Pure braid group actions on category O modules

TL;DR

This paper proves that quantum Weyl group operators induce a canonical pure braid group action on category O modules for symmetrizable Kac-Moody algebras, linking monodromy of the Casimir connection with quantum group symmetries.

Contribution

It establishes a canonical pure braid group action on category O modules via quantum Weyl group operators, answering a question of Etingof and extending monodromy results.

Findings

Pure braid group acts on category O modules via quantum Weyl group operators.

Monodromy of the Casimir connection corresponds to the braid group action.

Results extend to parabolic category O and parabolic pure braid groups.

Abstract

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.