Exercise 7, § 4

Let $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ be a continuous representation of G on a finite dimensional real vector space. Assume that for all $\lambda \in X(T)$, we have dim$_{\mathbf{C}}\widetilde{V}_{\lambda}\leq 1$.

a) Show that the representation $\tau$ is the direct sum of a finite family $(\tau_i)_{1\leq i\leq s}$ of irreducible representations, mutually non-isomorphic, whose commutants $K_i(1\leq i\leq s)$ are isomorphic to $\mathbf{R}$ or $\mathbf{C}$.

b) There exists a $\mathbf{C}$-vector space structure on V for which the operations of $\tau (G)$ are $\mathbf{C}$-linear if and only if $K_1, . . . ,K_s$ are isomorphic to $\mathbf{C}$; there are then $2^s$ structures of this type.