Pure braid group actions on category O modules

Let g be a symmetrisable Kac-Moody algebra and U_h(g) its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of U_h(g) give rise to a canonical action of the pure braid group of g on any category O (not necessarily integrable) U_h(g)-module V. By relying on our recent results in arXiv:1512.03041, we show that this action describes the monodromy of the rational Casimir connection on the g-module corresponding to V under the Etingof-Kazhdan equivalence of category O for g and U_h(g). We also extend these results to yield equivalent quantum Weyl group and monodromic representations of parabolic pure braid groups on parabolic category O for U_h(g) and g.

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