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In this note, we present some results that partially answer the followingquestion. Let $G$ be a simple real Lie group; what is the set ofrepresentations $V$ of $G$ in which the longest element $w_0$ of the restrictedWeyl group $W$ acts nontrivially on the subspace $V^L$ of $V$ formed by vectorsthat are invariant by $L$, the centralizer of a maximal split torus of $G$? Wegive a conjectural answer to that question, as well as the experimental resultsthat 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.

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