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  1. ; (, International Mathematics Research Notices)
    Abstract We compute the monodromy of the mirabolic $$\mathcal{D}$$-module for all values of the parameters $$(\vartheta ,c)$$ in rank 1 and outside an explicit codimension 2 set of values in ranks 2 and higher. This shows in particular that the Finkelberg–Ginzburg conjecture, which is known to hold for generic values of $$(\vartheta ,c)$$, fails at special values even in rank 1. Our main tools are Opdam’s shift operators and intertwiners for the extended affine Weyl group, which allow for the resolution of resonances outside the codimension two set. 
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  2. ; (, Inventiones mathematicae)
    Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group BW on V. We then prove that this action is canonically equivalent to the quantum Weyl group action of BW on a quantum deformation of V, that is an integrable, category O module V over the quantum group Uhg such that V/hV is isomorphic to V. This extends a result of the second author which is valid for g semisimple. 
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  3. ; (, Pure and Applied Mathematics Quarterly)
    Let be a symmetrisable Kac–Moody algebra and Uhg its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of Uhg give rise to a canonical action of the pure braid group of g on any category O (not necessarily integrable) module. By relying on our recent results, we show that this action describes the monodromy of the rational Casimir connection on the Uhg-module corresponding to V. We also extend these results to yield equivalent representations of parabolic pure braid groups on parabolic category for Uhg and g. 
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