Quotient manifold of a finite group action

Question

Let R be a (crystallographic) root system on an Euclidian space $(E,⟨−,−⟩)$ and $$W:=gen\{σ_r∣r∈R\}=gen\{σ_r∣r∈R^+\}$$ its associated reflection group. Taking $$M=E-\cup_{r\in R^+} H_r,$$ where $H_r$ is the hyperplane associated to $\sigma_r$, $W$ acts freely on $M$ (which is a smooth manifold since it is an open subset) and therefore the orbit space $M/W$ is a smooth manifold. I would to know when $M/W$ is a compact and orientable manifold without boundary or at least an example for which this happends.

Answer 1

This can never be true. The Weyl group acts freely on $M$ and an open fundamental domain is given by any Weyl chamber in the root system, which thus maps homeomorphically to the quotient. Unless I misunderstand your definition of root system this is a positive cone in euclidean space, taking a sequence of points going out to infinity along a ray shows that this space is never compact.