Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $G=U(n)$ a compact Lie group and $T$ a maximal torus in $G$ (subgroup of diagonal matrix). We define $W=N(T)/T$ the Weyl group where $N(T)$ is the normalizer of $T \in G$. I have to prove that $W \rightarrow G/T \rightarrow G/N(T)$ is a covering space. So I think I have to prove that the action of $W$ on $G/T$ is free. Is there a simple proof of this?

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

You just have to write down the action of $W$ on $G/T$ :

$N(T)$ acts on $G$ by multiplication from the right : $x\cdot g = gx^{-1}$ and as you can easily see, the action descends to a well defined action of $W$ on $G/T$ by : $(xT)\cdot (gT) = (gx^{-1})T$.

Since the Weyl group $W$ of $G$ is finite, you only have to check that the action is in fact free which is straightforward (every stabilizer subgroup is trivial).

Hence, the map that you were looking for is a covering : $G/T\rightarrow (G/T)/(N(T)/T)\simeq G/N(T)$. This is exactly the space of maximal tori of $G$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.