Monodromy of the Casimir connection of a symmetrisable Kac–Moody algebra

Let $\mathfrak{g}$ $\mathfrak {g}$ be a symmetrisable Kac–Moody algebra and $V$ $V$ an integrable $\mathfrak{g}$ $\mathfrak {g}$ –module in category $\mathcal{O}$ $\mathcal {O}$ . We show that the monodromy of the (normally ordered) rational Casimir connection on $V$ $V$ can be made equivariant with respect to the Weyl group $W$ $W$ of $\mathfrak{g}$ $\mathfrak {g}$ , and therefore defines an action of the braid group ${\mathcal{B}}_W$ $\mathcal {B}_{W}$ on $V$ $V$ . We then prove that this action is canonically equivalent to the quantum Weyl group action of ${\mathcal{B}}_W$ $\mathcal {B}_{W}$ on a quantum deformation of $V$ $V$ , that is an integrable, category $\mathcal{O}$ $\mathcal {O}$ module $\mathcal{V}$ $\mathcal {V}$ over the quantum group $U_{ħ}\mathfrak{g}$ $U_{\hbar }\mathfrak {g}$ such that $\mathcal{V}/ħ\mathcal{V}$ $\mathcal {V}/\hbar \mathcal {V}$ is isomorphic to $V$ $V$ . This extends a result of the second author which is valid for $\mathfrak{g}$ $\mathfrak {g}$ semisimple.