Action of Weyl group on zero weight space

TL;DR

This paper classifies irreducible finite-dimensional representations of simple complex Lie groups based on how the Weyl group's longest element acts on their zero weight space, revealing a precise criterion involving the highest weight.

Contribution

It provides a complete classification of representations where the Weyl group's longest element acts nontrivially on the zero weight space, depending on the highest weight and fundamental weights.

Findings

Weyl group's longest element acts by ±Id on zero weight space under specific conditions.

The action depends on the highest weight being a multiple of a fundamental weight.

A bound related to the fundamental weight determines the action's nature.

Abstract

For any simple complex Lie group we classify irreducible finite-dimensional representations $\rho$ for which the longest element $w_0$ of the Weyl group acts nontrivially on the zero weight space. Among irreducible representations that have zero among their weights, $w_0$ acts by $\pm$Id if and only if the highest weight of $\rho$ is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.