# Polyhedra Dual to the Weyl Chamber Decomposition: A Précis

### Kyoji Saito

Kyoto University, Japan

## 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 $τ$ -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 $B_{W}$ 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* $V_{R}$ *which are dual to the Weyl chamber decomposition of* $V_{R}$.

As an application, we obtain two *geometric descriptions of generators for* $π_{1}((V//W)_{C})$, *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 $(V//W)_{R}$ over the simplicial cone $E_{W}$ in $T_{W,R}$.

## Cite this article

Kyoji Saito, Polyhedra Dual to the Weyl Chamber Decomposition: A Précis. Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, pp. 1337–1384

DOI 10.2977/PRIMS/1145475449