeter group $G$ on the complement $\mathcal M_G$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^*(\mathcal M_G, {\mathbb C})$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_n$, the Weyl groups of type $D_{2m+1}$, $E_6$ and dihedral groups $I_2 (2k+1).$ We discuss also the relations with the cohomology of Brieskorn's braid groups.