Polyhedra dual to the Weyl chamber decomposition: a précis (Q558622)

From author's abstract: Let \(V_{\mathbb{R}}\) be a real vector space with an irreducible action of a finite reflection group \(W\). The author constructs a one-parameter family of semialgebraic polyhedra \(K_W(\lambda)\) in \(V_\mathbb{R}\) which are dual to the Weyl chamber decomposition of \(V_\mathbb{R}\). The key of such a construction is a theorem on linearization of the tube domain in \((V/\!/W)_\mathbb{R}\) over the simplicial cone. As an application, the author obtains two geometric descriptions of generators for\break \(\pi_1 ((V/\!/W)_\mathbb{C}^{\text{reg}})\), satisfying the Artin braid relations.