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.