I would like to ask for a hint for exercise 22.5 in Bump's book "Lie groups". The setting is as follows:
Let $G$ be a (semisimple, connected, simply connected) compact Lie group, choose a set of positive roots $\Phi^+$. Denote by $\Lambda$ the weight lattice, $\mathcal{C}_+$ the positive Weyl chamber, $\rho$ Weyl vector, $W$ Weyl group. Take $\lambda\in \Lambda\cap \mathcal{C}_+$ dominant weight and $\nu\in \Lambda$ any weight.
I have a problem with the following claim: assume that there is $w\in W$ such that $w(\nu+\lambda+\rho)\in \mathcal{C}_+^{\circ}$, where $\mathcal{C}^{\circ}_+$ is the interior of $\mathcal{C}_+$. Then $w(\nu+\lambda+\rho)-\rho\in \mathcal{C}_+$.
If it was true that $w(\nu+\lambda+\rho)$ belongs to the root lattice $\Lambda_{root}=\mathbb{Z}\Phi\subseteq \Lambda$, then I would know how to prove the above claim, but I don't see a reason why that would be the case, as root lattice is in general smaller.
Let $\lambda \in C^\circ_+ \cap \Lambda$ be any integral weight inside the interior of the positive chamber. This means in particular that we have $\langle \alpha^\vee, \lambda \rangle > 0$ for all simple coroots $\alpha^\vee \in \Delta^\vee$. In particular, since integral weights pair with simple coroots to integers, we have $\langle \alpha^\vee, \lambda \rangle \geq 1$. Now since $\langle \alpha^\vee, \rho \rangle = 1$ for all simple coroots $\alpha^\vee$, we have $\langle \alpha^\vee, \lambda - \rho \rangle \geq 0$ and hence $\lambda - \rho \in C_+$.