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

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