$h_\chi$ in $V_\chi$. When $\chi \in X({{\mathbb C}}G)_\varepsilon$ with $\varepsilon=\pm$, let $c_\chi$ be the anti-linear automorphism of $V_\chi$ characterized by the property that $\alpha_\chi(x,y)=h_\chi(c_\chi(x), y)$. Note that $c_\chi$ is $G$-equivariant. Hence so is $c_\chi^2$. As $c_\chi^2$ is also ${{\mathbb C}}$-linear, Schur’s lemma (applied to the irreducible ${{\mathbb C}}G$-module $V_\chi$) implies that $c_\chi^2$ must be a scalar, $\lambda$, say. The identity $$\lambda h_\chi(x, x)=h_\chi(c_\chi^2(x), x)=\alpha_\chi(c_\chi(x),x)=\varepsilon \alpha_\chi(x, c_\chi(x))=\varepsilon h_\chi(c_\chi(x), c_\chi(x))$$ then shows that $\lambda$ is real and has the sign $\varepsilon$. Upon replacing $\alpha_\chi$ by $|\lambda|^{-1/2}\alpha_\chi$, we arrange that $c_\chi^2=\varepsilon 1$. In the orthogonal case ($c_\chi^2=1$), the fixed point space of $c_\chi$ is a real form $V_\chi({{\mathbb R}})$ of $V_\chi$ and so ${{\mathbb R}}G$ maps to ${\operatorname{End}}_{{\mathbb R}}(V_\chi({{\mathbb R}}))$. In the symplectic case ($c_\chi^2=-1$), $c_\chi$ endows $V_\chi$ with the structure of a right ${{\mathbb K}}$-module (recall that ${{\mathbb K}}$ denotes the Hamilton quaternions: here we let $c_\chi$ act as $j$), which extends the ${{\mathbb C}}$-vector space structure, so that ${{\mathbb R}}G$ maps to ${\operatorname{End}}_{{\mathbb K}}(V_\chi)$. It is known that the disjoint union of $\{V_\chi({{\mathbb R}})\}_{\chi\in X({{\mathbb C}}G)_+}$, $\{V_\chi\}_{\chi\in X({{\mathbb C}}G)_-}$ and $\{V_\chi\}_{\chi\in X({{\mathbb C}}G)_o}$ represent the distinct elements of $X({{\mathbb R}}G)$ (complexification as a ${{\mathbb R}}G$-module reproduces resp. $V_\chi$, $V_\chi\oplus V_\chi$ and 