The Weyl alternation operator

Let $W$ be the Weyl group of the root system of $\mathfrak{g}$, acting on ${\mathfrak{h}}^{\ast }$ by the root reflections of Root reflections and the Weyl group action; by The Weyl group is finite and faithful and Finite Weyl root system, lattice and chamber conventions, $W$ is a finite group and it permutes the roots, hence preserves the root lattice $Q$ and the weight lattice $P$ (Finite Weyl positive roots and simple reflections).

Action on finite support. Every $w\in W$ maps the finite subsets of ${\mathfrak{h}}^{\ast }$ to finite subsets, so the assignment $w\cdot e^{\mu }:=e^{w\mu }$ extends uniquely to a $\mathbb{Z}$-algebra automorphism $f\mapsto w\cdot f$ of the group ring of finite-support elements of the completed character ring $\mathcal{R}$ (The completed formal character ring), with inverse $w^{-1}\cdot (-)$; it restricts to a $\mathbb{Z}$-algebra automorphism of $\mathbb{Z}[P]$, because $W$ preserves the weight lattice $P$. Caveat. If $\mathrm{\Phi }\mathrm{\ne }\mathrm{\varnothing }$, this formula does not define an action on all of $\mathcal{R}$. For a simple root ${\alpha }_i$, the series ${\sum }_{k\ge 0}{e}^{-k{\alpha }_i}$ lies in $\mathcal{R}$, whereas its image under $s_i$ has support $\{k{\alpha }_i:k\ge 0\}$, outside every finite union of downward cones: in each cone the $i$th simple-root coordinate is bounded above. If $\mathrm{\Phi }=\mathrm{\varnothing }$, then $Q_{+}=\{0\}$, every cone is a point, $\mathcal{R}$ has only finite-support elements and $W=\{1\}$ acts trivially. Only finite-support elements are acted on below.

The alternation operator. For $\nu \in {\mathfrak{h}}^{\ast }$ define the Weyl alternation operator by $A(\nu ):={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w\nu },$ the sum being finite because $W$ is finite; here $\mathrm{\ell }$ is the length function of Finite Weyl root system, lattice and chamber conventions turned into the sign homomorphism of The sign of the Weyl length is multiplicative. By The sign of the Weyl length is multiplicative the coefficients $(-1{)}^{\mathrm{\ell }(w)}$ define the determinant sign of $w$ acting on the real span $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$ of the roots. Since the sum is finite, $A(\nu )$ is a finite-support element of $\mathcal{R}$ for every $\nu$, and when $\nu \in P$ it lies in $\mathbb{Z}[P]\subseteq \mathcal{R}$.