Why Demazure operator is an endomorphism of $\mathbb{Z}[P]$?

Question

Let $P$ be the weight lattice of some Lie algebra. Let $$ \Delta_{\alpha}(u) = \frac{u-s_{\alpha}\cdot u}{1-e^{-\alpha}}, $$ where $\alpha$ is a root, $u \in P$. In the article, it is said that Demazure operator is an endomorphism of $\mathbb{Z}[P]$. For $u \in \mathbb{P}$, why $\frac{u-s_{\alpha}\cdot u}{1-e^{-\alpha}}$ is in $\mathbb{Z}[P]$? It is easy to see that $u-s_{\alpha}\cdot u$ is in $\mathbb{Z}[P]$. But I don't know how to show that $\frac{u-s_{\alpha}\cdot u}{1-e^{-\alpha}}$ is in $\mathbb{Z}[P]$. Thank you very much.

Answer 1

To complement the excellent suggestions for computing an explicit formula made by @JyrkiLahtonen in the comments, here is a somewhat different strategy: show that the element $1-e^{-\alpha}$ generates the ideal of functions in $\mathbf{Z}[P]$ vanishing on the fixed space of $s_\alpha$ (acting on the Cartan), and then observe that $u-s_\alpha u$ is zero on the fixed space of $s_\alpha$.

This is the strategy one must use in somewhat more general contexts where the combinatorics of root systems is not available. But in any case, for computing with Demazure operators the explicit formulas are extremely important. I suggest you understand both approaches!