Closure of a Fundamental Weyl Chamber

Question

Can someone explain what a "closure" of a Fundamental Weyl Chamber means? I assume it is related to an algebraic closure, but I don't see how. In addition, how does the Weyl group act on it and why does it act that way?

Thank you for the help

Answer 1

It is the topological closure $\overline{C}$ ($C$ is the fundamental chamber). With respect to the topology of the ambient Euclidean space. The difference $\overline{C}\setminus C$ consist of walls, sections of the hyperplanes bordering $C$ (remember that those are in a bijective correspondence with the simple roots).

The Weyl group does not act on $\overline{C}$ at all in the sense it can move all the points out of the closure. More or less the exact opposite is true (presumably you are really asking about this). Each orbit of the Weyl group intersects $\overline{C}$ in a single point. The points in $C$ have trivial stabilizers. Any point $x$ on the boundary of $C$ has a non-trivial stabilizer. The stabilizer $\operatorname{Stab}_W(x)$ is generated by the simple reflections that keep $x$ fixed. In other words, these stabilizers are themselves Weyl groups of a lower rank root system.

Answer 2

I might understand it wrong, as I have not reviewed Lie theory for a while. I think the closure just means the inclusion of the fundamental Weyl chamber as well as the two lines(the two hyperplanes). When the Weyl group act on the fundamental Weyl chamber, it just permutes it to other different Weyl chambers. Looking at the Weyl group action on $A_{3}$ might be helpful, since this is a "typical" Lie group.