Brian Bi
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Remark: In the proof of Theorem 4.6.2, a claim is made that if \(B_1, B_2\) are two \(G\)-invariant positive definite Hermitian forms on \(V\), then \(B_1(v, w) = B_2(Av, w)\) for some homomorphism \(A : V \to V\). To see why, first observe that by nondegeneracy, \(B_1(v, w) = \varphi_v(w) = B_2(v', w)\) for some unique \(\varphi_v \in V^*, v' \in V\). Define \(A\) such that \(Av\) is that unique vector \(v'\) for each \(v \in V\). By linearity of \(B_1, B_2\) in the first argument, it follows that \(A\) is \(\mathbb{C}\)-linear. Furthermore, for all \(g \in G\), we have \(B_1(gv, w) = B_1(v, g^{-1}w) = B_2(Av, g^{-1}w) = B_2(gAv, w)\), therefore \(Agv = gAv\). Therefore \(A\) is a homomorphism of \(\mathbb{C}[G]\)-modules, as claimed.

It is also claimed that we can start with any positive definite form \(B\) and the form \(\overline{B}\) constructed from \(B\) will be Hermitian. It's not obvious to me how \(\overline{B}\) becomes Hermitian if \(B\) wasn't already Hermitian, but this doesn't matter, because it is easy to construct \(B\) to be Hermitian in the first place (and in fact this is already implied by positive definiteness).