For any simple Lie algebra $\mathfrak{g}$, is there a category $C$ of (possibly infinite-dimensional) representations of $\mathfrak{g}$ such the Weyl group $W$ of the corresponding root system acts in a nontrivial way on $C$? In other words, can we find such a category $C$ and for each $g \in W$, an endofunctor

$$ A(g) : C \to C $$

together with natural isomorphisms

$$ \alpha_{g,h} : A(g) A(h) \stackrel{\sim}{\rightarrow} A(g h) $$

perhaps obeying the obvious coherence laws?

I'd be even happier if certain weights $\lambda \in \mathfrak{h}^*$ (where $\mathfrak{h}$ is the Cartan) somehow gave rise to simple objects $R_\lambda \in C$ in such a way that

$$ A(g) R_\lambda \cong R_{g(\lambda)} $$

At one point I hoped the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ would do the job here, because for each weight $\lambda \in \mathfrak{h}^*$ I believe there's a simple object $L_\lambda \in \mathcal{O}$. But someone more knowledgeable than me persuaded me that no endofunctors $A(g)$ sending $L_\lambda$ to $L_{g(\lambda)}$, or perhaps just no *exact* such functors, exist on category $\mathcal{O}$. I would love to be wrong here, or at least to learn how close (or far) the Weyl group comes to being able to act on category $\mathcal{O}$ in such a way that

$$A(g) L_\lambda \cong L_{g(\lambda)}$$

Maybe we can't find such an action with exact or even right exact functors, but we can still do it with functors that preserve direct sums.