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.
$V_0$ is a representation of $G$, a real one. I don't understand what you misread. – Mariano Suárez-Alvarez Sep 28 '11 at 3:00
Do an example: take $V_0$ to be an irreducible real representation of degree two of the cyclic group of order three. If you do not know any such representation, leave Ex. 3.39 aside for a while, and find one. – Mariano Suárez-Alvarez Sep 28 '11 at 3:12
@Mariano: Dear Mariano, Perhaps in Fulton's book real representation means a complex representation which admits an underlying real structure? (Basing this on the OP's comment that "$V = V_0\otimes \mathbb C$ [is] the corresponding real representation", and nothing more.) Regards, – Matt E Sep 28 '11 at 3:17
@Mariano: As Fulton describes it, "If a group $G$ acts on a real vector space $V_0$, then we say the corresponding complex representation of $V=V_0 \otimes \Bbb C$ is real." – Nathan Sep 28 '11 at 3:22
Dear Nathan, Now that the terminology is sorted out, regarding your question: it could be (i.e. it will be in some cases, but not in others) that the real representation $V$ is irreducible. So you won't be able to automatically decompose it into two subreps. (as you tried to too with your $V = V_0 \oplus i V_0$ gambit). You will have to assume that $V$ contains a proper subrep. $W$ (i.e. assume that it is not irreducible), and then try to deduce that $V$ is a direct sum of $W$ and $\overline{W}$. For this, you should try to work out how $W$ interacts with $V_0$. (And you should ... – Matt E Sep 28 '11 at 3:29