Pure braid group actions on category $\mathcal{O}$ modules

Let $\mathfrak{g}$ be a symmetrisable Kac–Moody algebra and $U_\hbar \mathfrak{g}$ its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of $U_\hbar \mathfrak{g}$ give rise to a canonical action of the pure braid group of $\mathfrak{g}$ on any category $\mathcal{O}$ (not necessarily integrable) $U_\hbar \mathfrak{g}$-module $\mathcal{V}$. By relying on our recent results $\href{http://arxiv.org/abs/1512.03041}{[\textrm{ATL15}]}$, we show that this action describes the monodromy of the rational Casimir connection on the $\mathfrak{g}$-module $V$ corresponding to $\mathcal{V}$. We also extend these results to yield equivalent representations of parabolic pure braid groups on parabolic category $\mathcal{O}$ for $U_\hbar \mathfrak{g}$ and $\mathfrak{g}$.