For the split symmetric pair so_m ⊂ sl_m, the monodromy of the boundary Casimir connection is isomorphic to the iota-quantum Weyl group representation.
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A Kohno--Drinfeld Theorem for iquantum Weyl groups
For the split symmetric pair so_m ⊂ sl_m, the monodromy of the boundary Casimir connection is isomorphic to the iota-quantum Weyl group representation.