G-bundles, isomonodromy, and quantum Weyl groups

First an “irregular Riemann-Hilbert correspondence” is established for meromorphic connections on principal G-bundlesover a disc, where G is any connected complex reductive group. Secondly, in the case of poles of order two, isomonodromic deformations of such connections are considered and it is proved that the classical actions of quantum Weyl groups found by De Concini, Kac, and Procesi do arise from isomonodromy (and so have a purely geometrical origin). Finally, a certain flat connection appearing in the work of De Concini and Toledano Laredo is derived from isomonodromy, indicating that the above result is the classical analogue of their conjectural Kohno-Drinfeld theorem for quantum Weyl groups.