There is a special weight \begin{align*} \rho \coloneqq\varpi_{1}+\cdots+\varpi_{\ell} \in \Lambda^{+} = {1\over 2}\sum_{\alpha\in \Phi^+} \alpha ,\end{align*} i.e. the sum of fundamental weights or the half-sum of positive roots. It satisfies \begin{align*} \left\langle\rho, \alpha^{\vee}\right\rangle=1 \qquad\text{ and }\qquad s_{\alpha} \rho=\rho-\alpha \qquad \forall \alpha\in \Delta .\end{align*} It is the smallest regular dominant weight fixed by no nontrivial element of \(W\), and the associated line bundle on the flag variety \(G/B\) is ample, and is in fact a square root of the canonical bundle.