Proposition 1

Let $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ be an irreducible representation of G of highest weight $\lambda \in X_{++}$. Let $m$ be the integer $\sum_{\alpha\in R_+}\langle \lambda , K_{\alpha}\rangle$, and let $w_0$

be the element of the Weyl group such that $w_0(R_+) = R_-$ (Chap. VI, § 6, Cor. 3 of Prop. 17). There are three possible cases:

a$)w_0(\lambda ) =-\lambda$ and $m$ is even. Then there exists a G-invariant separating symmetric bilinear form on V; the representation $\tau$ is of real type (Appendix II).

b$)w_0(\lambda )\not=-\lambda$. Every G-invariant bilinear form on V is zero; the representation $\tau$ is of complex type (loc. cit.).

c$)w_0(\lambda ) =-\lambda$ and $m$ is odd. Then there exists a G-invariant separating alternating bilinear form on V; the representation $\tau$ is of quaternionic type (loc. cit.).

If the restriction of $\tau$ to $C(G)_0$ is not trivial, we are in case b).

A bilinear form B on V is invariant under G if and only if it is invariant under $\mathfrak{g}_{\mathbf{C}}$ (Chap. III, §6, no. 5, Cor. 3). Thus, if G is semi-simple, the proposition follows from Chap. VIII, §7, no. 5, Prop. 12 and Prop. 3 of Appendix II.

In the general case, put $C(G)_0= S$, and identify $X(T/S)$ with a subgroup of X(T) (stable under W). If $\tau (S) =\{1_V\},\tau$ induces by passage to the quotient a representation $\tau ': G/S\rightarrow \mathbf{G}\mathbf{L}(V)$ of highest weight $\lambda$; in this case the proposition follows from the preceding, applied to $G/S$.

Assume that $\tau (S)\not=\{1_V\}$. There exists a non-zero element $\nu$ of X(S) such that $\tau (s) =\nu (s)_V$ for all $s\in S$ (Remark 3). Then $\nu$ is the image of $\lambda$ under the restriction homomorphism $X(T)\rightarrow X(S)$; since W operates trivially on X(S), the equality $w_0(\lambda ) =-\lambda$ implies that $\nu =-\nu$, which is impossible: hence $w_0(\lambda )\not=-\lambda$. On the other hand, if B is a G-invariant bilinear form on V, we have, for all $x, y$ in V and $s$ in S,

which implies that B = 0, hence the proposition.

Let $\mathfrak{S}_{\mathbf{R}}(G)$ be the set of classes of irreducible continuous representations of G on finite dimensional real vector spaces. Prop. 1 and the results of Appendix II give a bijection $\Phi : X_{++}/\Sigma \rightarrow \mathfrak{S}_{\mathbf{R}}$(G), where $\Sigma$ denotes the subgroup $\{1,-w_0\}$ of Aut(X(T)). More precisely, let $\lambda \in X_{++}$, and let $E_{\lambda}$ be a representation of G of highest weight $\lambda$; then

$\Phi (\{\lambda ,-w_0(\lambda )\}) = E_{\lambda[\mathbf{R}]}$ if $\lambda \not=-w_0(\lambda )$ or if $\sum_{\alpha\in R_+}\langle \lambda , K_{\alpha}\rangle \notin2\mathbf{Z}$

$\Phi (\{\lambda ,-w_0(\lambda )\}) = E'_{\lambda}$ if $\lambda =-w_0(\lambda )$ and $\sum_{\alpha\in R_+}\langle \lambda , K_{\alpha}\rangle \in 2\mathbf{Z}$

where $E'_{\lambda}$ is an $\mathbf{R}$-structure on $E_{\lambda}$ invariant under G.