 Solution

For a finite-support formal character, use the same pullback rule $(wu)(\xi )=u(w^{-1}\xi )$; equivalently $w(e^{\lambda })=e^{w\lambda }$. This specifies the action on the denominator, which is a finite delta sum rather than a polynomial function. The whole infinite-support completion need not be Weyl-stable, so no such claim is required.

A simple reflection $s_{\alpha }$ permutes the positive roots other than $\alpha$ and sends $\alpha$ to $-\alpha$. In the product defining $D$, all factors are therefore permuted except the $\alpha$ factor, whose sign reverses. Thus $s_{\alpha }D=-D$. Simple reflections generate $W$, and each has determinant $-1$, so