In the proof of Lemma 3.35 in Fulton--Harris, Representation Theory, it is claimed that the identification $H(\phi^2(x),y)=H(x, \phi^2(y))$ implies that $\lambda$ is a positive real ($\phi^2$ is known to act by a scalar $\lambda$). I see why $\lambda$ must be real, but I do not see why $\lambda$ must be positive.
I googled this question, and I found that Noam Elkies came across the same subtlety, and resolved it by saying that $H$ can be assumed to be positive-definite, after which it follows that $H(\phi^2(x),x) = H(\phi(x), \phi(x))$, so $$\frac{H(\phi^2(x),x)}{H(x,x)}$$ is positive. I do not however understand why $H$ can be assumed positive-definite, and I suspect that the reason would involve the same manipulations necessary to prove the desired result in general.
Anyone care to help out?
To elaborate on Alex-omsk's comment, you can construct a $G$-invariant Hermitian form on a representation $V$ by "averaging" over $G$. (This is done in Fulton and Harris in the proof of Proposition 1.5.) That is, for any choice of Hermitian form $H_0$ on the underlying vector space $V$, set
$$ H(v,w) = \sum_{g \in G} H_0(gv,gw). $$
Then $H$ is a $G$-invariant Hermitian form on $V$. Now you can check that if we choose a form $H_0$ that is positive definite, then $H$ is positive definite as well.