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 $D_W$ in it and the $\tau$ -quotient affine variety $V//W//\tau$ with the bifurcation set $BW$ in it, where $\tau$ 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 : $D_W\to V//W//\tau$.

Our goal is the construction of a one-parameter family of the semi-algebraic polyhedra $K_W(\lambda )$ 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 ${\pi }_1((V//W{)}_C^{\text{reg}})$, satisfying the Artin braid relations.

The key of the construction of the polyhedra $K_W(\lambda )$ is a theorem on a linearization of the tube domain in $(V//W{)}_R$ over the simplicial cone $E_W$ in $T_{W,R}$.