1) Can I see infinite Grassmannian as the quotient of infinite Stiefel manifold for unitary group $V_{n}(\mathbb{C}^{\infty})/U(n)$?
2) 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?
