Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra

Question

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-X$ is still clear. What is known about the action of $W$ on $L$? Is anything known about the orbit structure? References would be most welcome.