G-bundles, isomonodromy and quantum Weyl groups
Int. Math. Res. Not. (2002) no.22 1129-1166

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. This extends previously known results for irregular connections on vector bundles.

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). (Whereas isomonodromic deformations of logarithmic connections on vector bundles over the Riemann sphere lead to actions of the the standard n-string braid group, we now obtain, from a completely geometrical perspective, an action of the braid group associated to G.)

Thirdly a certain flat connection appearing in work of De Concini, Felder, Markov, Tarasov, Varchenko, Toledano Laredo and Millson is derived from isomonodromy, indicating that the above result is the classical analogue of the conjectural Kohno-Drinfeld theorem for quantum Weyl groups.

Finally the appendix explains how the results of the author's previous paper may now be extended to the case of complex reductive groups G, in particular a natural holomorphic map g^* to G^* is defined for all such groups G and shown to be Poisson.

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