TITLE: Exercise 3.39 of Fulton & Harris
QUESTION [5 upvotes]: I would like some help with exercise 3.39 from Fulton & Harris' 'Representation Theory A First Course':
Let $V_0$ be a real vector space on which $G$ acts irreducibly, $V=V_0 \otimes \mathbb{C}$ is a complex representation of $G$. Show that if $V$ is not irreducible, then it has two irreducible factors, and they are conjugate complex representations of $G$.
First of all I would like to remark that conjugate complex representations were not covered before in the book, but after a bit of research I believe the statement in the exercise is equivalent to: $V=W \oplus W^*$ for some (complex) irreducible sub-representation $W \subset V$. If I am mistaken please do correct me!
This question was asked here: Question about Real Representations, but the post got bogged down with notations and since it was so old I decided to ask it again here. It was suggested in the comments of that post that we consider a irreducible (proper) sub-representation $W \subset V$, let $V=W \oplus U$, and show that $U=W^*$, but I don't really see how we can go on...
The only way I have learnt to compute the amount of irreducible factors inside a representation $V$ is by computing $(\chi_V,\chi_V)=\frac{1}{|G|} \sum_{g \in G} |\chi_V(g)|^2$, but since we don't have a corresponding formula (not one that I am aware of) in the real case, I'm not sure how we can prove there are 2 irreducible factors in $V$ here. I am also failing to see how the second statement about the relation between the two factors can follow from the first part.
Any help is appreciated!

REPLY [2 votes]: There is actually a much more elementary way to see this. We write elements of $V_0\otimes \mathbb{C}$ as $x+iy$ for $x,y\in V_0$. 
Claim: If W is a nonzero submodule of V, then $W\oplus W^* = V$. 
Proof: It is trivial that $W+ W^*$ is a submodule of $V$. Now suppose $x+iy\in W$ is nonzero, where $x,y\in V_0$. Note that $x-iy\in W^*$, so $(x+iy)+(x-iy) = 2x\in W+ W^*$. Then if $x\neq 0$, since $V_0$ is irreducible, this implies that $V_0\subseteq W+W^*$ (using the natural embedding of $V_0$ into $V$). Then $iV_0\subseteq W+ W^*$ and therefore $V = W+ W^*$. If instead $x = 0$, then since we assumed $x+iy\neq 0$, we have $y\neq 0$. So then irreducibility gives $iV_0\subseteq W$ and we proceed similarly to above. We are done proving this claim. 
Now if we assume that $W$ is a proper nonzero irreducible submodule of $V$, then since we can easily show $W\cap W^*$ is a submodule of $W$, we have either $W\cap W^* = W$ or $W\cap W^* = 0$. The first case would contradict the claim proven above, since we assumed $W$ was a proper submodule of $V$. So $W\cap W^* = 0$ and $V = W\oplus W^*$.