= Solution
An invariant bilinear form $B$ defines a linear map
$$
\Phi_B:V\longrightarrow V^*,
\qquad
\Phi_B(v)(w)=B(v,w).
$$
Invariance gives
$$
\Phi_B(gv)(w)=B(gv,w)=B(v,g^{-1}w)
=(g\Phi_B(v))(w),
$$
so $\Phi_B$ is an <intertwining operator>[$G$-homomorphism]. Its <kernel> is a $G$-subrepresentation of <irreducible representation>[irreducible] $V$. Hence either $\Phi_B=0$, so $B=0$, or $\Phi_B$ is <injective function>[injective] and therefore a <linear isomorphism>[linear isomorphism] $V\to V^*$. In the latter case $B$ is a <nondegenerate bilinear form>[nondegenerate].
If $B$ and $C$ are two nonzero invariant forms, then
$$
\Phi_C^{-1}\Phi_B\in\operatorname{End}_G(V).
$$
By <Schur lemma>, this endomorphism is scalar, so $B=\lambda C$. Cases involving the zero form are immediate. Thus any two invariant bilinear forms are proportional.
Back to article page
