Flat Connections and Quantum Groups

We review the Kohno–Drinfeld theorem and a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection ∇C on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture for the cases when g=sln or when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection ∇C and Cherednik's generalisation of the Knizhnik–Zamolodchikov connection to finite reflection groups.