Lemma 1

Let $\lambda$ be an element of X(T). The following conditions are equivalent:

(i) $\lambda -w(\lambda )\geq 0$ (resp. $>0$) for all $w\in W$ such that $w\not= 1$;

(ii) for all $w\in W$ such that $w\not= 1,\lambda -w(\lambda )$ is a linear combination with positive coefficients (resp. positive coefficients not all zero) of simple roots;

(iii) $\langle \lambda , K_{\alpha}\rangle \geq 0$ (resp. $>0$) for every positive root $\alpha$;

(iv) $\langle \lambda , K_{\alpha}\rangle \geq 0$ (resp. $>0$) for every simple root $\alpha$.

The equivalence if (iii) and (iv) is immediate. Since the set of the $K_{\alpha}$ can be identified with the inverse root system of $R(G,T) ($§4, no. 5), the equivalence of (i) and (iii) follows from Chap. VI, §1, no. 6, Prop. 18 and Cor. The implication (ii) $\Rightarrow$ (i) is trivial, and the opposite implication follows from loc. cit.

Denote by $X_{++}$ the set of elements of X(T) such that $\langle \lambda , K_{\alpha}\rangle \geq 0$ for every positive root $\alpha$. The elements of $X_{++}$ are said to be dominant. They form a fundamental domain for the operation of W on X(T) (Chap. VI, §1, no. 10). We have $X_{++}\subset X_+$.

If G is simply-connected, for each simple root $\alpha$ there exists an element $\varpi_{\alpha}$ of X(T) such that $\langle \varpi_{\beta}, K_{\alpha}\rangle =\delta_{\alpha \beta}$ for every simple root $\beta$, that is, $s_{\alpha}(\varpi_{\alpha}) =$ $\varpi_{\alpha}-\alpha ,s_{\beta}(\varpi_{\alpha}) =\varpi_{\alpha}$ for every simple root $\beta \not=\alpha$; the $\varpi_{\alpha}$ are called the fundamental dominant weights; they form a basis of the commutative group X(T) and of the commutative monoid $X_{++}$; more precisely, every element $\lambda$ of X(T) can be written in the form $\lambda =\sum_{\alpha}\langle \lambda , K_{\alpha}\rangle \varpi_{\alpha}$.

Denote by $\rho$ the element of $X(T)\otimes \mathbf{Q}$ such that

We have $\langle \rho , K_{\alpha}\rangle = 1$ for every simple root $\alpha$ (Chap. VI, §1, no. 10, Prop. 29). If G is simply-connected, $\rho$ is the sum of the fundamental dominant weights.