Tau Signatures and Characters of Weyl Groups

TL;DR

This paper explores the relationship between Weyl group characters, special nilpotent orbits, and Harish-Chandra modules, proposing algorithms to identify these orbits via descent sets.

Contribution

It introduces algorithms to determine special nilpotent orbits associated with representation cells using descent set analysis.

Findings

Algorithms for identifying special nilpotent orbits

Connection between cells and nilpotent orbits

Organization of Harish-Chandra modules

Abstract

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.