Pure braid group actions on category O modules

Let g be a symmetrisable Kac-Moody algebra and U(hg )its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of U ,g give rise to a canonical action of the pure braid group of g on any category O (not necessarily integrable) U-hg-module V. By relying on our recent results [ATL15], we show that this action describes the monodromy of the rational Casimir connection on the g-module V corresponding to V. We also extend these results to yield equivalent representations of parabolic pure braid groups on parabolic category O for U(hg )and g.