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 $\mathfrak{g}$.

Keywords

quantum groups, braid groups, Casimir connection, Coxeter category, Etingof–Kazhdan quantization

2010 Mathematics Subject Classification

Primary 81R50. Secondary 17B37, 20F36.

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To Corrado De Concini

The first-named author was partially supported by the Program FIL 2020 of the University of Parma and co-sponsored by the Fondazione Cariparma.

The second-named author was partially supported by the NSF grant DMS-1802412.

Received 9 November 2022

Accepted 11 September 2023

Published 26 March 2024