Let G be a compact Lie group with Weyl W. We give formula for the character of W on zero weight space any finite-dimensional representation G. The involves weighted partition functions, generalizing Kostant’s function. On elliptic set W, functions are trivial. regular set, is monomial product certain coroots, up to constant equal 0 or ± 1. This generalizes trace Coxeter element space. If long w...

 Suppose $f$ is a map from a non-empty finite set $X$ to a finite group $G$. Define the map $zeta^f_G: Glongrightarrow mathbb{N}cup {0}$ by $gmapsto |f^{-1}(g)|$. In this article, we show that for a suitable choice of $f$, the map $zeta^f_G$ is a character. We use our results to show that the solution function for the word equation $w(t_1,t_2,dots,t_n)=g$ ($gin G$) is a character, where $w(t_1,...