Tau Signatures and Characters of Weyl Groups

Let $G_{\mathbb R}$ be the set of real points of a complex linear reductive group and $\hat G_\lambda$ its classes of irreducible admissible representations with infinitesimal integral regular character $\lambda$. In this case each cell of representations is associated to a \emph{special} nilpotent orbit. This helps organize the corresponding set of irreducible Harish-Chandra modules. The goal of this paper is to is to describe algorithms for identifying the special nilpotent orbit attached to a cell in terms of descent sets appearing in the cell.