Proposition 7

a) For all $\lambda \in X(T)$, we have $\widetilde{V}_{\lambda}(T) = \widetilde{V}_{\delta(\lambda)}(\mathfrak{t}_{\mathbf{C}})$.

b) The map $\delta : X(T)\rightarrow$ Hom$_{\mathbf{C}}(\mathfrak{t}_{\mathbf{C}},\mathbf{C})$ induces a bijection from $P(\rho ,T)$ to the set of weights of $\mathfrak{t}_{\mathbf{C}}$ on $\widetilde{V}$.

Note first that, if W operates on $\mathfrak{t}_{\mathbf{C}}$ by associating to any element $w$ of W the endomorphism $L(w)_{(\mathbf{C})}$ of $\mathfrak{t}_{\mathbf{C}}$, the map $\delta$ is compatible with the operation of W on X(T) and Hom$_{\mathbf{C}}(\mathfrak{t}_{\mathbf{C}},\mathbf{C})$.

Assume now that $k=\mathbf{R}$. Denote by $\sigma$ the conjugation of $\widetilde{V}$ relative to V, defined by $\sigma (x+iy) =x-iy$ for $x, y$ in V; for every complex vector subspace E of $\widetilde{V}$, the smallest subspace of $\widetilde{V}$ rational over $\mathbf{R}$ and containing E is $E +\sigma (E)$. In particular, for all $\lambda \in X(T)$, there exists a real vector subspace $V(\lambda )$ of V such that the subspace $V(\lambda )_{(\mathbf{C})}$ of $\widetilde{V}$ is $\widetilde{V}_{\lambda}(T) + \widetilde{V}_{-\lambda}(T)$ (note that $\sigma ( \widetilde{V}_{\lambda}(T)) = \widetilde{V}_{-\lambda}(T)$). We have $V(\lambda ) = V(-\lambda )$, and the $V(\lambda )$ are the isotypical components of the representation of T on V induced by $\rho$.