Polyhedra Dual to the Weyl Chamber Decomposition: A Précis

Abstract

Let VR be a real vector space with an irreducible action of a finite reflection group W . We study the semi-algebraic geometry of the W -quotient affine variety V//W with the discriminant divisor DW in it and the τ -quotient affine variety V //W//τ with the bifurcation set BW in it, where τ is the Ga-action on V/ /W obtained by the integration of the primitive vector field D on V/ /W and BW is the discriminant divisor of the induced projection : DW → V //W//τ . Our goal is the construction of a one-parameter family of the semi-algebraic polyhedra KW (λ) in VR which are dual to the Weyl chamber decomposition of VR. As an application, we obtain two geometric descriptions of generators for π1(( V/ /W) reg ), satisfying the Artin braid relations. The key of the construction of the polyhedra KW (λ) is a theorem on a linearization of the tube domain in ( V/ /W)R over the simplicial cone EW in TW,R.