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.