Weyl group characters afforded by zero weight spaces

TL;DR

This paper derives a new character formula for the Weyl group acting on zero weight spaces of finite-dimensional representations of simple Lie groups, generalizing Kostant's partition function and providing insights into representation irreducibility.

Contribution

It introduces a novel character formula involving partition functions for Weyl group actions on zero weight spaces, extending Kostant's work and enabling the classification of certain irreducible representations.

Findings

The formula simplifies on the elliptic set of W.

On the elliptic regular set, the character is a monomial product of co-roots.

It provides a method to identify representations with irreducible zero weight spaces.

Abstract

Let $G$ be a simple complex Lie group with Weyl group $W$. We give a formula for the character of $W$ on the zero weight space of any finite dimensional representation of $G$. The formula involves partition functions, generalizing Kostant's partition function. On the elliptic set of $W$ the partition functions are trivial. On the elliptic regular set, the character formula is a monomial product of certain co-roots, up to a constant equal to $0$ or $\pm 1$. For a Coxeter element we recover Kostant's formula for this trace. If the long element $w_0=-1$, our formula leads to a method for determining all representations of $G$ for which the zero weight space is irreducible.