Definition 1

The weights of the representation $\rho$ of G, relative to a maximal torus T of G, are the elements $\lambda$ of X(T) such that $\widetilde{V}_{\lambda}(T)\not= 0$.

Denote by $P(\rho ,T)$, or by $P(\rho )$ if there is no possibility of confusion over the choice of T, the set of weights of $\rho$ relative to T. By Lemma 1,

Let $T'$ be another maximal torus of G and $g$ an element of G such that (Int $g$)$T = T'($§2, no. 2, Th. 2). For all $\lambda \in X(T)$,

$\widetilde{\rho}(g)( \widetilde{V}_{\lambda}(T)) = \widetilde{V}_{\lambda'}(T')$, where $\lambda '=$ X(Int $g^{-1}$)$(\lambda )$. (8)

Consequently,

X(Int $g$)$(P(\rho ,T')) = P(\rho ,T)$. (9)

The Weyl group $W = W_G(T)$ operates on the left on the $\mathbf{Z}$-module X(T) by $w \rightarrow X(w^{-1})$; thus, for $t\in T, \lambda \in X(T), w\in W$, we have $t^{w\lambda}= (w^{-1}(t))^{\lambda}$.