Polyhedra Dual to the Weyl Chamber Decomposition: A Précis

Abstract

Let V_ℝ 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 G_a-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 _V_ℝ which are dual to the Weyl chamber decomposition of V_ℝ. As an application, we obtain two geometric descriptions of generators for π1((V//W)ℂ_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 T__W,ℝ.