TITLE: Action of longest element of Weyl group on zero weight space
QUESTION [6 upvotes]: 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!

REPLY [3 votes]: The action of $w_0$ can be computed by Allen Knutson's suggestion (whether or not $-1\in W$). As Allen says, suppose $g\in N(T)$ is a representative of $w_0$. Then
$\text{trace}(\pi(w_0)) \text{ on } {V_0}=\text{trace}(\pi(g))\text{ on } V$
(since the only element of the root lattice fixed by $w_0$ is $0$). 
It is known that $g$ is conjugate to $\exp(\pi i\rho^\vee)$. Note that $g^2=\exp(2\pi i\rho^\vee)\in Z(G)$, and $\lambda(g^2)=e^{2\pi i\langle\lambda,\rho^\vee\rangle}=1$. Write $\Delta(V)$ for the set of weights of $V$. These are in the root lattice; write $\text{ht}(\mu)$ for the height (writing $\mu$ as $\pm$ a sum of simple roots). Then
$$\text{trace}(\pi(w_0))=|\{\mu\in\Delta(V)\mid \text{ht}(\mu)\text{ even}\}|-|\{\mu\in \Delta(V)\mid ht(\mu)\text{ odd}\}|$$
Since $\pi(w_0)$ on $V_0$ is diagonalizable with $\pm1$ on the diagonal this computes the number of $1/-1$ eigenvalues.