  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.
