# Covering space (Lie groups and their maximal tori)

Let $G$ be a compact Lie group and $T$ a maximal torus in $G$. We define the Weyl group $W$ as the quotient space ${N_{G}}(T)/T$, where ${N_{G}}(T)$ is the normalizer of $T$ in $G$. We thus have a map $$W \longrightarrow G/T \longrightarrow G/{N_{G}}(T).$$

1. How can I prove that this map is a covering? I think that if $W$ acts freely on $G/T$, then I can say something...

2. Is it true that $G/T$ and $G/{N_{G}}(T)$ are manifolds? Of course, $G/T$ is not a group because $T$ isn’t a normal subgroup of $G$.

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Your 2nd question is answered in, for example, Lee's book on smooh manifolds. Both $T$ and $N_G(T)$ are compact, so they are closed, and quotients by closed subgroups are always manifolds. –  Mariano Suárez-Alvarez Jan 20 '13 at 10:56
What can you say about the possible fixed points of the action of $W$ on $G/T$? –  Mariano Suárez-Alvarez Jan 20 '13 at 11:00
I think that tha action of $W$ on $G/T$ hasn't fixed points... but how can I prove it? –  ArthurStuart Jan 20 '13 at 11:16
Why if they are compact they are also closed? It dependes from topology... (I know that closed in compact is compact) –  ArthurStuart Jan 20 '13 at 11:21
You should probably review a bit of topology before going on. –  Mariano Suárez-Alvarez Jan 20 '13 at 11:29
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