Generalized Kostant partition function

In the formal completion of Kac Moody formal character completion, let $P$ be the root-multiplicity product of Kac Moody denominator product with root multiplicities. Define the generalized Kostant partition function by $P^{-1}={\sum }_{\beta \in Q^{+}}K(\beta )e^{-\beta },K(\gamma )=0(\gamma \mathrm{\notin }{Q}^{+})\mathrm{.}$

For a root of multiplicity $m$ its inverse factor is $(1-x{)}^{-m}=({\sum }_{a\ge 0}{x}^a{)}^m={\sum }_{a\ge 0}\left(\genfrac{}{}{0px}{}{m+a-1}{a}\right)x^a$. The coefficient counts the $m$ nonnegative integers summing to $a$: place $m-1$ separators among $a+m-1$ positions. Equivalently $K(\beta )$ counts collections of nonnegative integers $n_{\alpha ,j}$, with $1\le j\le \dim {\mathfrak{g}}_{\alpha }$, satisfying ${\sum }_{\alpha ,j}{n}_{\alpha ,j}\alpha =\beta$. Colors are these integer labels, requiring no choices of root-space bases.

For $\beta =\sum b_i{\alpha }_i$, only roots with coordinates between zero and $b_i$ can occur, a finite set; each has finite multiplicity and each count is bounded by $\sum b_i$. Thus this is a finite count and agrees with the formal inverse by multiplying the finite coefficients. At $\beta =0$ all counts are zero, giving $K(0)=1$. An empty root system therefore gives only this value. Imaginary roots receive all their multiplicity colors just as real roots do. No analytic convergence or AC is involved.