G-bundles, Isomonodromy and Quantum Weyl Groups

First an ‘irregular Riemann-Hilbert correspondence ’ is established for meromorphic connections on principal G-bundles over 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 [10, 11] do arise from isomonodromy (and so have a purely geometrical origin). Finally the relationship with a recent conjectural analogue of the Kohno-Drinfeld theorem for quantum Weyl groups is explained. 1.