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Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie algebra representation.

Exercise 8.17 in Fulton and Harris asks to show the following.

Problem: Show that if a subspace $W$ of $V$ is invariant by $\mathfrak{g}$ then it is invariant by $G$.

After attempting this problem and failing, I looked at other textbooks, and they all use non-trivial facts about the exponential map. However, at this point Fulton and Harris did not introduce the exponential map.

Is there a proof without using the exponential map?

However, as hinted, we may use the existence of universal covering groups and the one-to-one correspondence between Lie group homomorphisms and Lie algebra homomorphisms when the domain is simply connected.

So, let $\pi:\tilde{G}\to G$ be the universal cover of $G$. Then, $W$ is invariant by $G$ if it is invariant by $\tilde{G}$. Thus, we may assume that $G$ is simply connected. Now, I guess there is a way to use that $\varphi$ is fully determined by $\psi$, but I don't see how.

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    $\begingroup$ Using the existence of universal covers is way overkill (and the correspondence between Lie group and Lie algebra homomorphisms relies crucially on properties of the exponential map). You should be using the exponential map. I wouldn't necessarily trust Fulton and Harris to keep all relevant material in the appropriate order. $\endgroup$ – Qiaochu Yuan Jan 8 '16 at 22:17
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    $\begingroup$ @QiaochuYuan Thanks for the advice. Do you think Fulton and Harris is a good book in general? Would you recommend it for a first course in representation theory? $\endgroup$ – SHP Jan 8 '16 at 22:23
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    $\begingroup$ @QiaochuYuan I hould not trust Fulton and Harris, period. Sorry if this sounds too absolute. $\endgroup$ – guest Jan 8 '16 at 23:50
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    $\begingroup$ User: as an exposition on representation theory of complex Lie algebras, it is ok (but to do the exercises, if this is your first course, the first chapter you need to read is the... appendices). As a book on Rep theory of Lie groups, Lie group/algebra correspondence and generally, anything non-algebraic, in my opinion, no. Stick with Knapp's books. $\endgroup$ – guest Jan 8 '16 at 23:55

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