Corollary 2

Let $\mu\in P_{++}$ and $\nu \in Q_+$. If $\mu+\nu \in \mathscr{X}$, then $\mu\in \mathscr{X}$.

Write $\nu =\sum_{\alpha\in B}c_{\alpha}.\alpha$, where $c_{\alpha}\in \mathbf{N}$ for all $\alpha \in B$. The corollary is clear

when $\sum_{\alpha\in B}c_{\alpha}= 0$; assume that $\sum_{\alpha\in B}c_{\alpha}>0$ and argue by induction on $\sum_{\alpha\in B}c_{\alpha}$. Let $(\cdot | \cdot )$ be a W-invariant non-degenerate positive symmetric bilinear form on $\mathfrak{h}^*_{\mathbf{R}}$. Then $(\nu |\sum_{\alpha\in B}c_{\alpha}.\alpha )>0$, so there exists $\beta \in B$ such that $c_{\beta}\geq 1$ and

$(\nu |\beta )>0$, hence $\nu (H_{\beta})\geq 1$. Since $\mu\in P_{++}$, it follows that $(\mu+\nu )(H_{\beta})\geq 1$. By Cor. $1,\mu+ (\nu -\beta )\in \mathscr{X}$, and it suffices to apply the induction hypothesis.