# Two question on cohomology of Grassmannians and covering spaces

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?

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Neither of these is research-level. (1) is pretty tautological. (2) is easy so long as you make sure that the action of $W$ is on the right of $G/T$. –  Allen Knutson Jan 20 at 23:20