maximal tori and principal $N(T)$-bundles.

Question

Let $U(n)$ be the unitary group and $T^{n}= S^{1} \times \cdots \times S^{1}$ a maximal torus in $U(n)$. Let $N(T^{n})$ be the normalizer in $U(n)$ of $T^{n}$. How can i prove that $U(n) \rightarrow U(n)/N(T^{n})$ is a principal $N(T^{n})$-bundle?

Answer 1

This is a general fact of Lie groups : if $G$ is a Lie group, $H$ a closed subgroup of $G$ then the map $\pi:G\rightarrow G/H, g\mapsto gH$ is a $H$-principal bundle.

You can read a general proof of this in Representations of Compact Lie Groups, T. Bröcker and T.tom Dieck p34.

The idea is to use the exponential map and restrict the action of $H$ on a $\varepsilon$-neightborhood of $G$ in order to get local chart.