Action of $w_0$ on $V^L$ for orthogonal and exceptional groups

TL;DR

This paper investigates how the longest element of the restricted Weyl group acts on certain invariant subspaces of representations of real Lie groups, providing conjectures and experimental evidence for orthogonal and exceptional groups.

Contribution

It offers a conjectural characterization of representations where $w_0$ acts nontrivially on $V^L$ for orthogonal and exceptional groups, supported by experimental results.

Findings

Conjectural description of the action of $w_0$ on $V^L$

Experimental evidence supporting the conjecture for specific groups

Partial answers to the behavior of $w_0$ in representation theory

Abstract

In this note, we present some results that partially answer the following question. Let $G$ be a simple real Lie group; what is the set of representations $V$ of $G$ in which the longest element $w_0$ of the restricted Weyl group $W$ acts nontrivially on the subspace $V^L$ of $V$ formed by vectors that are invariant by $L$, the centralizer of a maximal split torus of $G$? We give a conjectural answer to that question, as well as the experimental results that back this conjecture, when $G$ is either an orthogonal group (real form of $\operatorname{SO}_n(\mathbb{C})$ for some $n$) or an exceptional group.