The completed formal character ring

Fix the finite Weyl root-system data of Finite Weyl root system, lattice and chamber conventions: the real span $E$ of the roots with its positive definite form, a positive system ${\mathrm{\Phi }}^{+}$ with base ${\alpha }_1,\dots ,{\alpha }_r$, the root group $Q={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}\alpha \subseteq P$, the weight lattice $P\subseteq E$, and the positive cone $Q_{+}={\sum }_{i=1}^r{\mathbb{Z}}_{\ge 0}{\alpha }_i$ (the set written $Q^{+}$ in The Grothendieck group and character of O). A downward cone is a set $\lambda -Q_{+}=\{\lambda -\beta :\beta \in Q_{+}\}$ with $\lambda \in {\mathfrak{h}}^{\ast }$.

The completed formal character ring $\mathcal{R}$ is the set of formal sums $f={\sum }_{\mu \in {\mathfrak{h}}^{\ast }}{c}_{\mu }{e}^{\mu },c_{\mu }\in \mathbb{Z},$ whose support $\mathrm{supp}f=\{\mu :c_{\mu }\mathrm{\ne }0\}$ is contained in a finite union of downward cones. Addition is coefficientwise, the product is the convolution $(fg{)}_{\eta }={\sum }_{\mu +\nu =\eta }{c}_{\mu }{d}_{\nu },$ and the unit is $e^0$, so that $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$. The product is well defined: if $\mathrm{supp}f$ lies in ${\bigcup }_{i=1}^m({\lambda }_i-Q_{+})$ and $\mathrm{supp}g$ in ${\bigcup }_{j=1}^n({\mu }_j-Q_{+})$, then a pair of exponents contributing to $\eta$ lies in some ${\lambda }_i-Q_{+}$ and some ${\mu }_j-Q_{+}$, and the solutions $\beta ,\gamma \in Q_{+}$ of $\beta +\gamma ={\lambda }_i+{\mu }_j-\eta$ are the elements of the box $0\le {\beta }_k\le ({\lambda }_i+{\mu }_j-\eta {)}_k$ in the simple-root coordinates, a finite set (empty unless ${\lambda }_i+{\mu }_j-\eta \in Q_{+}$). The coefficients are integers, and a finite union of downward cones is again such a union, so addition and multiplication make $\mathcal{R}$ a commutative $\mathbb{Z}$-algebra. In the notation of The Grothendieck group and character of O this is the ring denoted $\mathcal{R}$ there, where the character homomorphism of category $\mathcal{O}$ takes its values; the elements of finite support form the group ring $\mathbb{Z}[{\mathfrak{h}}^{\ast }]$ of the additive group ${\mathfrak{h}}^{\ast }$ and contain the subring $\mathbb{Z}[P]$ generated by the $e^{\mu }$ with $\mu \in P$.

By The formal character of a Verma module, $\mathrm{ch}M(\lambda )=e^{\lambda }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}$ is an element of $\mathcal{R}$: the geometric series ${\sum }_{k\ge 0}{e}^{-k\alpha }$ is supported in the downward cone $-Q_{+}$, and a finite product of elements of $\mathcal{R}$ lies in $\mathcal{R}$ by the convolution formula, while $e^{\lambda }$ is a single monomial. No convergence of any formal sum is asserted: all sums are formal, coefficients are compared coefficientwise, and every finite sum, product or finite product of geometric series below is interpreted in $\mathcal{R}$ by the rules just recorded.