Proposition 3

Assume that $\rho$ is irreducible.

a) The representation $\rho$ is of real type if and only if there exists a non-zero symmetric bilinear form B on W, invariant under G. In this case the form B is separating; the set V of $w\in W$ such that $H(w, x) = B(w, x)$ for all $x\in W$ is an $\mathbf{R}$-structure on W invariant under G.

b) The representation $\rho$ is of complex type if and only if there exists no non-zero bilinear form on W invariant under G.

c) The representation $\rho$ is of quaternionic type if and only if there exists a non-zero alternating bilinear form on W, invariant under G; such a form is necessarily separating.

For $\theta \in$ Hom$_{\mathbf{C}^{(G)}}(W,W)$ and $x, y\in W$, put $B_{\theta}(x, y) = H(\theta x, y)$. Then $B_{\theta}$ is a bilinear form on W, invariant under G, and separating if $\theta$ is non-zero. Denote by $\mathscr{B}(W)^G$ the space of bilinear forms on W invariant under G; the map $\theta \rightarrow B_{\theta}$ from Hom$_{\mathbf{C}^{(G)}}(W,W)$ to $\mathscr{B}(W)^G$ is an isomorphism of $\mathbf{C}$-vector spaces. This implies, in particular, assertion b).

Let $\theta$ be a $\mathbf{C}^{(G)}$-isomorphism from W to $\overline{W}$ such that $\theta \circ \theta =\alpha_W$, with $\alpha \in \{-1,+1\}$ (Prop. 2); since $\mathscr{B}(W)^G$ is of dimension 1, there exists $\varepsilon \in \mathbf{C}$ such that

$B_{\theta}(y, x) =\varepsilon B_{\theta}(x, y)$ for all $x, y$ in $W$.

Iterating, we obtain $B_{\theta}(y, x) =\varepsilon B_{\theta}(x, y) =\varepsilon^2B_{\theta}(y, x)$, so $\varepsilon^2$ = 1 and $\varepsilon \in \{-1,+1\}$. Moreover, for $x$ in W,

so $\varepsilon \alpha >0$ since H is positive, that is, $\varepsilon =\alpha$. Assertions a) and c) now follow from Prop. 2.

Denote by $dg$ the Haar measure of total mass 1 on G.