Polyhedra Dual to the Weyl Chamber Decomposition: A Précis

Abstract

Let V_\mathbb{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 D_W in it and the τ-quotient affine variety V//W//τ with the bifurcation set B_W in it, where τ is the \mathbb{G}_a-action on V//W obtained by the integration of the primitive vector field D on V//W and B_W is the discriminant divisor of the induced projection : D_W→V//W//τ.
Our goal is the construction of a one-parameter family of the semi-algebraic polyhedra K_W(λ) in V_\mathbb{R} which are dual to the Weyl chamber decomposition of V_\mathbb{R}.
As an application, we obtain two geometric descriptions of generators for π₁ ((V//W)_\mathbb{C}^reg), satisfying the Artin braid relations.
The key of the construction of the polyhedra K_W(λ) is a theorem on a linearization of the tube domain in (V//W)_\mathbb{R} over the simplicial cone E_W in T_{W, \mathbb{R}}.