Flat connections and quantum groups 
-- Valerio Toledano Laredo, January 31, 2003

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I will describe a flat connection D on the Cartan
subalgebra of a simple Lie algebra g with values
in any g-module V and logarithmic singularities
on the root hyperplanes obtained jointly with J.
Millson. Its monodromy gives a one-parameter family
of representations of the generalised braid group
Bg of type g which deforms the action of (a finite
extension of) the Weyl group of g on V. I will also
relate, in analogy with the Kohno-Drinfeld theorem,
this monodromy to the quantum Weyl group representations
of Bg obtained by Lusztig via the quantum group U_{q}g.