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Question 8.17 from Fulton's Representation Theory reads as follows:

Let $V$ be a representation of a connected Lie group $G$ and $\rho: \frak{g} \to$ $ \operatorname{End}(V)$ the corresponding map of Lie algebras. Show that a subspace $W$ of $V$ is invariant by $G$ if and only if it is carried into itself under the action of the Lie algebra $\frak g$, i.e., $\rho(X)(W) \subset W$ for all $X$ in $\frak g$.

The hint we are given is "Use statement (ii), noting that $W$ is $G$-invariant if it is $\widetilde G$-invariant, with $\widetilde G$ the universal covering of $G$." Also, statement (ii) is "if $G$ and $H$ are Lie groups with $G$ connected and simply connected, the maps from $G$ to $H$ are in one-to-one correspondence with maps of the associated Lie algebras, by associating to $\rho: G \to H$ its differential $(d\rho)_e: \frak g \to h$."

I am very unsure as to how to attack this problem. I assume that we somehow show that $W$ is carried into itself under the action of $\frak g$, then this implies that $W$ under the universal cover $\widetilde G$ of $G$, which is not only connected but also simply connected, is invariant, so it is also $G$-invariant. However, I'm unsure of how to show this, even using (ii). And I'm also unsure as to how to prove the forward direction as well. Thanks for any help.

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Suppose $W$ is $G$-invariant. Then $d\rho_e(X)w$, for $w \in W$ and $X \in \mathfrak g$, is, for example, the derivative of the path in $V$ that you get when you push $w$ around by $\rho$ applied to a path in $G$. Convince yourself that this path is actually in $W$ and use the fact that $W$ is a vector space.

On the other hand, suppose $W$ is $\mathfrak g$-invariant, so that $d\rho$ maps $\mathfrak g$ to $\mathfrak{gl}(W).$ Then statement (ii) gives you a map $\widetilde G \to GL(W)$, which you want to see as inducing the action of $\rho(G)$ on $W$...

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