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I am stuck on a problem in Fulton's Representation Theory: A First Course. Exercise 3.39 states:

Let $V_0$ be a real vector space on which $G$ acts irreducibly, $V=V_0 \otimes \Bbb C$ the corresponding real representation of $G$. Show that if $V$ is not irreducible, then it has exactly two irreducible factors, and they are conjugate complex representations of $G$.

I had originally misread the problem and taken $V_0$ to be not only a real $G$-invariant vector space, but a representation of $G$ itself. Proceeding from there, I showed that $V= V_0 \oplus iV_0$, but this is clearly wrong, as these $V_0$ is not complex and these two representations are not conjugate. However, given that $V_0$ is only a real $G$-invariant vector space, I'm not sure how to proceed.

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    Does anyone have any suggestions on how to do this problem? Here's what I've come up with so far: Since $V=V_0\otimes_{\mathbb{R}}\mathbb{C}$ is reducible, it has a nontrivial irreducible subrepresentation $W$. Now, our goal is to show that $V=W\oplus\overline{W}$. To do so, note that the spaces $W+\overline{W}$ and $W\cap\overline{W}$ are invariant under conjugation. This should somehow allow us to identify $W+\overline{W}$ and $W\cap\overline{W}$ with subrepresentations of $V_0$ and, using the irreducibility of $V_0$, this should give that $W+\overline{W}=V_0$ and $W\cap\overline{W}=0$. I'm2011-10-05

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