Let:

- $G$ be a
*real*semisimple Lie group; - $\rho$ be an irreducible representation of $G$ on a finite-dimensional real vector space;
- $A$ be a "Cartan subspace" of $G$ (a Lie subalgebra which is abelian, composed of hyperbolic elements, and maximal);
- $L$ be the centralizer of $A$ in $G$ (this group is often called "$MA$");
- $W = N_G(A)/Z_G(A)$ be the restricted Weyl group of $G$, and $w_0$ its longest element.

I would like to classify the representations that satisfy the following two conditions:

- the space $V^L$ of points of $V$ fixed by every element of $L$ is nonzero;
- the action of $w_0$ on $V^L$ is nontrivial.

An important particular case (which it would already be nice to have) is when $G$ is split. In that case, $V^L$ is simply the zero weight space, which is nonzero whenever the highest weight is an integer combination of roots. So we just need to find the action of $w_0$ on the zero weight space.

This looks like it should have already been done somewhere, but I have been unable to find any references so far. Any pointers would be appreciated!