# 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 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.