Is there a category of representations of a simple Lie algebra on which its Weyl group naturally acts?

Question

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.

Answer 1

To follow up on what Tobias and Chuck wrote, I think you should reconsider category $\mathcal{O}$ or better yet, its principal block $\mathcal{O}_0$, but with the proviso that you have to make some sacrifices. One sacrifice is that you have to work in the derived category, not the abelian, and the other sacrifice is that you must accept a braid group action, not a Weyl group one.

Once you do this, there are a beautiful pair of commuting braid group actions on $D^b(\mathcal{O}_0)$ given by shuffling and twisting functors. If you identify the Grothendieck group of $D^b(\mathcal{O}_0)$ with $\mathbb{Z}[W]$ by sending the class of the Verma module $M_{w\cdot \rho-\rho}$ to $w\in \mathbb{Z}[W]$, then these categorify the left and right actions of right and left actions of $W$ on itself, except that the functors satisfy the braid relations, not the Weyl group relations. One secret reason for this is that they really categorify the right and left action of the Hecke algebra on itself, so you can't really expect more than the braid relations.

This action also really thinks that it is a Weyl group action in the following sense: the most natural way of defining them actually assigns a functor in a each element of the Weyl group, and the proof of the braid relations is really that $T_wT_{w'}=T_{ww'}$ if $\ell(w)+\ell(w')=\ell(ww')$. Perhaps this is slicing it a little finely, but to me that says you should really think of it as a Weyl group action that got upgraded a bit.

EDIT: Yes, I mean the Artin group, which you can define as the group freely generated by the elements of the Weyl group, modulo the relation $T_wT_{w'}=T_{ww'}$ if $\ell(w)+\ell(w')=\ell(ww')$.

If you want a monoidal category, then maybe you should think about the category of Harish-Chandra bimodules, discussed in work of Bernstein and Gelfand (that's BGG with lower Gelfand multiplicity). Let $U(\mathfrak{g})_0=U(\mathfrak{g})/I_0$ denote $U(\mathfrak{g})$ modulo the ideal $I_0$ generated by central elements that act trivially on the trivial representation. The category I have is essentially the category of bimodules you get over $U(\mathfrak{g})_0$ by looking taking all sums, quotients and extensions of the bimodules $B_V$ gotten from $U(\mathfrak{g})\otimes V$ for all finite dimensional reps $V$ by killing the left and right action of $I_0$. The right action on $U(\mathfrak{g})\otimes V$ is the obvious one ignoring $V$, and the left action is by the coproduct (this is the bimodule where tensor product of it with a left module over $U(\mathfrak{g})$ is the same as usual Hopf algebra tensoring).

The derived category of HC-bimodules is monoidal under the usual tensor product, and there is a functor from the braid group (thought of as a monoidal category) to this category, which corresponds to the shuffling functors.

Answer 2

I can cook up such category on my Foreman grill. Let $\tilde{U}(g)=S(h)\otimes_{S(h)^W}U(g)$ be the extended enveloping algebra of $g$. $W$ acts on $\tilde{U}(g)$, hence, it acts on the category of its representations, all of whom are $g$-modules.

This action is rather silly on blocks with generic central character: $W$ just moves the blocks around. However, if the central character has a non-trivial stabilizer in $W$, the action has non-trivial content.