  This note constitutes a brief survey of our recent work on the problem of
determining, for a given real Lie group~$G$, the set of representations~$V$ 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~$G$.
