I am trying to prove something about the action of a particular Lie algebra on a particular representation (it's the starred claim on page 7 of this paper for those interested). My friend showed me a sketch of a proof that involved acting on an element of the Lie algebra by an element of its Weyl group $W$. However, I was not even aware that the Weyl group had a reasonable action on its Lie algebra.
I am aware that one can use the Killing form to identify a Cartan subalgebra $\mathfrak h$ with its dual $\mathfrak h^*$ and thus obtain an action of $W$ on $\mathfrak h$, but how do we extend this action to all of $\mathfrak g$?
In my case, my Lie algebra is a Kac-Moody algebra, but it behaves so similarly to a complex semisimple Lie algebra that I believe answering this question assuming that $\mathfrak g$ is complex semisimple will be enough.
Additional related question: Is there a reasonable sense in which the Weyl group acts on a representation of $\mathfrak g$?